RegionStandardDescriptionLevel
AlabamaK.CC.1Students will: Count to 100 by ones and by tens.Kindergarten
AlabamaK.CC.2Students will: Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Kindergarten
AlabamaK.CC.3Students will: Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).Kindergarten
AlabamaK.CC.4Students will: Understand the relationship between numbers and quantities; connect counting to cardinality.Kindergarten
AlabamaK.CC.5Students will: Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
AlabamaK.CC.6Students will: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.Kindergarten
AlabamaK.CC.7Students will: Compare two numbers between 1 and 10 presented as written numerals.Kindergarten
AlabamaK.G.17Students will: Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.Kindergarten
AlabamaK.G.18Students will: Correctly name shapes regardless of their orientations or overall size.Kindergarten
AlabamaK.G.19Students will: Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).Kindergarten
AlabamaK.G.20Students will: Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices or “corners”), and other attributes (e.g., having sides of equal length).Kindergarten
AlabamaK.G.21Students will: Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.Kindergarten
AlabamaK.G.22Students will: Compose simple shapes to form larger shapes.Kindergarten
AlabamaK.MD.14Students will: Describe measurable attributes of objects such as length or weight. Describe several measurable attributes of a single object.Kindergarten
AlabamaK.MD.15Students will: Directly compare two objects, with a measurable attribute in common, to see which object has “more of” or “less of” the attribute, and describe the difference.Kindergarten
AlabamaK.MD.16Students will: Classify objects into given categories; count the number of objects in each category, and sort the categories by count.Kindergarten
AlabamaK.NBT.13Students will: Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Kindergarten
AlabamaK.OA.8Students will: Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
AlabamaK.OA.9Students will: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.Kindergarten
AlabamaK.OA.10Students will: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).Kindergarten
AlabamaK.OA.11Students will: For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.Kindergarten
AlabamaK.OA.12Students will: Fluently add and subtract within 5.Kindergarten
Alabama1.G.19Students will: Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.Grade 1
Alabama1.G.20Students will: Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.Grade 1
Alabama1.G.21Students will: Partition circles and rectangles into two and four equal shares; describe the shares using the words halves, fourths, and quarters; and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.Grade 1
Alabama1.MD.15Students will: Order three objects by length; compare the lengths of two objects indirectly by using a third object.Grade 1
Alabama1.MD.16Students will: Express the length of an object as a whole number of length units by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.Grade 1
Alabama1.MD.17Students will: Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
Alabama1.MD.18Students will: Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
Alabama1.NBT.9Students will: Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
Alabama1.NBT.10Students will: Understand that the two digits of a two-digit number represent amounts of tens and ones.Grade 1
Alabama1.NBT.11Students will: Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.Grade 1
Alabama1.NBT.12Students will: Add within 100, including adding a two-digit number and a one-digit number and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method, and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
Alabama1.NBT.13Students will: Given a two-digit number, mentally find 10 more or 10 less than the number without having to count; explain the reasoning used.Grade 1
Alabama1.NBT.14Students will: Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method, and explain the reasoning used.Grade 1
Alabama1.OA.1Students will: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Grade 1
Alabama1.OA.2Students will: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Grade 1
Alabama1.OA.3Students will: Apply properties of operations as strategies to add and subtract.Grade 1
Alabama1.OA.5Students will: Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).Grade 1
Alabama1.OA.6Students will: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 − 4 = 13 − 3 − 1 = 10 − 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 − 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Grade 1
Alabama1.OA.7Students will: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.Grade 1
Alabama1.OA.8Students will: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.Grade 1
Alabama2.G.24Students will: Recognize and draw shapes having specified attributes such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Grade 2
Alabama2.G.25Students will: Partition a rectangle into rows and columns of same-size squares, and count to find the total number of them.Grade 2
Alabama2.G.26Students will: Partition circles and rectangles into two, three, or four equal shares; describe the shares using the words halves, thirds, half of, a third of, etc.; and describe the whole as two halves, three thirds, or four fourths. Recognize that equal shares of identical wholes need not have the same shape.Grade 2
Alabama2.MD.14Students will: Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.Grade 2
Alabama2.MD.15Students will: Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.Grade 2
Alabama2.MD.16Students will: Estimate lengths using units of inches, feet, centimeters, and meters.Grade 2
Alabama2.MD.17Students will: Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.Grade 2
Alabama2.MD.18Students will: Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.Grade 2
Alabama2.MD.19Students will: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100 on a number line diagram.Grade 2
Alabama2.MD.20Students will: Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Grade 2
Alabama2.MD.21Students will: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¢ symbols appropriately.Grade 2
Alabama2.MD.22Students will: Generate measurement data by measuring lengths of several objects to the nearest whole unit or by making repeated measurements of the same object. Show the measurements by making a line plot where the horizontal scale is marked off in whole-number units.Grade 2
Alabama2.MD.23Students will: Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.Grade 2
Alabama2.NBT.5Students will: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones.Grade 2
Alabama2.NBT.6Students will: Count within 1000; skip-count by 5s, 10s, and 100s.Grade 2
Alabama2.NBT.7Students will: Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.Grade 2
Alabama2.NBT.8Students will: Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits using >, =, and < symbols to record the results of comparisons.Grade 2
Alabama2.NBT.9Students will: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 2
Alabama2.NBT.10Students will: Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
Alabama2.NBT.11Students will: Add and subtract within 1000 using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Grade 2
Alabama2.NBT.12Students will: Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.Grade 2
Alabama2.NBT.13Students will: Explain why addition and subtraction strategies work, using place value and the properties of operations.Grade 2
Alabama2.OA.1Students will: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 2
Alabama2.OA.2Students will: Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.Grade 2
Alabama2.OA.3Students will: Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.Grade 2
Alabama2.OA.4Students will: Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.Grade 2
Alabama3.G.24Students will: Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Grade 3
Alabama3.G.25Students will: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.Grade 3
Alabama3.MD.16Students will: Tell and write time to the nearest minute, and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.Grade 3
Alabama3.MD.17Students will: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.Grade 3
Alabama3.MD.18Students will: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.Grade 3
Alabama3.MD.19Students will: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot where the horizontal scale is marked off in appropriate units–whole numbers, halves, or quarters.Grade 3
Alabama3.MD.20Students will: Recognize area as an attribute of plane figures, and understand concepts of area measurement.Grade 3
Alabama3.MD.21Students will: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).Grade 3
Alabama3.MD.22Students will: Relate area to the operations of multiplication and addition.Grade 3
Alabama3.MD.23Students will: Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.Grade 3
Alabama3.NBT.10Students will: Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
Alabama3.NBT.11Students will: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
Alabama3.NBT.12Students will: Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.Grade 3
Alabama3.NF.13Students will: Understand a fraction 1/𝘣 as the quantity formed by 1 part when a whole is partitioned into 𝘣 equal parts; understand a fraction 𝘢/𝑏 as the quantity formed by 𝘢 parts and size 1/𝘣.Grade 3
Alabama3.NF.14Students will: Understand a fraction as a number on the number line; represent fractions on a number line diagram.Grade 3
Alabama3.NF.15Students will: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.Grade 3
Alabama3.OA.1Students will: Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.Grade 3
Alabama3.OA.2Students will: Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.Grade 3
Alabama3.OA.3Students will: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 3
Alabama3.OA.4Students will: Determine the unknown whole number in a multiplication or division equation relating three whole numbers.Grade 3
Alabama3.OA.5Students will: Apply properties of operations as strategies to multiply and divide.Grade 3
Alabama3.OA.6Students will: Understand division as an unknown-factor problem.Grade 3
Alabama3.OA.7Students will: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
Alabama3.OA.8Students will: Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 3
Alabama3.OA.9Students will: Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.Grade 3
Alabama4.G.26Students will: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Grade 4
Alabama4.G.27Students will: Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.Grade 4
Alabama4.G.28Students will: Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.Grade 4
Alabama4.MD.19Students will: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
Alabama4.MD.20Students will: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
Alabama4.MD.21Students will: Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.Grade 4
Alabama4.MD.22Students will: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
Alabama4.MD.23Students will: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.Grade 4
Alabama4.MD.24Students will: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Grade 4
Alabama4.MD.25Students will: Recognize angle measure as additive. When an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Grade 4
Alabama4.NBT.6Students will: Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.Grade 4
Alabama4.NBT.7Students will: Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
Alabama4.NBT.8Students will: Use place value understanding to round multi-digit whole numbers to any place.Grade 4
Alabama4.NBT.9Students will: Fluently add and subtract multi-digit whole numbers using the standard algorithm.Grade 4
Alabama4.NBT.10Students will: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Alabama4.NBT.11Students will: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Alabama4.NF.12Students will: Explain why a fraction 𝘢/𝘣 is equivalent to a fraction (𝘯 × 𝘢)/(𝘯 × 𝘣) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
Alabama4.NF.13Students will: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.Grade 4
Alabama4.NF.14Students will: Understand a fraction 𝘢/𝘣 with 𝘢 > 1 as a sum of fractions 1/𝘣.Grade 4
Alabama4.NF.15Students will: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.Grade 4
Alabama4.NF.16Students will: Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.Grade 4
Alabama4.NF.17Students will: Use decimal notation for fractions with denominators 10 or 100.Grade 4
Alabama4.NF.18Students will: Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.Grade 4
Alabama4.OA.1Students will: Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
Alabama4.OA.2Students will: Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Grade 4
Alabama4.OA.3Students will: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 4
Alabama4.OA.4Students will: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.Grade 4
Alabama4.OA.5Students will: Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.Grade 4
Alabama5.G.23Students will: Use a pair of perpendicular number lines, called axes, to define a coordinate system with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., 𝘹-axis and 𝘹-coordinate, 𝘺-axis and 𝘺-coordinate).Grade 5
Alabama5.G.24Students will: Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
Alabama5.G.25Students will: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.Grade 5
Alabama5.G.26Students will: Classify two-dimensional figures in a hierarchy based on properties.Grade 5
Alabama5.MD.18Students will: Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real-world problems.Grade 5
Alabama5.MD.19Students will: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.Grade 5
Alabama5.MD.20Students will: Recognize volume as an attribute of solid figures, and understand concepts of volume measurement.Grade 5
Alabama5.MD.21Students will: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.Grade 5
Alabama5.MD.22Students will: Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.Grade 5
Alabama5.NBT.4Students will: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
Alabama5.NBT.5Students will: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.Grade 5
Alabama5.NBT.7Students will: Use place value understanding to round decimals to any place.Grade 5
Alabama5.NBT.8Students will: Fluently multiply multi-digit whole numbers using the standard algorithm.Grade 5
Alabama5.NBT.9Students will: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 5
Alabama5.NBT.10Students will: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method, and explain the reasoning used.Grade 5
Alabama5.NF.11Students will: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.Grade 5
Alabama5.NF.12Students will: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally, and assess the reasonableness of answers.Grade 5
Alabama5.NF.13Students will: Interpret a fraction as division of the numerator by the denominator (𝘢/𝘣 = 𝘢 ÷ 𝘣). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.Grade 5
Alabama5.NF.14Students will: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
Alabama5.NF.15Students will: Interpret multiplication as scaling (resizing).Grade 5
Alabama5.NF.16Students will: Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.Grade 5
Alabama5.NF.17Students will: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Grade 5
Alabama5.OA.1Students will: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Grade 5
Alabama5.OA.2Students will: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.Grade 5
Alabama5.OA.3Students will: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.Grade 5
Alabama6.EE.12Students will: Write and evaluate numerical expressions involving whole-number exponents.Grade 6
Alabama6.EE.13Students will: Write, read, and evaluate expressions in which letters stand for numbers.Grade 6
Alabama6.EE.14Students will: Apply the properties of operations to generate equivalent expressions.Grade 6
Alabama6.EE.15Students will: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).Grade 6
Alabama6.EE.16Students will: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
Alabama6.EE.17Students will: Use variables to represent numbers, and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or, depending on the purpose at hand, any number in a specified set.Grade 6
Alabama6.EE.18Students will: Solve real-world and mathematical problems by writing and solving equations of the form 𝘹 + 𝘱 = 𝘲 and 𝘱𝘹 = 𝘲 for cases in which 𝘱, 𝘲, and 𝘹 are all nonnegative rational numbers.Grade 6
Alabama6.EE.19Students will: Write an inequality of the form 𝘹 > 𝘤 or 𝘹 < 𝘤 to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form 𝘹 > 𝘤 or 𝘹 < 𝘤 have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
Alabama6.EE.20Students will: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.Grade 6
Alabama6.G.21Students will: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Alabama6.G.22Students will: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas 𝘝 = 𝘭𝘸𝘩 and 𝘝 = 𝐵𝘩 to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.Grade 6
Alabama6.G.23Students will: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Alabama6.G.24Students will: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Alabama6.RP.1Students will: Understand the concept of a ratio, and use ratio language to describe a ratio relationship between two quantities.Grade 6
Alabama6.RP.2Students will: Understand the concept of a unit rate 𝘢/𝘣 associated with a ratio 𝘢:𝘣 with 𝘣 ≠ 0, and use rate language in the context of a ratio relationship.Grade 6
Alabama6.RP.3Students will: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Grade 6
Alabama6.SP.25Students will: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.Grade 6
Alabama6.SP.26Students will: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.Grade 6
Alabama6.SP.27Students will: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.Grade 6
Alabama6.SP.28Students will: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.Grade 6
Alabama6.SP.29Students will: Summarize numerical data sets in relation to their context.Grade 6
Alabama6.NS.4Students will: Interpret and compute quotients of fractions, and solve word problems involving division of fractions, e.g., by using visual fraction models and equations to represent the problem.Grade 6
Alabama6.NS.5Students will: Fluently divide multi-digit numbers using the standard algorithm.Grade 6
Alabama6.NS.6Students will: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Grade 6
Alabama6.NS.7Students will: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.Grade 6
Alabama6.NS.8Students will: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts explaining the meaning of 0 in each situation.Grade 6
Alabama6.NS.9Students will: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
Alabama6.NS.10Students will: Understand ordering and absolute value of rational numbers.Grade 6
Alabama6.NS.11Students will: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
Alabama7.EE.7Students will: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Grade 7
Alabama7.EE.8Students will: Understand that rewriting an expression in different forms in a problem context can shed light on the problem, and how the quantities in it are related.Grade 7
Alabama7.EE.9Students will: Solve multistep real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form, convert between forms as appropriate, and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
Alabama7.EE.10Students will: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Grade 7
Alabama7.G.11Students will: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
Alabama7.G.12Students will: Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
Alabama7.G.13Students will: Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Grade 7
Alabama7.G.14Students will: Know the formulas for the area and circumference of a circle, and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.Grade 7
Alabama7.G.15Students will: Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.Grade 7
Alabama7.G.16Students will: Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Grade 7
Alabama7.RP.1Students will: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units.Grade 7
Alabama7.RP.2Students will: Recognize and represent proportional relationships between quantities.Grade 7
Alabama7.RP.3Students will: Use proportional relationships to solve multistep ratio and percent problems.Grade 7
Alabama7.SP.17Students will: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.Grade 7
Alabama7.SP.18Students will: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.Grade 7
Alabama7.SP.19Students will: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.Grade 7
Alabama7.SP.20Students will: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.Grade 7
Alabama7.SP.21Students will: Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.Grade 7
Alabama7.SP.22Students will: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Grade 7
Alabama7.SP.23Students will: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.Grade 7
Alabama7.SP.24Students will: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Grade 7
Alabama7.NS.4Students will: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.Grade 7
Alabama7.NS.5Students will: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Grade 7
Alabama7.NS.6Students will: Solve real-world and mathematical problems involving the four operations with rational numbers.Grade 7
Alabama8.EE.3Students will: Know and apply the properties of integer exponents to generate equivalent numerical expressions.Grade 8
Alabama8.EE.4Students will: Use square root and cube root symbols to represent solutions to equations of the form 𝘹² = 𝘱 and 𝘹³ = 𝘱, where 𝘱 is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.Grade 8
Alabama8.EE.5Students will: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.Grade 8
Alabama8.EE.6Students will: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.Grade 8
Alabama8.EE.7Students will: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.Grade 8
Alabama8.EE.8Students will: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation 𝘺 = 𝘮𝘹 for a line through the origin and the equation 𝘺 = 𝘮𝘹 + 𝘣 for a line intercepting the vertical axis at 𝘣.Grade 8
Alabama8.EE.9Students will: Solve linear equations in one variable.Grade 8
Alabama8.EE.10Students will: Analyze and solve pairs of simultaneous linear equations.Grade 8
Alabama8.F.11Students will: Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.Grade 8
Alabama8.F.12Students will: Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Grade 8
Alabama8.F.13Students will: Interpret the equation 𝘺 = 𝘮𝘹 + 𝘣 as defining a linear function whose graph is a straight line; give examples of functions that are not linear.Grade 8
Alabama8.F.14Students will: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (𝘹, 𝘺) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.Grade 8
Alabama8.F.15Students will: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.Grade 8
Alabama8.G.16Students will: Verify experimentally the properties of rotations, reflections, and translations:Grade 8
Alabama8.G.17Students will: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
Alabama8.G.18Students will: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Grade 8
Alabama8.G.19Students will: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Grade 8
Alabama8.G.20Students will: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.Grade 8
Alabama8.G.21Students will: Explain a proof of the Pythagorean Theorem and its converse.Grade 8
Alabama8.G.22Students will: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
Alabama8.G.23Students will: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
Alabama8.G.24Students will: Know the formulas for the volumes of cones, cylinders, and spheres, and use them to solve real-world and mathematical problems.Grade 8
Alabama8.SP.25Students will: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
Alabama8.SP.26Students will: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
Alabama8.SP.27Students will: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Grade 8
Alabama8.SP.28Students will: Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.Grade 8
Alabama8.NS.1Students will: Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.Grade 8
Alabama8.NS.2Students will: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).Grade 8
AlabamaA-SSE.7Students will: Interpret expressions that represent a quantity in terms of its context.Algebra I
AlabamaA-SSE.8Students will: Use the structure of an expression to identify ways to rewrite it.Algebra I
AlabamaA-SSE.9Students will: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Algebra I
AlabamaA-APR.10Students will: Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Algebra I
AlabamaA-APR.11Students will: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.Algebra I
AlabamaA-CED.12Students will: Create equations and inequalities in one variable, and use them to solve problems.Algebra I
AlabamaA-CED.13Students will: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra I
AlabamaA-CED.14Students will: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context.Algebra I
AlabamaA-CED.15Students will: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Algebra I
AlabamaA-REI.16Students will: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.Algebra I
AlabamaA-REI.17Students will: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Algebra I
AlabamaA-REI.18Students will: Solve quadratic equations in one variable.Algebra I
AlabamaA-REI.19Students will: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.Algebra I
AlabamaA-REI.20Students will: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Algebra I
AlabamaA-REI.21Students will: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.Algebra I
AlabamaA-REI.22Students will: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).Algebra I
AlabamaA-REI.23Students will: Explain why the 𝘹-coordinates of the points where the graphs of the equations 𝘺 = 𝘧(𝘹) and 𝘺 = 𝑔(𝘹) intersect are the solutions of the equation 𝘧(𝘹) = 𝑔(𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘧(𝘹) and/or 𝑔(𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Algebra I
AlabamaA-REI.24Students will: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.Algebra I
AlabamaF-IF.25Students will: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝘧 is a function and 𝘹 is an element of its domain, then 𝘧(𝘹) denotes the output of 𝘧 corresponding to the input 𝘹. The graph of 𝘧 is the graph of the equation 𝘺 = 𝘧(𝘹).Algebra I
AlabamaF-IF.26Students will: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Algebra I
AlabamaF-IF.27Students will: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.Algebra I
AlabamaF-IF.28Students will: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Algebra I
AlabamaF-IF.29Students will: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Algebra I
AlabamaF-IF.30Students will: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Algebra I
AlabamaF-IF.31Students will: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra I
AlabamaF-IF.32Students will: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Algebra I
AlabamaF-IF.33Students will: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Algebra I
AlabamaF-BF.34Students will: Write a function that describes a relationship between two quantities.Algebra I
AlabamaF-BF.35Students will: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.Algebra I
AlabamaF-BF.36Students will: Identify the effect on the graph of replacing 𝘧(𝘹) by 𝘧(𝘹) + 𝘬, 𝘬𝘧(𝘹), 𝘧(𝘬𝘹), and 𝘧(𝘹 + 𝘬) for specific values of 𝘬 (both positive and negative); find the value of 𝘬 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.Algebra I
AlabamaF-LE.37Students will: Distinguish between situations that can be modeled with linear functions and with exponential functions.Algebra I
AlabamaF-LE.38Students will: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).Algebra I
AlabamaF-LE.39Students will: Observe, using graphs and tables, that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.Algebra I
AlabamaF-LE.40Students will: Interpret the parameters in a linear or exponential function in terms of a context.Algebra I
AlabamaN-RN.1Students will: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.Algebra I
AlabamaN-RN.2Students will: Rewrite expressions involving radicals and rational exponents using the properties of exponents.Algebra I
AlabamaN-RN.3Students will: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.Algebra I
AlabamaN-Q.4Students will: Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.Algebra I
AlabamaN-Q.5Students will: Define appropriate quantities for the purpose of descriptive modeling.Algebra I
AlabamaN-Q.6Students will: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.Algebra I
AlabamaS-ID.41Students will: Represent data with plots on the real number line (dot plots, histograms, and box plots).Algebra I
AlabamaS-ID.42Students will: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.Algebra I
AlabamaS-ID.43Students will: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).Algebra I
AlabamaS-ID.44Students will: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.Algebra I
AlabamaS-ID.45Students will: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.Algebra I
AlabamaS-ID.46Students will: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Algebra I
AlabamaS-CP.47Students will: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.Algebra I
AlabamaG-CO.1Students will: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Geometry
AlabamaG-CO.2Students will: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).Geometry
AlabamaG-CO.3Students will: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.Geometry
AlabamaG-CO.4Students will: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.Geometry
AlabamaG-CO.5Students will: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Geometry
AlabamaG-CO.6Students will: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Geometry
AlabamaG-CO.7Students will: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Geometry
AlabamaG-CO.8Students will: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.Geometry
AlabamaG-CO.9Students will: Prove theorems about lines and angles.Geometry
AlabamaG-CO.10Students will: Prove theorems about triangles.Geometry
AlabamaG-CO.11Students will: Prove theorems about parallelograms.Geometry
AlabamaG-CO.12Students will: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Geometry
AlabamaG-CO.13Students will: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.Geometry
AlabamaG-SRT.14Students will: Verify experimentally the properties of dilations given by a center and a scale factor.Geometry
AlabamaG-SRT.15Students will: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.Geometry
AlabamaG-SRT.16Students will: Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar.Geometry
AlabamaG-SRT.17Students will: Prove theorems about triangles.Geometry
AlabamaG-SRT.18Students will: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.Geometry
AlabamaG-SRT.19Students will: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles.Geometry
AlabamaG-SRT.20Students will: Explain and use the relationship between the sine and cosine of complementary angles.Geometry
AlabamaG-SRT.21Students will: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Geometry
AlabamaG-SRT.22Students will: Prove the Law of Sines and the Law of Cosines and use them to solve problems.Geometry
AlabamaG-SRT.23Students will: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).Geometry
AlabamaG-C.24Students will: Prove that all circles are similar.Geometry
AlabamaG-C.25Students will: Identify and describe relationships among inscribed angles, radii, and chords.Geometry
AlabamaG-C.26Students will: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.Geometry
AlabamaG-C.27Students will: Construct a tangent line from a point outside a given circle to the circle.Geometry
AlabamaG-C.28Students will: Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.Geometry
AlabamaG-GPE.29Students will: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.Geometry
AlabamaG-GPE.30Students will: Use coordinates to prove simple geometric theorems algebraically.Geometry
AlabamaG-GPE.31Students will: Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).Geometry
AlabamaG-GPE.32Students will: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.Geometry
AlabamaG-GPE.33Students will: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.Geometry
AlabamaG-GPE.34Students will: Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics.Geometry
AlabamaG-GMD.35Students will: Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone.Geometry
AlabamaG-GMD.36Students will: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.Geometry
AlabamaG-GMD.37Students will: Determine the relationship between surface areas of similar figures and volumes of similar figures.Geometry
AlabamaG-GMD.38Students will: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.Geometry
AlabamaG-MG.39Students will: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).Geometry
AlabamaG-MG.40Students will: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).Geometry
AlabamaG-MG.41Students will: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).Geometry
AlabamaS-MD.42Students will: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Geometry
AlabamaS-MD.43Students will: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Geometry
AlabamaA-MOD.1Students will: Create algebraic models for application-based problems by developing and solving equations and inequalities, including those involving direct, inverse, and joint variation.Algebraic Connections
AlabamaA-MOD.2Students will: Solve application-based problems by developing and solving systems of linear equations and inequalities.Algebraic Connections
AlabamaA-MOD.3Students will: Use formulas or equations of functions to calculate outcomes of exponential growth or decay.Algebraic Connections
AlabamaA-GRA.4Students will: Determine maximum and minimum values of a function using linear programming procedures.Algebraic Connections
AlabamaA-GRA.5Students will: Determine approximate rates of change of nonlinear relationships from graphical and numerical data.Algebraic Connections
AlabamaA-GRA.6Students will: Use the extreme value of a given quadratic function to solve applied problems.Algebraic Connections
AlabamaA-FIN.7Students will: Use analytical, numerical, and graphical methods to make financial and economic decisions, including those involving banking and investments, insurance, personal budgets, credit purchases, recreation, and deceptive and fraudulent pricing and advertising.Algebraic Connections
AlabamaG-MOD.8Students will: Determine missing information in an application-based situation using properties of right triangles, including trigonometric ratios and the Pythagorean Theorem.Algebraic Connections
AlabamaG-SYM.9Students will: Analyze aesthetics of physical models for line symmetry, rotational symmetry, or the golden ratio.Algebraic Connections
AlabamaG-MEA.10Students will: Critique measurements in terms of precision, accuracy, and approximate error.Algebraic Connections
AlabamaG-MEA.11Students will: Use ratios of perimeters, areas, and volumes of similar figures to solve applied problems.Algebraic Connections
AlabamaS-GRA.12Students will: Create a model of a set of data by estimating the equation of a curve of best fit from tables of values or scatter plots.Algebraic Connections
AlabamaA-SSE.12Students will: Interpret expressions that represent a quantity in terms of its context.Algebra II
AlabamaA-SSE.13Students will: Use the structure of an expression to identify ways to rewrite it.Algebra II
AlabamaA-SSE.14Students will: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.Algebra II
AlabamaA-APR.15Students will: Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Algebra II
AlabamaA-APR.16Students will: Know and apply the Remainder Theorem: For a polynomial 𝘱(𝘹) and a number 𝘢, the remainder on division by 𝘹 – 𝘢 is 𝘱(𝘢), so 𝘱(𝘢) = 0 if and only if (𝘹 – 𝘢) is a factor of 𝘱(𝘹).Algebra II
AlabamaA-APR.17Students will: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Algebra II
AlabamaA-APR.18Students will: Prove polynomial identities and use them to describe numerical relationships.Algebra II
AlabamaA-APR.19Students will: Rewrite simple rational expressions in different forms; write 𝘢(𝘹)/𝘣(𝘹) in the form 𝘲(𝘹) + 𝘳(𝘹)/𝘣(𝘹), where 𝘢(𝘹), 𝘣(𝘹), 𝘲(𝘹), and 𝘳(𝘹) are polynomials with the degree of 𝘳(𝘹) less than the degree of 𝘣(𝘹), using inspection, long division, or for the more complicated examples, a computer algebra system.Algebra II
AlabamaA-CED.20Students will: Create equations and inequalities in one variable and use them to solve problems.Algebra II
AlabamaA-CED.21Students will: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra II
AlabamaA-CED.22Students will: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.Algebra II
AlabamaA-CED.23Students will: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Algebra II
AlabamaA-REI.24Students will: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.Algebra II
AlabamaA-REI.25Students will: Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b.Algebra II
AlabamaA-REI.26Students will: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).Algebra II
AlabamaA-REI.27Students will: Explain why the 𝘹-coordinates of the points where the graphs of the equations 𝘺 = 𝘧(𝘹) and 𝘺 = 𝑔(𝘹) intersect are the solutions of the equation 𝘧(𝘹) = 𝑔(𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘧(𝘹) and/or 𝑔(𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Algebra II
AlabamaA-CS.28Students will: Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations.Algebra II
AlabamaF-IF.29Students will: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Algebra II
AlabamaF-IF.30Students will: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra II
AlabamaF-IF.31Students will: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Algebra II
AlabamaF-IF.32Students will: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Algebra II
AlabamaF-BF.33Students will: Write a function that describes a relationship between two quantities.Algebra II
AlabamaF-BF.34Students will: Identify the effect on the graph of replacing 𝘧(𝘹) by 𝘧(𝘹) + 𝘬, 𝘬 𝘧(𝘹), 𝘧(𝘬𝘹), and 𝘧(𝘹 + 𝘬) for specific values of 𝘬 (both positive and negative); find the value of 𝘬 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Algebra II
AlabamaF-BF.35Students will: Find inverse functions.Algebra II
AlabamaF-LE.36Students will: For exponential models, express as a logarithm the solution to 𝘢𝘣 to the 𝘤𝘵 power = 𝘥 where 𝘢, 𝘤, and 𝘥 are numbers and the base 𝘣 is 2, 10, or 𝘦; evaluate the logarithm using technology.Algebra II
AlabamaN-CN.1Students will: Know there is a complex number 𝘪 such that 𝘪² = –1, and every complex number has the form 𝘢 + 𝘣𝘪 with 𝘢 and 𝘣 real.Algebra II
AlabamaN-CN.2Students will: Use the relation 𝘪² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.Algebra II
AlabamaN-CN.3Students will: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.Algebra II
AlabamaN-CN.4Students will: Solve quadratic equations with real coefficients that have complex solutions.Algebra II
AlabamaN-CN.5Students will: Extend polynomial identities to the complex numbers.Algebra II
AlabamaN-CN.6Students will: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.Algebra II
AlabamaN-VM.7Students will: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.Algebra II
AlabamaN-VM.8Students will: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.Algebra II
AlabamaN-VM.9Students will: Add, subtract, and multiply matrices of appropriate dimensions.Algebra II
AlabamaN-VM.10Students will: Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.Algebra II
AlabamaN-VM.11Students will: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.Algebra II
AlabamaS-MD.37Students will: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Algebra II
AlabamaS-MD.38Students will: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Algebra II
AlabamaS-CP.39Students will: Describe events as subsets of a sample space (the set of outcomes), using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).Algebra II
AlabamaS-CP.40Students will: Understand the conditional probability of 𝘈 given 𝘉 as 𝘗(𝘈 and 𝘉)/𝘗(𝘉), and interpret independence of 𝘈 and 𝘉 as saying that the conditional probability of 𝘈 given 𝘉 is the same as the probability of 𝘈, and the conditional probability of 𝘉 given 𝘈 is the same as the probability of 𝘉.Algebra II
AlabamaS-CP.41Students will: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.Algebra II
AlabamaS-CP.42Students will: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.Algebra II
AlabamaS-CP.43Students will: Find the conditional probability of 𝘈 given 𝘉 as the fraction of 𝘉’s outcomes that also belong to 𝘈, and interpret the answer in terms of the model.Algebra II
AlabamaS-CP.44Students will: Apply the Addition Rule, 𝘗(𝘈 or 𝘉) = 𝘗(𝘈) + 𝘗(𝘉) – 𝘗(𝘈 and 𝘉), and interpret the answer in terms of the model.Algebra II
AlabamaS-CP.45Students will: Apply the general Multiplication Rule in a uniform probability model, 𝘗(𝘈 and 𝘉) = 𝘗(𝘈)𝘗(𝘉|𝘈) = 𝘗(𝘉)𝘗(𝘈|𝘉), and interpret the answer in terms of the model.Algebra II
AlabamaS-CP.46Students will: Use permutations and combinations to compute probabilities of compound events and solve problems.Algebra II
AlabamaA-SSE.12Students will: Interpret expressions that represent a quantity in terms of its context.Algebra II With Trigonometry
AlabamaA-SSE.13Students will: Use the structure of an expression to identify ways to rewrite it.Algebra II With Trigonometry
AlabamaA-SSE.14Students will: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.Algebra II With Trigonometry
AlabamaA-APR.15Students will: Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Algebra II With Trigonometry
AlabamaA-APR.16Students will: Know and apply the Remainder Theorem: For a polynomial 𝘱(𝘹) and a number 𝘢, the remainder on division by 𝘹 – 𝘢 is 𝘱(𝘢), so 𝘱(𝘢) = 0 if and only if (𝘹 – 𝘢) is a factor of 𝘱(𝘹).Algebra II With Trigonometry
AlabamaA-APR.17Students will: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Algebra II With Trigonometry
AlabamaA-APR.18Students will: Prove polynomial identities and use them to describe numerical relationships.Algebra II With Trigonometry
AlabamaA-APR.19Students will: Rewrite simple rational expressions in different forms; write 𝘢(𝘹)/𝘣(𝘹) in the form 𝘲(𝘹) + 𝘳(𝘹)/𝘣(𝘹), where 𝘢(𝘹), 𝘣(𝘹), 𝘲(𝘹), and 𝘳(𝘹) are polynomials with the degree of 𝘳(𝘹) less than the degree of 𝘣(𝘹), using inspection, long division, or for the more complicated examples, a computer algebra system.Algebra II With Trigonometry
AlabamaA-CED.20Students will: Create equations and inequalities in one variable and use them to solve problems.Algebra II With Trigonometry
AlabamaA-CED.21Students will: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra II With Trigonometry
AlabamaA-CED.22Students will: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.Algebra II With Trigonometry
AlabamaA-CED.23Students will: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Algebra II With Trigonometry
AlabamaA-REI.24Students will: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.Algebra II With Trigonometry
AlabamaA-REI.25Students will: Recognize when the quadratic formula gives complex solutions, and write them as a ± bi for real numbers a and b.Algebra II With Trigonometry
AlabamaA-REI.26Students will: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).Algebra II With Trigonometry
AlabamaA-REI.27Students will: Explain why the 𝘹-coordinates of the points where the graphs of the equations 𝘺 = 𝘧(𝘹) and 𝘺 = 𝑔(𝘹) intersect are the solutions of the equation 𝘧(𝘹) = 𝑔(𝘹); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝘧(𝘹) and/or 𝑔(𝘹) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Algebra II With Trigonometry
AlabamaA-CS.28Students will: Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations.Algebra II With Trigonometry
AlabamaF-IF.29Students will: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Algebra II With Trigonometry
AlabamaF-IF.30Students will: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra II With Trigonometry
AlabamaF-IF.31Students will: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Algebra II With Trigonometry
AlabamaF-IF.32Students will: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Algebra II With Trigonometry
AlabamaF-BF.33Students will: Write a function that describes a relationship between two quantities.Algebra II With Trigonometry
AlabamaF-BF.34Students will: Identify the effect on the graph of replacing 𝘧(𝘹) by 𝘧(𝘹) + 𝘬, 𝘬𝘧(𝘹), 𝘧(𝘬𝘹), and 𝘧(𝘹 + 𝘬) for specific values of 𝘬 (both positive and negative); find the value of 𝘬 given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Algebra II With Trigonometry
AlabamaF-BF.35Students will: Find inverse functions.Algebra II With Trigonometry
AlabamaF-LE.36Students will: For exponential models, express as a logarithm the solution to 𝘢𝘣 to the 𝘤𝘵 power = 𝘥 where 𝘢, 𝘤, and 𝘥 are numbers and the base 𝘣 is 2, 10, or 𝘦; evaluate the logarithm using technology.Algebra II With Trigonometry
AlabamaF-TF.37Students will: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.Algebra II With Trigonometry
AlabamaF-TF.38Students will: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.Algebra II With Trigonometry
AlabamaF-TF.39Students will: Define the six trigonometric functions using ratios of the sides of a right triangle, coordinates on the unit circle, and the reciprocal of other functions.Algebra II With Trigonometry
AlabamaF-TF.40Students will: Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.Algebra II With Trigonometry
AlabamaN-CN.1Students will: Know there is a complex number 𝘪 such that 𝘪² = –1, and every complex number has the form 𝘢 + 𝘣𝘪 with 𝘢 and 𝘣 real.Algebra II With Trigonometry
AlabamaN-CN.2Students will: Use the relation 𝘪² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.Algebra II With Trigonometry
AlabamaN-CN.3Students will: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.Algebra II With Trigonometry
AlabamaN-CN.4Students will: Solve quadratic equations with real coefficients that have complex solutions.Algebra II With Trigonometry
AlabamaN-CN.5Students will: Extend polynomial identities to the complex numbers.Algebra II With Trigonometry
AlabamaN-CN.6Students will: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.Algebra II With Trigonometry
AlabamaN-VM.7Students will: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.Algebra II With Trigonometry
AlabamaN-VM.8Students will: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.Algebra II With Trigonometry
AlabamaN-VM.9Students will: Add, subtract, and multiply matrices of appropriate dimensions.Algebra II With Trigonometry
AlabamaN-VM.10Students will: Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.Algebra II With Trigonometry
AlabamaN-VM.11Students will: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.Algebra II With Trigonometry
AlabamaS-MD.41Students will: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Algebra II With Trigonometry
AlabamaS-MD.42Students will: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Algebra II With Trigonometry
AlabamaS-CP.43Students will: Describe events as subsets of a sample space (the set of outcomes), using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).Algebra II With Trigonometry
AlabamaS-CP.44Students will: Understand the conditional probability of 𝘈 given 𝘉 as 𝘗(𝘈 and 𝘉)/𝘗(𝘉), and interpret independence of 𝘈 and 𝘉 as saying that the conditional probability of 𝘈 given 𝘉 is the same as the probability of 𝘈, and the conditional probability of 𝘉 given 𝘈 is the same as the probability of 𝘉.Algebra II With Trigonometry
AlabamaS-CP.45Students will: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.Algebra II With Trigonometry
AlabamaS-CP.46Students will: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.Algebra II With Trigonometry
AlabamaS-CP.47Students will: Find the conditional probability of 𝘈 given 𝘉 as the fraction of 𝘉’s outcomes that also belong to 𝘈, and interpret the answer in terms of the model.Algebra II With Trigonometry
AlabamaS-CP.48Students will: Apply the Addition Rule, 𝘗(𝘈 or 𝘉) = 𝘗(𝘈) + 𝘗(𝘉) – 𝘗(𝘈 and 𝘉), and interpret the answer in terms of the model.Algebra II With Trigonometry
AlabamaS-CP.49Students will: Apply the general Multiplication Rule in a uniform probability model, 𝘗(𝘈 and 𝘉) = 𝘗(𝘈)𝘗(𝘉|𝘈) = 𝘗(𝘉)𝘗(𝘈|𝘉), and interpret the answer in terms of the model.Algebra II With Trigonometry
AlabamaS-CP.50Students will: Use permutations and combinations to compute probabilities of compound events and solve problems.Algebra II With Trigonometry
AlabamaA.7.aDevelop optimal solutions of application-based problems using existing and student-created algorithms.Discrete Mathematics
AlabamaA.8.aUse shortest path techniques to find optimal shipping routes.Discrete Mathematics
AlabamaA.5.aCreate a Fibonacci sequence when given two initial integers.Mathematical Investigations
AlabamaA.5.bInvestigate Tartaglia’s formula for solving cubic equations.Mathematical Investigations
AlabamaA.7.aSummarize the significance of René Descartes’ Cartesian coordinate system.Mathematical Investigations
AlabamaA.7.bInterpret the foundation of analytic geometry with regard to geometric curves and algebraic relationships.Mathematical Investigations
AlabamaG.9.aSummarize the historical development of perspective in art and architecture.Mathematical Investigations
AlabamaG.10.aConstruct multiple proofs of the Pythagorean Theorem.Mathematical Investigations
AlabamaG.10.bSolve problems involving figurate numbers, including triangular and pentagonal numbers.Mathematical Investigations
AlabamaN.1.aDetermine relationships among mathematical achievements of ancient peoples, including the Sumerians, Babylonians, Egyptians, Mesopotamians, Chinese, Aztecs, and Incas.Mathematical Investigations
AlabamaN.1.bExplain origins of the Hindu-Arabic numeration system.Mathematical Investigations
AlabamaN.2.aDetermine lengths of strings necessary to produce harmonic tones as in Pythagorean tuning.Mathematical Investigations
AlabamaN.3.aIdentify transcendental numbers.Mathematical Investigations
AlabamaN.4.aAnalyze contributions to the number system by well-known mathematicians, including Archimedes, John Napier, René Descartes, Sir Isaac Newton, Johann Carl Friedrich Gauss, and Julius Wilhelm Richard Dedekind.Mathematical Investigations
AlabamaA-SSE.12Students will: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.Precalculus
AlabamaA-APR.13Students will: Know and apply the Binomial Theorem for the expansion of (𝘹 + 𝘺)ⁿ in powers of 𝘹 and y for a positive integer 𝘯, where 𝘹 and 𝘺 are any numbers, with coefficients determined, for example, by Pascal’s Triangle.Precalculus
AlabamaA-REI.14Students will: Represent a system of linear equations as a single matrix equation in a vector variable.Precalculus
AlabamaA-CS.15Students will: Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from second-degree equations.Precalculus
AlabamaF-IF.16Students will: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Precalculus
AlabamaF-IF.17Students will: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Precalculus
AlabamaF-IF.18Students will: Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Precalculus
AlabamaF-BF.19Students will: Compose functions.Precalculus
AlabamaF-BF.20Students will: Determine the inverse of a function and a relation.Precalculus
AlabamaF-BF.21Students will: Verify by composition that one function is the inverse of another.Precalculus
AlabamaF-BF.22Students will: Read values of an inverse function from a graph or a table, given that the function has an inverse.Precalculus
AlabamaF-BF.23Students will: Produce an invertible function from a non-invertible function by restricting the domain.Precalculus
AlabamaF-BF.24Students will: Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.Precalculus
AlabamaF-BF.25Students will: Compare effects of parameter changes on graphs of transcendental functions.Precalculus
AlabamaF-TF.26Students will: Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses.Precalculus
AlabamaF-TF.27Students will: Use the sum, difference, and half-angle identities to find the exact value of a trigonometric function.Precalculus
AlabamaF-TF.28Students will: Utilize parametric equations by graphing and by converting to rectangular form.Precalculus
AlabamaF-TF.29Students will: Use special triangles to determine geometrically the values of sine, cosine, and tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π – 𝘹, π + 𝘹, and 2π – 𝘹 in terms of their values for 𝘹, where 𝘹 is any real number.Precalculus
AlabamaF-TF.30Students will: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.Precalculus
AlabamaF-TF.31Students will: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.Precalculus
AlabamaF-TF.32Students will: Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.Precalculus
AlabamaF-TF.33Students will: Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.Precalculus
AlabamaF-TF.34Students will: Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.Precalculus
AlabamaG-SRT.35Students will: Derive the formula 𝐴 = 1/2 𝘢𝘣 sin(𝐶) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.Precalculus
AlabamaG-GPE.36Students will: Derive the equation of a parabola given a focus and directrix.Precalculus
AlabamaG-GPE.37Students will: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.Precalculus
AlabamaG-GPE.38Students will: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.Precalculus
AlabamaN-CN.1Students will: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.Precalculus
AlabamaN-CN.2Students will: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.Precalculus
AlabamaN-CN.3Students will: Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.Precalculus
AlabamaN-L.4Students will: Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.Precalculus
AlabamaN-VM.5Students will: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., 𝙫, |𝙫|, ||𝙫||, 𝘷).Precalculus
AlabamaN-VM.6Students will: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.Precalculus
AlabamaN-VM.7Students will: Solve problems involving velocity and other quantities that can be represented by vectors.Precalculus
AlabamaN-VM.8Students will: Add and subtract vectors.Precalculus
AlabamaN-VM.9Students will: Multiply a vector by a scalar.Precalculus
AlabamaN-VM.10Students will: Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.Precalculus
AlabamaN-VM.11Students will: Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.Precalculus
AlabamaS-ID.39Students will: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.Precalculus
AlabamaS-ID.40Students will: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).Precalculus
AlabamaS-ID.41Students will: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.Precalculus
AlabamaS-ID.42Students will: Compute (using technology) and interpret the correlation coefficient of a linear fit.Precalculus
AlabamaS-ID.43Students will: Distinguish between correlation and causation.Precalculus
AlabamaS-IC.44Students will: Understand statistics as a process for making inferences about population parameters based on a random sample from that population.Precalculus
AlabamaS-IC.45Students will: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.Precalculus
AlabamaS-IC.46Students will: Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.Precalculus
AlabamaS-IC.47Students will: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.Precalculus
AlabamaS-IC.48Students will: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.Precalculus
AlabamaS-IC.49Students will: Evaluate reports based on data.Precalculus
AlabamaS-MD.50Students will: Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.Precalculus
AlabamaS-MD.51Students will: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.Precalculus
AlabamaS-MD.52Students will: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.Precalculus
AlabamaS-MD.53Students will: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.Precalculus
AlabamaS-MD.54Students will: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.Precalculus
Alabama8.aCritiquing and comparing options for purchasing an automobile including leasing, purchasing by cash, and purchasing by loanAlgebra with Finance
Alabama8.bInterpreting and analyzing various functions, graphs, graphs, and data analysis in order to make a responsible automobile purchase and to maintain the operation of an automobileAlgebra with Finance
Alabama8.cComputing braking distance using the formula BD = 5(.1s)²Algebra with Finance
Alabama8.dComputing distance, rate, and time using D = RT, R = D/T, and = T = D/RAlgebra with Finance
Alabama8.eUsing geometry theorems involving chords intersecting in a circle and radii perpendicular to chords to determine yaw mark arc lengthAlgebra with Finance
Alabama8.fComputing total stopping distance of an automobileAlgebra with Finance
Alabama8.gCalculating miles per gallon and distance using the formula D = MPG(G)Algebra with Finance
Alabama1.aCalculating cost of credit card interest with benefitsAlgebra with Finance
Alabama1.bUtilizing and understanding amortization tables for loansAlgebra with Finance
Alabama3.aDeriving formulas and use iteration to compute compound interestAlgebra with Finance
Alabama3.bCreating, interpreting, and analyzing a graph, table, and equation to compare compound interest and simple interestAlgebra with Finance
Alabama3.cApplying findings to short-term, long-term, single deposit and periodic deposit accountsAlgebra with Finance
Alabama3.dInterpreting the limit notationAlgebra with Finance
Alabama3.eModeling an infinite series and finding a finite sum for an infinite series with common ratio ½Algebra with Finance
Alabama3.fComputing limits of polynomial functions as x→∞Algebra with Finance
Alabama3.gComputing Annual Percentage Yield (APY) where APY = (1 + r/n)ⁿ − 1, given the Annual Percentage Rate (APR)Algebra with Finance
Alabama3.hAdapting algebra from banking formulas for input into a spreadsheetAlgebra with Finance
Alabama15.aCreating, evaluating, and interpreting algebraic proportionsAlgebra with Finance
Alabama15.bDetermining the curve of best fit using linear, quadratic, or cubic regression equationsAlgebra with Finance
Alabama15.cUsing exponential growth and decay equations that model given relationships between quantitiesAlgebra with Finance
Alabama15.dCalculating finance charge at various percentagesAlgebra with Finance
Alabama5.aCritiquing gross pay and net pay to determine total salary deductionsAlgebra with Finance
Alabama7.aIdentifying continuous and discontinuous functions by their graphsAlgebra with Finance
Alabama7.bGraphing pay schedulesAlgebra with Finance
Alabama7.cGraphing continuously polynomial functions with multiple slopes and cuspsAlgebra with Finance
Alabama17.aEvaluating the various mortgage products availableAlgebra with Finance
Alabama17.bComputing monthly mortgage payments at various terms and interest ratesAlgebra with Finance
Alabama17.cComparing mortgage payments and increasing resale value of a home using a future value of a periodic deposit formulaAlgebra with Finance
Alabama17.dModeling rent increases using exponential relationshipsAlgebra with Finance
Alabama18.aDetermining surface area and volume of irregular shapes including spheres, cylinders, or conesAlgebra with Finance
Alabama18.bDetermining the circumferences of circlesAlgebra with Finance
Alabama18.cDetermining area of various shapes including rectangles, squares, parallelograms, triangles, trapezoids, circles, regular polygons, irregular polygonsAlgebra with Finance
Alabama4.aConstructing, interpreting, and analyzing scatterplots by utilizing linear, quadratic, and regression equations to see a complete picture of supply, demand, revenue, and profitAlgebra with Finance
Alabama4.bConstructing algebraic ratios and proportionsAlgebra with Finance
Alabama4.cRecognizing, representing, and solving proportional relationships using equationsAlgebra with Finance
Alabama4.dDetermining percent increase/decrease of monetary amountsAlgebra with Finance
Alabama4.eConstructing and interpreting scatterplotsAlgebra with Finance
Alabama4.fIdentifying form, direction, and strength from a scatterplotAlgebra with Finance
Alabama4.gEvaluating and using functions to model relationships between quantitiesAlgebra with Finance
Alabama4.hTranslating verbal situations into algebraic linear functions and quadratic functionAlgebra with Finance
Alabama4.iCreating algebraic formulas for use in spreadsheetsAlgebra with Finance
Alabama4.jEvaluating and using functions to model relationships between algebraic fractions, ratios, and proportionsAlgebra with Finance
Alabama9.aUsing mathematical operations including addition and subtraction using negative numbersAlgebra with Finance
Alabama9.bSolving problems that require multiple mathematical operationsAlgebra with Finance
Alabama10.aFinding a common denominator in fractionsAlgebra with Finance
Alabama10.bFinding equivalent fractions in lowest termsAlgebra with Finance
Alabama10.cMultiplying mixed numbersAlgebra with Finance
Alabama12.aConverting units of money and time from one form to anotherAlgebra with Finance
Alabama19.aAnalyzing overall debt, cash flow, and resources to determine net worthAlgebra with Finance
Alabama19.bUsing the future value of a periodic investment formula to predict balances in future yearsAlgebra with Finance
Alabama19.cIdentifying the effect that a change in multipliers has to the value of an algebraic expressionAlgebra with Finance
Alabama19.dCreating rational expressions to represent increase over timeAlgebra with Finance
Alabama19.eCreating and interpreting a graph showing linear and a piecewise function and determining the point of intersectionAlgebra with Finance
Alabama19.fInterpreting points on a budget line graph in the context of their relationship to the budget lineAlgebra with Finance
AlabamaA-SSE.13Students will: Use the laws of Boolean Algebra to describe true/false circuits. Simplify Boolean expressions using the relationships between conjunction, disjunction, and negation operations.Analytical Mathematics
AlabamaA-SSE.14Students will: Use logic symbols to write truth tables.Analytical Mathematics
AlabamaA-APR.15Students will: Reduce the degree of either the numerator or denominator of a rational function by using partial fraction decomposition or partial fraction expansion.Analytical Mathematics
AlabamaF-TF.16Students will: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.Analytical Mathematics
AlabamaF-TF.17Students will: Prove the Law of Sines and the Law of Cosines and use them to solve problems. Understand Law of Sines = 2r, where r is the radius of the circumscribed circle of the triangle. Apply the Law of Tangents.Analytical Mathematics
AlabamaF-TF.18Students will: Apply Euler’s and deMoivre’s formulas as links between complex numbers and trigonometry.Analytical Mathematics
AlabamaN-VM.1Students will: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, │v│, ││v││), including the use of eigen-values and eigen-vectors.Analytical Mathematics
AlabamaN-VM.2Students will: Solve problems involving velocity and other quantities that can be represented by vectors, including navigation (e.g., airplane, aerospace, oceanic).Analytical Mathematics
AlabamaN-VM.3Students will: Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. Find the dot product and the cross product of vectors.Analytical Mathematics
AlabamaN-VM.4Students will: Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum, including vectors in complex vector spaces.Analytical Mathematics
AlabamaN-VM.5Students will: Understand vector subtraction v – w as v + (–w), where (–w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise, including vectors in complex vector spaces.Analytical Mathematics
AlabamaN-VM.6Students will: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network, including linear programming.Analytical Mathematics
AlabamaN-VM.7Students will: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled, including rotation matrices.Analytical Mathematics
AlabamaN-VM.8Students will: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. Solve matrix equations using augmented matrices.Analytical Mathematics
AlabamaN-VM.9Students will: Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors, including matrices larger than 2 × 2.Analytical Mathematics
AlabamaN-VM.10Students will: Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. Solve matrix application problems using reduced row echelon form.Analytical Mathematics
AlabamaN-CN.11Students will: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Understand the importance of using complex numbers in graphing functions on the Cartesian or complex plane.Analytical Mathematics
AlabamaN-L.12Students will: Calculate the limit of a sequence, of a function, and of an infinite series.Analytical Mathematics
Alabama7.aUtilize mathematical skills for trouble-shooting in business and industrial applications.Career Mathematics
Alabama9.aCalculate operation cost to maximize profit.Career Mathematics
Alabama9.bCalculate appropriate materials to use for an application.Career Mathematics
Alabama13.aFormulate tables from occupational outlook data to predict employment rates in various industrial areas.Career Mathematics
Alabama13.bConstruct scatter plots to analyze data and develop a plan that is most suitable for the application.Career Mathematics
Alabama14.aMake decisions basis on probabilities.Career Mathematics
Alabama3.aCreate graphs and tables related to personal finance and economics. The use of appropriate technology is encouraged for numerical and graphical investigations.Career Mathematics
Alabama3.bAnalyze job opportunities and career pathways related to business or industry.Career Mathematics
Alabama3.cEvaluate the economics of establishing and owning a business.Career Mathematics
Alabama3.dMake inferences and justify conclusions from economic conditions that can affect hiring and layoff decisions.Career Mathematics
Alabama4.aInterpret depreciation cost of decay relationships.Career Mathematics
Alabama5.aGraph functions expressed in tables, equations, or classroom-generated data to model consumer costs and to predict future outcomes.Career Mathematics
Alabama5.bAnalyze interest rates, depreciation, and tax rates in order to determine how each affects the cost of owning and/or operating a business.Career Mathematics
Alabama6.aPredict trends about population change that will affect employment rate.Career Mathematics
Alabama6.bCalculate pay scale based on occupational outlook projections.Career Mathematics
Alabama6.cCalculate operating costs, including cost of materials, supplies, equipment, license fees, and insurance fees.Career Mathematics
Alabama6.dConstruct charts that reflect current demographics in various industries.Career Mathematics
Alabama6.eForecast growth and decline of various career fields by interpreting data from charts and graphs.Career Mathematics
Alabama10.aDetermine overall angles or dimensions while working with various materials.Career Mathematics
Alabama10.bUse trigonometric ratios to apply properties of a right triangle to drawings or blueprints.Career Mathematics
Alabama11.aDesign drawings or blueprints to include pictorial, top, front, sides, back, and detailed views.Career Mathematics
Alabama11.bConstruct a project from designed drawings.Career Mathematics
Alabama12.aDetermine allowable geometric tolerance in various industrial applications.Career Mathematics
Alabama1.aDetermine dimensions by scaling plans or blueprints.Career Mathematics
Alabama1.bApply knowledge of fractions for reading a ruler to 1/16 inch.Career Mathematics
Alabama1.cConvert decimals to fractions for interpreting blue prints and measuring materials.Career Mathematics
Alabama1.dCompare Metric and English systems of measurements used in industry.Career Mathematics
Alabama1.eIdentify various measuring tools and demonstrate their use to verify precision, accuracy, and approximate error.Career Mathematics
Alabama2.aCalculate area utilizing the Pythagorean Theorem.Career Mathematics
Alabama2.bDemonstrate an understanding of blueprints and drawings.Career Mathematics
Alabama2.cCalculate estimates for construction or repair projects.Career Mathematics
AlaskaK.CC.1Count to 100 by ones and by tens.Kindergarten
AlaskaK.CC.2Count forward beginning from a given number within the known sequence.Kindergarten
AlaskaK.CC.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0 - 20 (with 0 representing a count of no objects).Kindergarten
AlaskaK.CC.4Understand the relationship between numbers and quantities; connect counting to cardinality.Kindergarten
AlaskaK.CC.5Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
AlaskaK.CC.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group (e.g., by using matching, counting, or estimating strategies).Kindergarten
AlaskaK.CC.7Compare and order two numbers between 1 and 10 presented as written numerals.Kindergarten
AlaskaK.G.1Describe objects in the environment using names of shapes and describe their relative positions (e.g., above, below, beside, in front of, behind, next to).Kindergarten
AlaskaK.G.2Name shapes regardless of their orientation or overall size.Kindergarten
AlaskaK.G.3Identify shapes as two-dimensional (flat) or three-dimensional (solid).Kindergarten
AlaskaK.G.4Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices), and other attributes (e.g., having sides of equal lengths).Kindergarten
AlaskaK.G.5Build shapes (e.g., using sticks and clay) and draw shapes.Kindergarten
AlaskaK.G.6Put together two-dimensional shapes to form larger shapes (e.g., join two triangles with full sides touching to make a rectangle).Kindergarten
AlaskaK.MD.1Describe measurable attributes of objects (e.g., length or weight). Match measuring tools to attribute (e.g., ruler to length). Describe several measurable attributes of a single object.Kindergarten
AlaskaK.MD.2Make comparisons between two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference.Kindergarten
AlaskaK.MD.3Classify objects into given categories (attributes). Count the number of objects in each category (limit category counts to be less than or equal to 10).Kindergarten
AlaskaK.MD.4Name in sequence the days of the week.Kindergarten
AlaskaK.MD.5Tell time to the hour using both analog and digital clocks.Kindergarten
AlaskaK.NBT.1Compose and decompose numbers from 11 to 19 into ten ones and some further ones (e.g., by using objects or drawings) and record each composition and decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight or nine ones.Kindergarten
AlaskaK.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps) acting out situations, verbal explanations, expressions, or equations.Kindergarten
AlaskaK.OA.2Add or subtract whole numbers to 10 (e.g., by using objects or drawings to solve word problems).Kindergarten
AlaskaK.OA.3Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., by using objects or drawings, and record each decomposition by a drawing or equation).Kindergarten
AlaskaK.OA.4For any number from 1- 4, find the number that makes 5 when added to the given number and, for any number from 1- 9, find the number that makes 10 when added to the given number (e.g., by using objects, drawings or 10 frames) and record the answer with a drawing or equation.Kindergarten
AlaskaK.OA.6Recognize, identify and continue simple patterns of color, shape, and size.Kindergarten
Alaska1.CC.2Use ordinal numbers correctly when identifying object position (e.g., first, second, third, etc.).Grade 1
Alaska1.CC.3Order numbers from 1 - 100. Demonstrate ability in counting forward and backward.Grade 1
Alaska1.CC.4Count a large quantity of objects by grouping into 10s and counting by 10s and 1s to find the quantity.Grade 1
Alaska1.CC.5Use the symbols for greater than, less than or equal to when comparing two numbers or groups of objects.Grade 1
Alaska1.CC.6Estimate how many and how much in a given set to 20 and then verify estimate by counting.Grade 1
Alaska1.G.1Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes. Identify shapes that have non-defining attributes (e.g., color, orientation, overall size). Build and draw shapes given specified attributes.Grade 1
Alaska1.G.2Compose (put together) two-dimensional or three-dimensional shapes to create a larger, composite shape, and compose new shapes from the composite shape.Grade 1
Alaska1.G.3Partition circles and rectangles into two and four equal shares. Describe the shares using the words, halves, fourths, and quarters and phrases half of, fourth of and quarter of. Describe the whole as two of or four of the shares. Understand for these examples that decomposing (break apart) into more equal shares creates smaller shares.Grade 1
Alaska1.MD.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.Grade 1
Alaska1.MD.3Tell and write time in half hours using both analog and digital clocks.Grade 1
Alaska1.MD.6Identify values of coins (e.g., nickel = 5 cents, quarter = 25 cents). Identify equivalent values of coins up to \$1 (e.g., 5 pennies = 1 nickel, 5 nickels = 1 quarter).Grade 1
Alaska1.NBT.1Count to 120. In this range, read, write and order numerals and represent a number of objects with a written numeral.Grade 1
Alaska1.NBT.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, <.Grade 1
Alaska1.NBT.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1
Alaska1.OA.1Use addition and subtraction strategies to solve word problems (using numbers up to 20), involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions, using a number line (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.Grade 1
Alaska1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.Grade 1
Alaska1.OA.3Apply properties of operations as strategies to add and subtract. (Students need not know the name of the property.)Grade 1
Alaska1.OA.7Understand the meaning of the equal sign (e.g., read equal sign as “same as”) and determine if equations involving addition and subtraction are true or false.Grade 1
Alaska1.OA.9Identify, continue and label patterns (e.g., aabb, abab). Create patterns using number, shape, size, rhythm or color.Grade 1
Alaska2.G.1Identify and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces compared visually, not by measuring. Identify triangles, quadrilaterals, pentagons, hexagons and cubes.Grade 2
Alaska2.G.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.Grade 2
Alaska2.G.3Partition circles and rectangles into shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.Grade 2
Alaska2.MD.1Measure the length of an object by selecting and using standard tools such as rulers, yardsticks, meter sticks, and measuring tapes.Grade 2
Alaska2.MD.2Measure the length of an object twice using different length units for the two measurements. Describe how the two measurements relate to the size of the unit chosen.Grade 2
Alaska2.MD.3Estimate, measure and draw lengths using whole units of inches, feet, yards, centimeters and meters.Grade 2
Alaska2.MD.4Measure to compare lengths of two objects, expressing the difference in terms of a standard length unit.Grade 2
Alaska2.MD.5Solve addition and subtraction word problems using numbers up to 100 involving length that are given in the same units (e.g., by using drawings of rulers). Write an equation with a symbol for the unknown to represent the problem.Grade 2
Alaska2.MD.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1,2, …, and represent whole-number sums and differences within 100 on a number line diagram.Grade 2
Alaska2.MD.7Tell and write time to the nearest five minutes using a.m. and p.m. from analog and digital clocks.Grade 2
Alaska2.MD.8Solve word problems involving dollar bills and coins using the \$ and ¢ symbols appropriately.Grade 2
Alaska2.MD.9Collect, record, interpret, represent, and describe data in a table, graph or line plot.Grade 2
Alaska2.MD.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart and compare problems using information presented in a bar graph.Grade 2
Alaska2.NBT.4Compare two three-digit numbers based on the meanings of the hundreds, tens and ones digits, using >, =, < symbols to record the results.Grade 2
Alaska2.NBT.6Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
Alaska2.NBT.8Mentally add 10 or 100 to a given number 100-900 and mentally subtract 10 or 100 from a given number.Grade 2
Alaska2.NBT.9Explain or illustrate the processes of addition or subtraction and their relationship using place value and the properties of operations.Grade 2
Alaska2.OA.1Use addition and subtraction strategies to estimate, then solve one- and two-step word problems (using numbers up to 100) involving situations of adding to, taking from, putting together, taking apart and comparing, with unknowns in all positions (e.g., by using objects, drawings and equations). Record and explain using equation symbols and a symbol for the unknown number to represent the problem.Grade 2
Alaska2.OA.2Fluently add and subtract using numbers up to 20 using mental strategies. Know from memory all sums of two one-digit numbers.Grade 2
Alaska2.OA.3Determine whether a group of objects (up to 20) is odd or even (e.g., by pairing objects and comparing, counting by 2s). Model an even number as two equal groups of objects and then write an equation as a sum of two equal addends.Grade 2
Alaska2.OA.4Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Write an equation to express the total as repeated addition (e.g., array of 4 by 5 would be 5 + 5 + 5 + 5 = 20).Grade 2
Alaska2.OA.5Identify, continue and label number patterns (e.g., aabb, abab). Describe a rule that determines and continues a sequence or pattern.Grade 2
Alaska3.G.1Categorize shapes by different attribute classifications and recognize that shared attributes can define a larger category. Generalize to create examples or non-examples.Grade 3
Alaska3.G.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.Grade 3
Alaska3.MD.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes or hours (e.g., by representing the problem on a number line diagram or clock).Grade 3
Alaska3.MD.2Estimate and measure liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (Excludes compound units such as cm³ and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve and create one-step word problems involving masses or volumes that are given in the same units (e.g., by using drawings, such as a beaker with a measurement scale, to represent the problem). (Excludes multiplicative comparison problems [problems involving notions of “times as much.”])Grade 3
Alaska3.MD.3Select an appropriate unit of English, metric, or non-standard measurement to estimate the length, time, weight, or temperature (L)Grade 3
Alaska3.MD.4Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs.Grade 3
Alaska3.MD.5Measure and record lengths using rulers marked with halves and fourths of an inch. Make a line plot with the data, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.Grade 3
Alaska3.MD.6Explain the classification of data from real-world problems shown in graphical representations. Use the terms minimum and maximum. (L)Grade 3
Alaska3.MD.7Recognize area as an attribute of plane figures and understand concepts of area measurement.Grade 3
Alaska3.MD.8Measure areas by tiling with unit squares (square centimeters, square meters, square inches, square feet, and improvised units).Grade 3
Alaska3.NBT.1Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
Alaska3.NBT.2Use strategies and/or algorithms to fluently add and subtract with numbers up to 1000, demonstrating understanding of place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
Alaska3.NBT.3Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 10 x 60) using strategies based on place value and properties of operations.Grade 3
Alaska3.NF.1Understand a fraction 1/𝑏 (e.g., 1/4) as the quantity formed by 1 part when a whole is partitioned into 𝑏 (e.g., 4) equal parts; understand a fraction 𝑎/𝑏 (e.g., 2/4) as the quantity formed by 𝑎 (e.g., 2) parts of size 1/𝑏. (e.g., 1/4)Grade 3
Alaska3.NF.2Understand a fraction as a number on the number line; represent fractions on a number line diagram.Grade 3
Alaska3.OA.1Interpret products of whole numbers (e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each).Grade 3
Alaska3.OA.2Interpret whole-number quotients of whole numbers (e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each).Grade 3
Alaska3.OA.3Use multiplication and division numbers up to 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem).Grade 3
Alaska3.OA.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers.Grade 3
Alaska3.OA.5Make, test, support, draw conclusions and justify conjectures about properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.)Grade 3
Alaska3.OA.7Fluently multiply and divide numbers up to 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 ×5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
Alaska3.OA.8Solve and create two-step word problems using any of the four operations. Represent these problems using equations with a symbol (box, circle, question mark) standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 3
Alaska3.OA.9Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations.Grade 3
Alaska4.G.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular, parallel, and intersecting line segments. Identify these in two-dimensional (plane) figures.Grade 4
Alaska4.G.2Classify two-dimensional (plane) figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.Grade 4
Alaska4.G.3Recognize a line of symmetry for a two-dimensional (plane) figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.Grade 4
Alaska4.MD.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
Alaska4.MD.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
Alaska4.MD.3Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.Grade 4
Alaska4.MD.5Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
Alaska4.MD.6Explain the classification of data from real-world problems shown in graphical representations including the use of terms range and mode with a given set of data.Grade 4
Alaska4.MD.7Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint.Grade 4
Alaska4.MD.8Measure and draw angles in whole-number degrees using a protractor. Estimate and sketch angles of specified measure.Grade 4
Alaska4.MD.9Recognize angle measure as additive. When an angle is divided into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems (e.g., by using an equation with a symbol for the unknown angle measure).Grade 4
Alaska4.NBT.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.Grade 4
Alaska4.NBT.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on the value of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
Alaska4.NBT.3Use place value understanding to round multi-digit whole numbers to any place using a variety of estimation methods; be able to describe, compare, and contrast solutions.Grade 4
Alaska4.NBT.4Fluently add and subtract multi-digit whole numbers using any algorithm. Verify the reasonableness of the results.Grade 4
Alaska4.NBT.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Alaska4.NBT.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Alaska4.NF.1Explain why a fraction 𝑎/𝑏 is equivalent to a fraction (𝑛 × 𝑎)/(𝑛 × 𝑏) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
Alaska4.NF.2Compare two fractions with different numerators and different denominators (e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2). Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions (e.g., by using a visual fraction model).Grade 4
Alaska4.NF.3Understand a fraction 𝑎/𝑏 with 𝑎 > 1 as a sum of fractions 1/𝑏.Grade 4
Alaska4.NF.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.Grade 4
Alaska4.NF.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.Grade 4
Alaska4.NF.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions (e.g., by using a visual model).Grade 4
Alaska4.OA.1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 groups of 7 and 7 groups of 5 (Commutative property). Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
Alaska4.OA.2Multiply or divide to solve word problems involving multiplicative comparison (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem or missing numbers in an array). Distinguish multiplicative comparison from additive comparison.Grade 4
Alaska4.OA.3Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 4
Alaska4.OA.4Find all factor pairs for a whole number in the range 1–100.Grade 4
Alaska4.OA.5Generate a number, shape pattern, table, t-chart, or input/output function that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. Be able to express the pattern in algebraic terms.Grade 4
Alaska4.OA.6Extend patterns that use addition, subtraction, multiplication, division or symbols, up to 10 terms, represented by models (function machines), tables, sequences, or in problem situations.Grade 4
Alaska5.G.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., 𝑥-axis and 𝑥-coordinate, 𝑦-axis and 𝑦-coordinate).Grade 5
Alaska5.G.2Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
Alaska5.G.3Understand that attributes belonging to a category of two-dimensional (plane) figures also belong to all subcategories of that category.Grade 5
Alaska5.G.4Classify two-dimensional (plane) figures in a hierarchy based on attributes and properties.Grade 5
Alaska5.MD.1Identify, estimate measure, and convert equivalent measures within systems English length (inches, feet, yards, miles) weight (ounces, pounds, tons) volume (fluid ounces, cups, pints, quarts, gallons) temperature (Fahrenheit) Metric length (millimeters, centimeters, meters, kilometers) volume (milliliters, liters), temperature (Celsius), (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems using appropriate tools.Grade 5
Alaska5.MD.3Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving information presented in line plots.Grade 5
Alaska5.MD.4Explain the classification of data from real-world problems shown in graphical representations including the use of terms mean and median with a given set of data.Grade 5
Alaska5.MD.5Recognize volume as an attribute of solid figures and understand concepts of volume measurement.Grade 5
Alaska5.MD.6Estimate and measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-standard units.Grade 5
Alaska5.MD.7Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.Grade 5
Alaska5.NBT.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
Alaska5.NBT.2Explain and extend the patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain and extend the patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.Grade 5
Alaska5.NBT.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, number lines, real life situations, and/or area models.Grade 5
Alaska5.NBT.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between the operations. Relate the strategy to a written method and explain their reasoning in getting their answers.Grade 5
Alaska5.NF.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.Grade 5
Alaska5.NF.2Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators (e.g., by using visual fraction models or equations to represent the problem). Use benchmark fractions and number sense of fractions to estimate mentally and check the reasonableness of answers.Grade 5
Alaska5.NF.3Interpret a fraction as division of the numerator by the denominator (𝑎/𝑏 = 𝑎 ÷ 𝑏). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers (e.g., by using visual fraction models or equations to represent the problem).Grade 5
Alaska5.NF.4Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
Alaska5.NF.6Solve real-world problems involving multiplication of fractions and mixed numbers (e.g., by using visual fraction models or equations to represent the problem).Grade 5
Alaska5.NF.7Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Grade 5
Alaska5.OA.1Use parentheses to construct numerical expressions, and evaluate numerical expressions with these symbols.Grade 5
Alaska5.OA.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.Grade 5
Alaska5.OA.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.Grade 5
Alaska6.EE.3Apply the properties of operations to generate equivalent expressions. Model (e.g., manipulatives, graph paper) and apply the distributive, commutative, identity, and inverse properties with integers and variables by writing equivalent expressions.Grade 6
Alaska6.EE.4Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).Grade 6
Alaska6.EE.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
Alaska6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.Grade 6
Alaska6.EE.7Solve real-world and mathematical problems by writing and solving equations of the form 𝑥 + 𝑝 = 𝑞 and 𝑝𝑥 = 𝑞 for cases in which 𝑝, 𝑞 and 𝑥 are all nonnegative rational numbers.Grade 6
Alaska6.EE.8Write an inequality of the form 𝑥 > 𝑐 or 𝑥 < 𝑐 to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form 𝑥 > 𝑐 or 𝑥 < 𝑐 have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
Alaska6.EE.9Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.Grade 6
Alaska6.G.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing or decomposing into other polygons (e.g., rectangles and triangles). Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Alaska6.G.2Apply the standard formulas to find volumes of prisms. Use the attributes and properties (including shapes of bases) of prisms to identify, compare or describe three-dimensional figures including prisms and cylinders.Grade 6
Alaska6.G.3Draw polygons in the coordinate plane given coordinates for the vertices; determine the length of a side joining the coordinates of vertices with the same first or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Alaska6.G.4Represent three-dimensional figures (e.g., prisms) using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Alaska6.RP.1Write and describe the relationship in real life context between two quantities using ratio language.Grade 6
Alaska6.RP.2Understand the concept of a unit rate (𝑎/𝑏 associated with a ratio 𝑎:𝑏 with 𝑏 ≠ 0, and use rate language in the context of a ratio relationship) and apply it to solve real-world problems (e.g., unit pricing, constant speed).Grade 6
Alaska6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems (e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations).Grade 6
Alaska6.SP.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.Grade 6
Alaska6.SP.2Understand that a set of data has a distribution that can be described by its center (mean, median, or mode), spread (range), and overall shape and can be used to answer a statistical question.Grade 6
Alaska6.SP.3Recognize that a measure of center (mean, median, or mode) for a numerical data set summarizes all of its values with a single number, while a measure of variation (range) describes how its values vary with a single number.Grade 6
Alaska6.SP.4Display numerical data in plots on a number line, including dot or line plots, histograms and box (box and whisker) plots.Grade 6
Alaska6.SP.5Summarize numerical data sets in relation to their context, such as by:Grade 6
Alaska6.SP.6Analyze whether a game is mathematically fair or unfair by explaining the probability of all possible outcomes.Grade 6
Alaska6.SP.7Solve or identify solutions to problems involving possible combinations (e.g., if ice cream sundaes come in 3 flavors with 2 possible toppings, how many different sundaes can be made using only one flavor of ice cream with one topping?)Grade 6
Alaska6.NS.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions (e.g., by using visual fraction models and equations to represent the problem).Grade 6
Alaska6.NS.2Fluently multiply and divide multi-digit whole numbers using the standard algorithm. Express the remainder as a whole number, decimal, or simplified fraction; explain or justify your choice based on the context of the problem.Grade 6
Alaska6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Express the remainder as a terminating decimal, or a repeating decimal, or rounded to a designated place value.Grade 6
Alaska6.NS.4Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor.Grade 6
Alaska6.NS.5Understand that positive and negative numbers describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explain the meaning of 0 in each situation.Grade 6
Alaska6.NS.6Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
Alaska6.NS.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
Alaska7.EE.1Apply properties of operations as strategies to add, subtract, factor, expand and simplify linear expressions with rational coefficients.Grade 7
Alaska7.EE.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.Grade 7
Alaska7.EE.3Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
Alaska7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct multi-step equations and inequalities to solve problems by reasoning about the quantities.Grade 7
Alaska7.G.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
Alaska7.G.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes including polygons and circles with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
Alaska7.G.3Describe the two-dimensional figures, i.e., cross-section, that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Grade 7
Alaska7.G.4Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.Grade 7
Alaska7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Grade 7
Alaska7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Grade 7
Alaska7.RP.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.Grade 7
Alaska7.RP.2Recognize and represent proportional relationships between quantities. Make basic inferences or logical predictions from proportional relationships.Grade 7
Alaska7.SP.1Understand that statistics can be used to gain information about a population by examining a reasonably sized sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.Grade 7
Alaska7.SP.2Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.Grade 7
Alaska7.SP.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.Grade 7
Alaska7.SP.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.Grade 7
Alaska7.SP.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.Grade 7
Alaska7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Grade 7
Alaska7.SP.7Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.Grade 7
Alaska7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Grade 7
Alaska7.NS.1Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.Grade 7
Alaska7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers and use equivalent representations.Grade 7
Alaska7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)Grade 7
Alaska8.EE.1Apply the properties (product, quotient, power, zero, negative exponents, and rational exponents) of integer exponents to generate equivalent numerical expressions.Grade 8
Alaska8.EE.2Use square root and cube root symbols to represent solutions to equations of the form 𝑥² = 𝑝 and 𝑥³ = 𝑝, where 𝑝 is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.Grade 8
Alaska8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.Grade 8
Alaska8.EE.4Perform operations with numbers expressed in scientific notation, including problems where both standard notation and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.Grade 8
Alaska8.EE.5Graph linear equations such as 𝑦 = 𝑚𝑥 + 𝑏, interpreting 𝑚 as the slope or rate of change of the graph and 𝑏 as the 𝑦-intercept or starting value. Compare two different proportional relationships represented in different ways.Grade 8
Alaska8.EE.6Use similar triangles to explain why the slope 𝑚 is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation 𝑦 = 𝑚𝑥 for a line through the origin and the equation 𝑦 = 𝑚𝑥 + 𝑏 for a line intercepting the vertical axis at 𝑏.Grade 8
Alaska8.F.1Understand that a function is a rule that assigns to each input (the domain) exactly one output (the range). The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. For example, use the vertical line test to determine functions and non-functions.Grade 8
Alaska8.F.2Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Grade 8
Alaska8.F.3Interpret the equation 𝑦 = 𝑚𝑥 + 𝑏 as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.Grade 8
Alaska8.F.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (𝑥, 𝑦) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Grade 8
Alaska8.F.5Given a verbal description between two quantities, sketch a graph. Conversely, given a graph, describe a possible real-world example.Grade 8
Alaska8.G.1Through experimentation, verify the properties of rotations, reflections, and translations (transformations) to figures on a coordinate plane).Grade 8
Alaska8.G.2Demonstrate understanding of congruence by applying a sequence of translations, reflections, and rotations on two-dimensional figures. Given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
Alaska8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Grade 8
Alaska8.G.4Demonstrate understanding of similarity, by applying a sequence of translations, reflections, rotations, and dilations on two-dimensional figures. Describe a sequence that exhibits the similarity between them.Grade 8
Alaska8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
Alaska8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
Alaska8.G.9Identify and apply the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Grade 8
Alaska8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
Alaska8.SP.2Explain why straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
Alaska8.SP.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and 𝑦-intercept.Grade 8
Alaska8.SP.4Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects and use relative frequencies to describe possible association between the two variables.Grade 8
Alaska8.NS.1Classify real numbers as either rational (the ratio of two integers, a terminating decimal number, or a repeating decimal number) or irrational.Grade 8
Alaska8.NS.2Order real numbers, using approximations of irrational numbers, locating them on a number line.Grade 8
Alaska8.NS.3Identify or write the prime factorization of a number using exponents.Grade 8
AlaskaN-Q.1Use units as a way to understand problems and to guide the solution of multi‐step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.High School
AlaskaN-Q.2Define appropriate quantities for the purpose of descriptive modeling.High School
AlaskaN-Q.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.High School
AlaskaN-CN.1Know there is a complex number 𝑖 such that 𝑖² = –1, and every complex number has the form 𝑎 + 𝑏𝑖 with 𝑎 and 𝑏 real.High School
AlaskaN-CN.2Use the relation 𝑖² = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.High School
AlaskaN-CN.3Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.High School
AlaskaN-CN.4Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.High School
AlaskaN-CN.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.High School
AlaskaN-CN.6Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.High School
AlaskaN-CN.8Extend polynomial identities to the complex numbers. For example, rewrite 𝑥² + 4 as (𝑥 + 2𝑖)(𝑥 – 2𝑖).High School
AlaskaN-CN.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.High School
AlaskaN-RN.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.High School
AlaskaN-RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.High School
AlaskaN-RN.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.High School
AlaskaN-VM.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., 𝙫, |𝙫|, ||𝙫||, 𝙫).High School
AlaskaN-VM.2Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.High School
AlaskaN-VM.3Solve problems involving velocity and other quantities that can be represented by vectors.High School
AlaskaN-VM.5Multiply a vector by a scalar.High School
AlaskaN-VM.6Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.High School
AlaskaN-VM.7Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.High School
AlaskaN-VM.9Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.High School
AlaskaN-VM.10Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.High School
AlaskaN-VM.11Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.High School
AlaskaN-VM.12Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.High School
AlaskaA-APR.1Add, subtract, and multiply polynomials. Understand that polynomials form a system similar to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.High School
AlaskaA-APR.2Know and apply the Remainder Theorem: For a polynomial 𝑝(𝑥) and a number 𝑎, the remainder on division by 𝑥 – 𝑎 is 𝑝(𝑎), so 𝑝(𝑎) = 0 if and only if (𝑥 – 𝑎) is a factor of 𝑝(𝑥).High School
AlaskaA-APR.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.High School
AlaskaA-APR.4Prove polynomial identities and use them to describe numerical relationships.High School
AlaskaA-APR.5Know and apply the Binomial Theorem for the expansion of (𝑥 + 𝑦)ⁿ in powers of 𝑥 and 𝑦 for a positive integer 𝑛, where 𝑥 and 𝑦 are any numbers, with coefficients determined for example by Pascal’s Triangle.High School
AlaskaA-APR.6Rewrite simple rational expressions in different forms; write 𝑎(𝑥)/𝑏(𝑥) in the form 𝑞(𝑥) + 𝑟(𝑥)/𝑏(𝑥), where 𝑎(𝑥), 𝑏(𝑥), 𝑞(𝑥), and 𝑟(𝑥) are polynomials with the degree of 𝑟(𝑥) less than the degree of 𝑏(𝑥), using inspection, long division, or, for the more complicated examples, a computer algebra system.High School
AlaskaA-APR.7Add, subtract, multiply, and divide rational expressions. Understand that rational expressions form a system similar to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression.High School
AlaskaA-CED.1Create equations and inequalities in one variable and use them to solve problems.High School
AlaskaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.High School
AlaskaA-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.High School
AlaskaA-CED.4Rearrange formulas (literal equations) to highlight a quantity of interest, using the same reasoning as in solving equations.High School
AlaskaA-REI.1Apply properties of mathematics to justify steps in solving equations in one variable.High School
AlaskaA-REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.High School
AlaskaA-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.High School
AlaskaA-REI.5Show that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.High School
AlaskaA-REI.6Solve systems of linear equations exactly and approximately, e.g., with graphs or algebraically, focusing on pairs of linear equations in two variables.High School
AlaskaA-REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.High School
AlaskaA-REI.8Represent a system of linear equations as a single matrix equation in a vector variable.High School
AlaskaA-REI.9Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).High School
AlaskaA-REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).High School
AlaskaA-REI.11Explain why the 𝑥‐coordinates of the points where the graphs of the equations 𝑦 = 𝑓(𝑥) and 𝑦 = 𝑔(𝑥) intersect are the solutions of the equation 𝑓(𝑥) = 𝑔(𝑥); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where 𝑓(𝑥) and/or 𝑔(𝑥) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.High School
AlaskaA-REI.12Graph the solutions to a linear inequality in two variables as a half‐plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half‐planes.High School
AlaskaA-SSE.1Interpret expressions that represent a quantity in terms of its context.High School
AlaskaA-SSE.2Use the structure of an expression to identify ways to rewrite it.High School
AlaskaA-SSE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.High School
AlaskaA-SSE.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.High School
AlaskaF-BF.1Write a function that describes a relationship between two quantities.High School
AlaskaF-BF.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.High School
AlaskaF-BF.3Identify the effect on the graph of replacing 𝑓(𝑥) by 𝑓(𝑥) + 𝑘, 𝑘 𝑓(𝑥), 𝑓(𝑘𝑥), and 𝑓(𝑥 + 𝑘) for specific values of 𝑘 (both positive and negative); find the value of 𝑘 given the graphs.High School
AlaskaF-BF.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.High School
AlaskaF-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If 𝑓 is a function and 𝑥 is an element of its domain, then 𝑓(𝑥) denotes the output of 𝑓 corresponding to the input 𝑥. The graph of 𝑓 is the graph of the equation 𝑦 = 𝑓(𝑥).High School
AlaskaF-IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.High School
AlaskaF-IF.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.High School
AlaskaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.High School
AlaskaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.High School
AlaskaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.High School
AlaskaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.High School
AlaskaF-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.High School
AlaskaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically, in tables, or by verbal descriptions).High School
AlaskaF-LE.1Distinguish between situations that can be modeled with linear functions and with exponential functions.High School
AlaskaF-LE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or input‐output table of values.High School
AlaskaF-LE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.High School
AlaskaF-LE.4For exponential models, express as a logarithm the solution to 𝑎𝑏𝑐𝑡 = 𝑑 where 𝑎, 𝑐, and 𝑑 are numbers and the base 𝑏 is 2, 10, or 𝑒; evaluate the logarithm using technology.High School
AlaskaF-LE.5Interpret the parameters in a linear or exponential function in terms of a context.High School
AlaskaF-TF.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.High School
AlaskaF-TF.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.High School
AlaskaF-TF.3Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for π – 𝑥, π + 𝑥, and 2π – 𝑥 in terms of their values for 𝑥, where 𝑥 is any real number.High School
AlaskaF-TF.4Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.High School
AlaskaF-TF.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.High School
AlaskaF-TF.6Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.High School
AlaskaF-TF.7Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.High School
AlaskaF-TF.8Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to calculate trigonometric ratios.High School
AlaskaF-TF.9Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.High School
AlaskaG-C.1Prove that all circles are similar.High School
AlaskaG-C.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.High School
AlaskaG-C.4Construct a tangent line from a point outside a given circle to the circle.High School
AlaskaG-C.5Use and apply the concepts of arc length and areas of sectors of circles. Determine or derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.High School
AlaskaG-CO.1Demonstrates understanding of key geometrical definitions, including angle, circle, perpendicular line, parallel line, line segment, and transformations in Euclidian geometry. Understand undefined notions of point, line, distance along a line, and distance around a circular arc.High School
AlaskaG-CO.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).High School
AlaskaG-CO.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.High School
AlaskaG-CO.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.High School
AlaskaG-CO.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.High School
AlaskaG-CO.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.High School
AlaskaG-CO.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.High School
AlaskaG-CO.8Explain how the criteria for triangle congruence (ASA, SAS, SSS, AAS, and HL) follow from the definition of congruence in terms of rigid motions.High School
AlaskaG-CO.9Using methods of proof including direct, indirect, and counter examples to prove theorems about lines and angles.High School
AlaskaG-CO.10Using methods of proof including direct, indirect, and counter examples to prove theorems about triangles.High School
AlaskaG-CO.11Using methods of proof including direct, indirect, and counter examples to prove theorems about parallelograms.High School
AlaskaG-CO.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.High School
AlaskaG-CO.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.High School
AlaskaG-GPE.1Determine or derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.High School
AlaskaG-GPE.2Determine or derive the equation of a parabola given a focus and directrix.High School
AlaskaG-GPE.3Derive the equations of ellipses and hyperbolas given foci and directrices.High School
AlaskaG-GPE.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).High School
AlaskaG-GPE.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.High School
AlaskaG-GPE.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.High School
AlaskaG-GMD.1Explain how to find the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.High School
AlaskaG-GMD.2Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.High School
AlaskaG-GMD.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.High School
AlaskaG-GMD.4Identify the shapes of two‐dimensional cross‐sections of three‐dimensional objects, and identify three‐dimensional objects generated by rotations of two‐dimensional objects.High School
AlaskaG-MG.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).High School
AlaskaG-MG.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).High School
AlaskaG-MG.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).High School
AlaskaG-SRT.1Verify experimentally the properties of dilations given by a center and a scale factor:High School
AlaskaG-SRT.2Given two figures, use the definition of similarity in terms of transformations to explain whether or not they are similar.High School
AlaskaG-SRT.3Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.High School
AlaskaG-SRT.5Apply congruence and similarity properties and prove relationships involving triangles and other geometric figures.High School
AlaskaG-SRT.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.High School
AlaskaG-SRT.7Explain and use the relationship between the sine and cosine of complementary angles.High School
AlaskaG-SRT.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.High School
AlaskaG-SRT.9Derive the formula 𝐴 = 1/2 𝑎𝑏 sin(𝐶) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.High School
AlaskaG-SRT.10Prove the Laws of Sines and Cosines and use them to solve problems.High School
AlaskaG-SRT.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non‐right triangles (e.g., surveying problems, resultant forces).High School
AlaskaS-CP.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).High School
AlaskaS-CP.2Understand that two events 𝐴 and 𝐵 are independent if the probability of 𝐴 and 𝐵 occurring together is the product of their probabilities, and use this characterization to determine if they are independent.High School
AlaskaS-CP.3Understand the conditional probability of 𝐴 given 𝐵 as 𝑃(𝐴 and 𝐵)/𝑃(𝐵), and interpret independence of 𝐴 and 𝐵 as saying that the conditional probability of 𝐴 given 𝐵 is the same as the probability of 𝐴, and the conditional probability of 𝐵 given 𝐴 is the same as the probability of 𝐵.High School
AlaskaS-CP.4Construct and interpret two‐way frequency tables of data when two categories are associated with each object being classified. Use the two‐way table as a sample space to decide if events are independent and to approximate conditional probabilities.High School
AlaskaS-CP.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.High School
AlaskaS-CP.6Find the conditional probability of 𝐴 given 𝐵 as the fraction of 𝐵’s outcomes that also belong to 𝐴, and interpret the answer in terms of the model.High School
AlaskaS-CP.7Apply the Addition Rule, 𝑃(𝐴 or 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) – 𝑃(𝐴 and 𝐵), and interpret the answer in terms of the model.High School
AlaskaS-CP.8Apply the general Multiplication Rule in a uniform probability model, 𝑃(𝐴 and 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) = 𝑃(𝐵)𝑃(𝐴|𝐵), and interpret the answer in terms of the model.High School
AlaskaS-CP.9Use permutations and combinations to compute probabilities of compound events and solve problems.High School
AlaskaS-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).High School
AlaskaS-ID.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.High School
AlaskaS-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).High School
AlaskaS-ID.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.High School
AlaskaS-ID.5Summarize categorical data for two categories in two‐way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.High School
AlaskaS-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.High School
AlaskaS-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.High School
AlaskaS-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.High School
AlaskaS-ID.9Distinguish between correlation and causation.High School
AlaskaS-IC.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.High School
AlaskaS-IC.2Decide if a specified model is consistent with results from a given data‐generating process, e.g., using simulation.High School
AlaskaS-IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.High School
AlaskaS-IC.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.High School
AlaskaS-IC.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.High School
AlaskaS-IC.6Evaluate reports based on data.High School
AlaskaS-MD.1Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.High School
AlaskaS-MD.2Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.High School
AlaskaS-MD.3Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.High School
AlaskaS-MD.4Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.High School
AlaskaS-MD.5Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.High School
AlaskaS-MD.6Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).High School
AlaskaS-MD.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).High School
ArizonaA-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Algebra
ArizonaA-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra
ArizonaA-SSE.A.2Use the structure of an expression to identify ways to rewrite it.Algebra
ArizonaA-SSE.B.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Algebra
ArizonaF-BF.A.1Write a function that describes a relationship between two quantities.Algebra
ArizonaF-IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Algebra
ArizonaF-IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Algebra
ArizonaF-IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra
ArizonaS-ID.B.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.Algebra
Arizona1.MD.B.3Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
Arizona1.MD.C.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
Arizona1.NBT.A.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
Arizona1.NBT.B.2Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: 10 can be thought of as a bundle of ten ones - called a 'ten.'. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).Grade 1
Arizona1.NBT.B.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with symbols.Grade 1
Arizona1.NBT.C.4Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
Arizona1.NBT.C.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1
Arizona1.NBT.C.6Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 1
Arizona1.OA.B.3Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)Grade 1
Arizona1.OA.B.4Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.Grade 1
Arizona1.OA.C.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Grade 1
Arizona1.OA.D.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.Grade 1
Arizona1.OA.D.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.Grade 1
Arizona2.G.A.1Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Grade 2
Arizona2.MD.C.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Grade 2
Arizona2.MD.D.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.Grade 2
Arizona2.MD.D.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.Grade 2
Arizona2.NBT.A.1Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: 100 can be thought of as a bundle of ten tens - called a 'hundred.'. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).Grade 2
Arizona2.NBT.A.2Count within 1000; skip-count by 5s, 10s, and 100s.Grade 2
Arizona2.NBT.A.3Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.Grade 2
Arizona2.NBT.A.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using symbols to record the results of comparisons.Grade 2
Arizona2.NBT.B.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 2
Arizona2.NBT.B.6Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
Arizona2.NBT.B.7Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Grade 2
Arizona2.NBT.B.8Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.Grade 2
Arizona2.OA.A.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 2
Arizona2.OA.B.2Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.Grade 2
Arizona3.G.A.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Grade 3
Arizona3.MD.A.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.Grade 3
Arizona3.MD.B.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step √±how many more√Æ and √±how many less√Æ problems using information presented in scaled bar graphs.Grade 3
Arizona3.MD.B.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.Grade 3
Arizona3.MD.C.5Recognize area as an attribute of plane figures and understand concepts of area measurement.Grade 3
Arizona3.MD.C.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).Grade 3
Arizona3.NBT.A.1Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
Arizona3.NBT.A.2Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
Arizona3.NBT.A.3Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 _ 80, 5 _ 60) using strategies based on place value and properties of operations.Grade 3
Arizona3.NF.A.1Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.Grade 3
Arizona3.NF.A.2Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.Grade 3
Arizona3.NF.A.3Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, orGrade 3
Arizona3.OA.A.1Interpret products of whole numbers, e.g., interpret 5 _ 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 _ 7.Grade 3
Arizona3.OA.A.2Interpret whole-number quotients of whole numbers, e.g., interpret 56 ¬Ö 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ¬Ö 8.Grade 3
Arizona3.OA.A.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 3
Arizona3.OA.A.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 _ ? = 48, 5 = _ ¬Ö 3, 6 _ 6 = ?Grade 3
Arizona3.OA.B.5Apply properties of operations as strategies to multiply and divide. Examples: If 6 _ 4 = 24 is known, then 4 _ 6 = 24 is also known. (Commutative property of multiplication.) 3 _ 5 _ 2 can be found by 3 _ 5 = 15, then 15 _ 2 = 30, or by 5 _ 2 = 10, then 3 _ 10 = 30. (Associative property of multiplication.) Knowing that 8 _ 5 = 40 and 8 _ 2 = 16, one can find 8 _ 7 as 8 _ (5 + 2) = (8 _ 5) + (8 _ 2) = 40 + 16 = 56. (Distributive property.)Grade 3
Arizona3.OA.B.6Understand division as an unknown-factor problem. For example, find 32 ¬Ö 8 by finding the number that makes 32 when multiplied by 8.Grade 3
Arizona3.OA.C.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 _ 5 = 40, one knows 40 ¬Ö 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
Arizona4.G.A.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Grade 4
Arizona4.G.A.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.Grade 4
Arizona4.MD.A.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
Arizona4.MD.A.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
Arizona4.MD.B.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
Arizona4.MD.C.5Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:Grade 4
Arizona4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Grade 4
Arizona4.MD.C.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Grade 4
Arizona4.NBT.A.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 / 70 = 10 by applying concepts of place value and division.Grade 4
Arizona4.NBT.A.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
Arizona4.NBT.A.3Use place value understanding to round multi-digit whole numbers to any place.Grade 4
Arizona4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm.Grade 4
Arizona4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Arizona4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Arizona4.NF.A.1Explain why a fraction a/b is equivalent to a fraction (n x a) / (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
Arizona4.NF.A.2Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, orGrade 4
Arizona4.NF.B.3Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.Grade 4
Arizona4.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4). Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x(2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a) / b.) Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?Grade 4
Arizona4.NF.C.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.Grade 4
Arizona4.NF.C.6Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.Grade 4
Arizona4.NF.C.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, orGrade 4
Arizona4.OA.A.1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
Arizona4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Grade 4
Arizona4.OA.B.4Find all factor pairs for a whole number in the range 1 - 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 ¬Ñ 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 - 100 is prime or composite.Grade 4
Arizona4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule 'Add 3' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.Grade 4
Arizona5.G.A.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and the given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).Grade 5
Arizona5.G.A.2Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
Arizona5.G.B.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.Grade 5
Arizona5.G.B.4Classify two-dimensional figures in a hierarchy based on properties.Grade 5
Arizona5.MD.B.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.Grade 5
Arizona5.NBT.A.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
Arizona5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.Grade 5
Arizona5.NBT.A.4Use place value understanding to round decimals to any place.Grade 5
Arizona5.NBT.B.5Fluently multiply multi-digit whole numbers using the standard algorithm.Grade 5
Arizona5.NBT.B.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 5
Arizona5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 5
Arizona5.NF.B.3Interpret a fraction as division of the numerator by the denominator (a/b = a / b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?Grade 5
Arizona5.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
Arizona5.NF.B.5Interpret multiplication as scaling (resizing), by comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.Grade 5
Arizona5.NF.B.6Solve real world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem.Grade 5
Arizona5.NF.B.7Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Grade 5
Arizona5.OA.A.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Grade 5
Arizona5.OA.B.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule √±Add 3√Æ and the starting number 0, and given the rule √±Add 6√Æ and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.Grade 5
Arizona6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Grade 6
Arizona6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Grade 6
Arizona6.EE.A.3Apply the properties of operations to generate equivalent expressions.Grade 6
Arizona6.EE.B.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
Arizona6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.Grade 6
Arizona6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
Arizona6.G.A.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.Grade 6
Arizona6.NS.A.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.Grade 6
Arizona6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Grade 6
Arizona6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Grade 6
Arizona6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.Grade 6
Arizona6.NS.C.6Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
Arizona6.NS.C.7Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.Grade 6
Arizona6.NS.C.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
Arizona6.NS.C.9Convert between expressions for positive rational numbers, including fractions, decimals, and percents.Grade 6
Arizona6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Grade 6
Arizona6.RP.A.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. For example, This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.Grade 6
Arizona6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams or equations.Grade 6
Arizona7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Grade 7
Arizona7.EE.B.3Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
Arizona7.G.A.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
Arizona7.G.A.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
Arizona7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Grade 7
Arizona7.NS.A.1Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.Grade 7
Arizona7.NS.A.2Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.Grade 7
Arizona7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.Grade 7
Arizona7.RP.A.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/3 hour, compute the unit rate as the complex fraction 1/2 divided by 1/4 per hour, equivalently 2 miles per hour.Grade 7
Arizona7.RP.A.2Recognize and represent proportional relationships between quantities.Grade 7
Arizona7.RP.A.3Use proportional relationships to solve multistep ratio and percent problems.Grade 7
Arizona8.EE.A.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much ones is than the other.Grade 8
Arizona8.EE.A.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.Grade 8
Arizona8.EE.B.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.Grade 8
Arizona8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.Grade 8
Arizona8.EE.C.7Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).Grade 8
Arizona8.EE.C.8Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.Grade 8
Arizona8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.Grade 8
Arizona8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.Grade 8
Arizona8.F.A.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.Grade 8
Arizona8.F.B.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Grade 8
Arizona8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.Grade 8
Arizona8.G.A.1Verify experimentally the properties of rotations, reflections, and translations:Grade 8
Arizona8.G.A.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
Arizona8.G.A.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Grade 8
Arizona8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
Arizona8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
Arizona8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
Arizona8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
ArizonaK.CC.A.1Count to 100 by ones and by tensKindergarten
ArizonaK.CC.A.2Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Kindergarten
ArizonaK.CC.A.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).Kindergarten
ArizonaK.CC.B.4Understand the relationship between numbers and quantities; connect counting to cardinality. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. Understand that each successive number name refers to a quantity that is one larger.Kindergarten
ArizonaK.CC.B.5Count to answer 'how many' questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
ArizonaK.CC.C.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.Kindergarten
ArizonaK.CC.C.7Compare two numbers between 1 and 10 presented as written numerals.Kindergarten
ArizonaK.NBT.A.1Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Kindergarten
ArizonaK.OA.A.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
ArizonaK.OA.A.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.Kindergarten
ArizonaK.OA.A.3Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).Kindergarten
ArizonaK.OA.A.4For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.Kindergarten
ArizonaK.OA.A.5Fluently add and subtract within 5.Kindergarten
ArkansasK.CC.A.1Count to 100 by ones, fives, and tens.Kindergarten
ArkansasK.CC.A.2Count forward, by ones, from any given number up to 100.Kindergarten
ArkansasK.CC.B.4Understand the relationship between numbers and quantities; connect counting to cardinality. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. Understand that each successive number name refers to a quantity that is one larger.Kindergarten
ArkansasK.CC.B.5Count to answer how many; count up to 20 objects in an arrangement, count up to 10 objects in a scattered configuration, given a number from 1-20 count out that many objects.Kindergarten
ArkansasK.CC.C.6Identify whether the number of objects in one group from 0-10 is greater than (more, most), less than (less, fewer, least), or equal to (same as) the number of objects in another group of 0-10.Kindergarten
ArkansasK.CC.C.7Compare two numbers between 0 and 20 presented as written numerals.Kindergarten
ArkansasK.CC.C.8Quickly identify a number of items in a set from 0-10 without counting (e.g., dominoes, dot cubes, tally marks, ten-frames).Kindergarten
ArkansasK.G.A.1Describe the positions of objects in the environment and geometric shapes in space using names of shapes, and describe the relative positions of these objects.Kindergarten
ArkansasK.G.A.2Correctly name shapes regardless of their orientations or overall size.Kindergarten
ArkansasK.G.A.3Identify shapes as two-dimensional or three-dimensional.Kindergarten
ArkansasK.G.B.4Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/‚Äúcorners‚Äù) and other attributes (e.g., having sides of equal length).Kindergarten
ArkansasK.G.B.5Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.Kindergarten
ArkansasK.G.B.6Compose two-dimensional shapes to form larger two-dimensional shapes.Kindergarten
ArkansasK.MD.A.1Describe several measurable attributes of a single object, including but not limited to length, weight, height, and temperature .Kindergarten
ArkansasK.MD.A.2Describe the difference when comparing two objects (side-by-side) with a measurable attribute in common, to see which object has more of or less of the common attribute.Kindergarten
ArkansasK.MD.B.3Classify, sort, and count objects using both measureable and non-measureable attributes such as size, number, color, or shape.Kindergarten
ArkansasK.MD.C.4Understand concepts of time including morning, afternoon, evening, today, yesterday, tomorrow, day, week, month and year. Understand that clocks, both analog and digital, and calendars are tools thatmeasure time.Kindergarten
ArkansasK.MD.C.6Identify pennies, nickels, and dimes, and know the vlaue of each.Kindergarten
ArkansasK.NBT.A.1Develop initial understanding of place value and the base-ten number system by showing equivalent forms of whole numbers from 11 to 19 as groups of tens and ones using objects and drawings.Kindergarten
ArkansasK.OA.A.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
ArkansasK.OA.A.2Solve real-world problems that involve addition and subtraction within 10 (e.g., by using objects or drawings to represent the problem).Kindergarten
ArkansasK.OA.A.3Use objects or drawings to decompose (break apart) numbers less than or equal to 10 into pairs in more than one way, and record each decomposition (part) by a drawing or an equation.Kindergarten
ArkansasK.OA.A.4Find the number that makes 10 when added to the given number (e.g., by using objects or drawings) and record the answer with a drawing or equation.Kindergarten
ArkansasK.OA.A.5Fluently add and subtract within 10 by using various strategies and manipulatives.Kindergarten
Arkansas1.G.A.1Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.Grade 1
Arkansas1.G.A.2Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.Grade 1
Arkansas1.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.Grade 1
Arkansas1.MD.A.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.Grade 1
Arkansas1.MD.A.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.Grade 1
Arkansas1.MD.B.3Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
Arkansas1.MD.B.4Identify and know the value of a penny, nickel, dime and quarter.Grade 1
Arkansas1.MD.B.5Count collections of like coins (pennies, nickels, and dimes).Grade 1
Arkansas1.MD.C.6Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
Arkansas1.NBT.A.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
Arkansas1.NBT.B.2Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: 10 can be thought of as a bundle of ten ones - called a 'ten.'. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).Grade 1
Arkansas1.NBT.B.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with symbols.Grade 1
Arkansas1.NBT.C.4Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
Arkansas1.NBT.C.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1
Arkansas1.NBT.C.6Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 1
Arkansas1.OA.A.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions.Grade 1
Arkansas1.OA.A.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 .Grade 1
Arkansas1.OA.B.3Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)Grade 1
Arkansas1.OA.B.4Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.Grade 1
Arkansas1.OA.C.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Grade 1
Arkansas1.OA.D.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.Grade 1
Arkansas1.OA.D.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.Grade 1
Arkansas2.G.A.1Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Grade 2
Arkansas2.G.A.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.Grade 2
Arkansas2.G.A.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.Grade 2
Arkansas2.G.A.4Recognize that equal shares of identical wholes need not have the same shape.Grade 2
Arkansas2.MD.A.1Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.Grade 2
Arkansas2.MD.A.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.Grade 2
Arkansas2.MD.A.3Estimate lengths using units of inches, feet, centimeters, and meters.Grade 2
Arkansas2.MD.A.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.Grade 2
Arkansas2.MD.B.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.Grade 2
Arkansas2.MD.B.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100 on a number line diagram.Grade 2
Arkansas2.MD.C.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Grade 2
Arkansas2.MD.C.8Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¬¢ symbols appropriately.Grade 2
Arkansas2.MD.D.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.Grade 2
Arkansas2.MD.D.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.Grade 2
Arkansas2.NBT.A.1Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: 100 can be thought of as a bundle of ten tens - called a 'hundred.'. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).Grade 2
Arkansas2.NBT.A.2Count within 1000; skip-count by 5s, 10s, and 100s.Grade 2
Arkansas2.NBT.A.3Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.Grade 2
Arkansas2.NBT.A.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using symbols to record the results of comparisons.Grade 2
Arkansas2.NBT.B.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 2
Arkansas2.NBT.B.6Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
Arkansas2.NBT.B.7Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Grade 2
Arkansas2.NBT.B.8Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.Grade 2
Arkansas2.NBT.B.9Explain why addition and subtraction strategies work, using place value and the properties of operations.Grade 2
Arkansas2.OA.A.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 2
Arkansas2.OA.B.2Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.Grade 2
Arkansas2.OA.C.3Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.Grade 2
Arkansas2.OA.C.4Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.Grade 2
Arkansas3.G.A.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Grade 3
Arkansas3.G.A.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.Grade 3
Arkansas3.MD.A.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.Grade 3
Arkansas3.MD.A.2Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.Grade 3
Arkansas3.MD.B.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step ‚Äúhow many more‚Äù and ‚Äúhow many less‚Äù problems using information presented in scaled bar graphs.Grade 3
Arkansas3.MD.B.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.Grade 3
Arkansas3.MD.C.5Recognize area as an attribute of plane figures and understand concepts of area measurement.Grade 3
Arkansas3.MD.C.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).Grade 3
Arkansas3.MD.D.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.Grade 3
Arkansas3.NBT.A.1Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
Arkansas3.NBT.A.2Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
Arkansas3.NBT.A.3Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.Grade 3
Arkansas3.NBT.A.4Understand that the four digits of a four-digit number represent amounts of thousands, hundreds, tens, and ones.Grade 3
Arkansas3.NBT.A.5Read and write numbers to 10,000 using base-ten numerals, number names, and expanded form(s).Grade 3
Arkansas3.NBT.A.6Compare two four-digit numbers based on meanings of thousands, hundreds, tens, and ones digits using symbols (<, >, =) to record the results of comparisons.Grade 3
Arkansas3.NF.A.1Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.Grade 3
Arkansas3.NF.A.2Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.Grade 3
Arkansas3.NF.A.3Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.Grade 3
Arkansas3.OA.A.1Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.Grade 3
Arkansas3.OA.A.2Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.Grade 3
Arkansas3.OA.A.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 3
Arkansas3.OA.A.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ?Grade 3
Arkansas3.OA.B.5Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)Grade 3
Arkansas3.OA.B.6Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.Grade 3
Arkansas3.OA.C.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
Arkansas3.OA.D.8Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 3
Arkansas3.OA.D.9Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.Grade 3
Arkansas4.G.A.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Grade 4
Arkansas4.G.A.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.Grade 4
Arkansas4.G.A.3Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.Grade 4
Arkansas4.MD.A.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
Arkansas4.MD.A.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
Arkansas4.MD.A.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems.Grade 4
Arkansas4.MD.B.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
Arkansas4.MD.C.5Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.Grade 4
Arkansas4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Grade 4
Arkansas4.MD.C.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Grade 4
Arkansas4.NBT.A.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 / 70 = 10 by applying concepts of place value and division.Grade 4
Arkansas4.NBT.A.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
Arkansas4.NBT.A.3Use place value understanding to round multi-digit whole numbers to any place.Grade 4
Arkansas4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm.Grade 4
Arkansas4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Arkansas4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Arkansas4.NF.A.1Explain why a fraction a/b is equivalent to a fraction (n x a) / (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
Arkansas4.NF.A.2Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.Grade 4
Arkansas4.NF.B.3Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.Grade 4
Arkansas4.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4). Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x(2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a) / b.) Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?Grade 4
Arkansas4.NF.C.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.Grade 4
Arkansas4.NF.C.6Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.Grade 4
Arkansas4.NF.C.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.Grade 4
Arkansas4.OA.A.1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
Arkansas4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Grade 4
Arkansas4.OA.A.3Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 4
Arkansas4.OA.B.4Find all factor pairs for a whole number in the range 1 - 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 ‚Äî 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 - 100 is prime or composite.Grade 4
Arkansas4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule 'Add 3' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.Grade 4
Arkansas5.G.A.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and the given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).Grade 5
Arkansas5.G.A.2Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
Arkansas5.G.B.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.Grade 5
Arkansas5.G.B.4Classify two-dimensional figures in a hierarchy based on properties.Grade 5
Arkansas5.MD.A.1Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.Grade 5
Arkansas5.MD.B.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.Grade 5
Arkansas5.MD.C.3Recognize volume as an attribute of solid figures and understand concepts of volume measurement.Grade 5
Arkansas5.MD.C.4Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.Grade 5
Arkansas5.MD.C.5Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.Grade 5
Arkansas5.NBT.A.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
Arkansas5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. Grade 5
Arkansas5.NBT.A.4Use place value understanding to round decimals to any place.Grade 5
Arkansas5.NBT.B.5Fluently multiply multi-digit whole numbers using the standard algorithm.Grade 5
Arkansas5.NBT.B.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 5
Arkansas5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 5
Arkansas5.NF.A.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.Grade 5
Arkansas5.NF.A.2Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators by using visual models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.Grade 5
Arkansas5.NF.B.3Interpret a fraction as division of the numerator by the denominator (a/b = a / b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?Grade 5
Arkansas5.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
Arkansas5.NF.B.5Interpret multiplication as scaling (resizing), by comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.Grade 5
Arkansas5.NF.B.6Solve real world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem.Grade 5
Arkansas5.NF.B.7Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Grade 5
Arkansas5.OA.A.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Grade 5
Arkansas5.OA.A.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.Grade 5
Arkansas5.OA.B.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule ‚ÄúAdd 3‚Äù and the starting number 0, and given the rule ‚ÄúAdd 6‚Äù and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.Grade 5
Arkansas6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Grade 6
Arkansas6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Grade 6
Arkansas6.EE.A.3Apply the properties of operations to generate equivalent expressions.Grade 6
Arkansas6.EE.A.4Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).Grade 6
Arkansas6.EE.B.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
Arkansas6.EE.B.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.Grade 6
Arkansas6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.Grade 6
Arkansas6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
Arkansas6.EE.C.9Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.Grade 6
Arkansas6.G.A.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Arkansas6.G.A.2Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas ùòù = ùò≠ ùò∏ ùò© and ùòù = ùò£ ùò© to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.Grade 6
Arkansas6.G.A.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.Grade 6
Arkansas6.G.A.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Arkansas6.NS.A.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.Grade 6
Arkansas6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Grade 6
Arkansas6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Grade 6
Arkansas6.NS.B.4Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1‚Äì100 with a common factor as a multiple of a sum of two whole numbers with no common factor.Grade 6
Arkansas6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.Grade 6
Arkansas6.NS.C.6Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
Arkansas6.NS.C.7Understand ordering and absolute value of rational numbers.Grade 6
Arkansas6.NS.C.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
Arkansas6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Grade 6
Arkansas6.RP.A.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.Grade 6
Arkansas6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams or equations.Grade 6
Arkansas6.SP.A.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.Grade 6
Arkansas6.SP.A.2Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.Grade 6
Arkansas6.SP.A.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.Grade 6
Arkansas6.SP.B.4Display numerical data in plots on a number line, including dot plots, histograms, and box plots.Grade 6
Arkansas6.SP.B.5Summarize numerical data sets in relation to their context.Grade 6
Arkansas7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Grade 7
Arkansas7.EE.A.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.Grade 7
Arkansas7.EE.B.3Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
Arkansas7.EE.B.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Grade 7
Arkansas7.G.A.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
Arkansas7.G.A.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
Arkansas7.G.A.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Grade 7
Arkansas7.G.B.4Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.Grade 7
Arkansas7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Grade 7
Arkansas7.G.B.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Grade 7
Arkansas7.NS.A.1Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.Grade 7
Arkansas7.NS.A.2Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.Grade 7
Arkansas7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.Grade 7
Arkansas7.RP.A.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/3 hour, compute the unit rate as the complex fraction 1/2 divided by 1/4 per hour, equivalently 2 miles per hour.Grade 7
Arkansas7.RP.A.2Recognize and represent proportional relationships between quantities.Grade 7
Arkansas7.RP.A.3Use proportional relationships to solve multistep ratio and percent problems.Grade 7
Arkansas7.SP.A.1Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.Grade 7
Arkansas7.SP.A.2Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.Grade 7
Arkansas7.SP.B.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.Grade 7
Arkansas7.SP.B.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.Grade 7
Arkansas7.SP.C.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.Grade 7
Arkansas7.SP.C.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Grade 7
Arkansas7.SP.C.7Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.Grade 7
Arkansas7.SP.C.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Grade 7
Arkansas8.EE.A.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Grade 8
Arkansas8.EE.A.2Use square root and cube root symbols to represent solutions to equations of the form ùòπ¬≤ = ùò± and ùòπ¬≥ = ùò±, where ùò± is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that ‚àö2 is irrational.Grade 8
Arkansas8.EE.A.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much ones is than the other.Grade 8
Arkansas8.EE.A.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.Grade 8
Arkansas8.EE.B.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.Grade 8
Arkansas8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.Grade 8
Arkansas8.EE.C.7Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).Grade 8
Arkansas8.EE.C.8Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.Grade 8
Arkansas8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.Grade 8
Arkansas8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.Grade 8
Arkansas8.F.A.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.Grade 8
Arkansas8.F.B.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Grade 8
Arkansas8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.Grade 8
Arkansas8.G.A.1Verify experimentally the properties of rotations, reflections, and translations.Grade 8
Arkansas8.G.A.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
Arkansas8.G.A.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Grade 8
Arkansas8.G.A.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Grade 8
Arkansas8.G.A.5Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.Grade 8
Arkansas8.G.B.6Explain a proof of the Pythagorean Theorem and its converse.Grade 8
Arkansas8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
Arkansas8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
Arkansas8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Grade 8
Arkansas8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.Grade 8
Arkansas8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., œÄ¬≤).Grade 8
Arkansas8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
Arkansas8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
Arkansas8.SP.A.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Grade 8
Arkansas8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.Grade 8
ArkansasA-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Algebra
ArkansasA-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra
ArkansasA-REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Algebra
ArkansasA-REI.C.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = ‚Äì3x and the circle x^2 + y^2 = 3.Algebra
ArkansasA-SSE.A.2Use the structure of an expression to identify ways to rewrite it.Algebra
ArkansasA-SSE.B.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Algebra
ArkansasF-BF.A.1Write a function that describes a relationship between two quantities.Algebra
ArkansasF-IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Algebra
ArkansasF-IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Algebra
ArkansasF-IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra
ArkansasS-ID.B.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.Algebra
ArkansasS-ID.C.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Algebra
CaliforniaK.CC.1Count to 100 by ones and by tens.Kindergarten
CaliforniaK.CC.2Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Kindergarten
CaliforniaK.CC.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).Kindergarten
CaliforniaK.CC.4Understand the relationship between numbers and quantities; connect counting to cardinality.Kindergarten
CaliforniaK.CC.5Count to answer ‚Äúhow many?‚Äù questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
CaliforniaK.CC.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.Kindergarten
CaliforniaK.CC.7Compare two numbers between 1 and 10 presented as written numerals.Kindergarten
CaliforniaK.G.1Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.Kindergarten
CaliforniaK.G.2Correctly name shapes regardless of their orientations or overall size.Kindergarten
CaliforniaK.G.3Identify shapes as two-dimensional (lying in a plane, ‚Äúflat‚Äù) or three-dimensional (‚Äúsolid‚Äù).Kindergarten
CaliforniaK.G.4Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/‚Äúcorners‚Äù) and other attributes (e.g., having sides of equal length).Kindergarten
CaliforniaK.G.5Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.Kindergarten
CaliforniaK.G.6Compose simple shapes to form larger shapes.Kindergarten
CaliforniaK.MD.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.Kindergarten
CaliforniaK.MD.2Directly compare two objects with a measurable attribute in common, to see which object has ‚Äúmore of‚Äù/‚Äúless of‚Äù the attribute, and describe the difference.Kindergarten
CaliforniaK.MD.3Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.Kindergarten
CaliforniaK.NBT.1Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Kindergarten
CaliforniaK.OA.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
CaliforniaK.OA.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.Kindergarten
CaliforniaK.OA.3Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).Kindergarten
CaliforniaK.OA.4For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.Kindergarten
CaliforniaK.OA.5Fluently add and subtract within 5.Kindergarten
California1.G.1Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.Grade 1
California1.G.2Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.Grade 1
California1.G.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.Grade 1
California1.MD.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.Grade 1
California1.MD.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.Grade 1
California1.MD.3Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
California1.MD.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
California1.NBT.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
California1.NBT.2Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:Grade 1
California1.NBT.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.Grade 1
California1.NBT.4Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
California1.NBT.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1
California1.NBT.6Subtract multiples of 10 in the range 10‚Äì90 from multiples of 10 in the range 10‚Äì90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 1
California1.OA.1Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Grade 1
California1.OA.2Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Grade 1
California1.OA.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Grade 1
California1.OA.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.Grade 1
California1.OA.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.Grade 1
California2.G.1Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Grade 2
California2.G.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.Grade 2
California2.G.3Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.Grade 2
California2.MD.1Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.Grade 2
California2.MD.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.Grade 2
California2.MD.3Estimate lengths using units of inches, feet, centimeters, and meters.Grade 2
California2.MD.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.Grade 2
California2.MD.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.Grade 2
California2.MD.6Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, . . . , and represent whole-number sums and differences within 100 on a number line diagram.Grade 2
California2.MD.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Know relationships of time (e.g., minutes in an hour, days in a month, weeks in a year).Grade 2
California2.MD.8Solve word problems involving combinations of dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¬¢ symbols appropriately.Grade 2
California2.MD.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.Grade 2
California2.MD.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.Grade 2
California2.NBT.1Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:Grade 2
California2.NBT.2Count within 1000; skip-count by 2s, 5s, 10s, and 100s.Grade 2
California2.NBT.3Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.Grade 2
California2.NBT.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.Grade 2
California2.NBT.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 2
California2.NBT.6Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
California2.NBT.7Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Grade 2
California2.NBT.7.1Use estimation strategies to make reasonable estimates in problem solving.Grade 2
California2.NBT.8Mentally add 10 or 100 to a given number 100‚Äì900, and mentally subtract 10 or 100 from a given number 100‚Äì900.Grade 2
California2.NBT.9Explain why addition and subtraction strategies work, using place value and the properties of operations.Grade 2
California2.OA.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 2
California2.OA.2Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.Grade 2
California2.OA.3Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.Grade 2
California2.OA.4Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.Grade 2
California3.G.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Grade 3
California3.G.2Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.Grade 3
California3.MD.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.Grade 3
California3.MD.2Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.Grade 3
California3.MD.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step ‚Äúhow many more‚Äù and ‚Äúhow many less‚Äù problems using information presented in scaled bar graphs.Grade 3
California3.MD.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.Grade 3
California3.MD.5Recognize area as an attribute of plane figures and understand concepts of area measurement.Grade 3
California3.MD.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).Grade 3
California3.MD.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.Grade 3
California3.NBT.1Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
California3.NBT.2Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
California3.NBT.3Multiply one-digit whole numbers by multiples of 10 in the range 10‚Äì90 (e.g., 9 √ó 80, 5 √ó 60) using strategies based on place value and properties of operations.Grade 3
California3.NF.1Understand a fraction 1/ùò£ as the quantity formed by 1 part when a whole is partitioned into ùò£ equal parts; understand a fraction ùò¢/ùëè as the quantity formed by ùò¢ parts of size 1/ùò£.Grade 3
California3.NF.2Understand a fraction as a number on the number line; represent fractions on a number line diagram.Grade 3
California3.NF.3Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.Grade 3
California3.OA.1Interpret products of whole numbers, e.g., interpret 5 √ó 7 as the total number of objects in 5 groups of 7 objects each.Grade 3
California3.OA.2Interpret whole-number quotients of whole numbers, e.g., interpret 56 √∑ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.Grade 3
California3.OA.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 3
California3.OA.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers.Grade 3
California3.OA.5Apply properties of operations as strategies to multiply and divide.Grade 3
California3.OA.6Understand division as an unknown-factor problem.Grade 3
California3.OA.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 √ó 5 = 40, one knows 40 √∑ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
California3.OA.8Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 3
California3.OA.9Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.Grade 3
California4.G.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Grade 4
California4.G.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. (Two-dimensional shapes should include special triangles, e.g., equilateral, isosceles, scalene, and special quadrilaterals, e.g., rhombus, square, rectangle, parallelogram, trapezoid.)Grade 4
California4.G.3Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.Grade 4
California4.MD.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
California4.MD.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
California4.MD.3Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.Grade 4
California4.MD.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
California4.MD.5Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:Grade 4
California4.MD.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Grade 4
California4.MD.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Grade 4
California4.NBT.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.Grade 4
California4.NBT.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
California4.NBT.3Use place value understanding to round multi-digit whole numbers to any place.Grade 4
California4.NBT.4Fluently add and subtract multi-digit whole numbers using the standard algorithm.Grade 4
California4.NBT.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
California4.NBT.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
California4.NF.1Explain why a fraction ùò¢/ùò£ is equivalent to a fraction (ùòØ √ó ùò¢)/(ùòØ √ó ùò£) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
California4.NF.2Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.Grade 4
California4.NF.3Understand a fraction ùò¢/ùò£ with ùò¢ > 1 as a sum of fractions 1/ùò£.Grade 4
California4.NF.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.Grade 4
California4.NF.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.Grade 4
California4.NF.6Use decimal notation for fractions with denominators 10 or 100.Grade 4
California4.NF.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using the number line or another visual model.Grade 4
California4.OA.1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 √ó 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
California4.OA.2Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Grade 4
California4.OA.3Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 4
California4.OA.4Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.Grade 4
California4.OA.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.Grade 4
California5.G.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., ùòπ-axis and ùòπ-coordinate, ùò∫-axis and ùò∫-coordinate).Grade 5
California5.G.2Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
California5.G.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.Grade 5
California5.G.4Classify two-dimensional figures in a hierarchy based on properties.Grade 5
California5.MD.1Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems.Grade 5
California5.MD.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.Grade 5
California5.MD.3Recognize volume as an attribute of solid figures and understand concepts of volume measurement.Grade 5
California5.MD.4Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.Grade 5
California5.MD.5Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.Grade 5
California5.NBT.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
California5.NBT.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.Grade 5
California5.NBT.4Use place value understanding to round decimals to any place.Grade 5
California5.NBT.5Fluently multiply multi-digit whole numbers using the standard algorithm.Grade 5
California5.NBT.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 5
California5.NBT.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 5
California5.NF.1Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.Grade 5
California5.NF.2Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.Grade 5
California5.NF.3Interpret a fraction as division of the numerator by the denominator (ùò¢/ùò£ = ùò¢ √∑ ùò£). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.Grade 5
California5.NF.4Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
California5.NF.5Interpret multiplication as scaling (resizing), by:Grade 5
California5.NF.6Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.Grade 5
California5.NF.7Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Grade 5
California5.OA.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Grade 5
California5.OA.2Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.Grade 5
California5.OA.2.1Express a whole number in the range 2-50 as a product of its prime factors.Grade 5
California5.OA.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.Grade 5
California6.EE.1Write and evaluate numerical expressions involving whole-number exponents.Grade 6
California6.EE.2Write, read, and evaluate expressions in which letters stand for numbers.Grade 6
California6.EE.3Apply the properties of operations to generate equivalent expressions.Grade 6
California6.EE.4Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).Grade 6
California6.EE.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
California6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.Grade 6
California6.EE.7Solve real-world and mathematical problems by writing and solving equations of the form ùòπ + ùò± = ùò≤ and ùò±ùòπ = ùò≤ for cases in which ùò±, ùò≤ and ùòπ are all nonnegative rational numbers.Grade 6
California6.EE.8Write an inequality of the form ùòπ > ùò§ or ùòπ < ùò§ to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form ùòπ > ùò§ or ùòπ < ùò§ have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
California6.EE.9Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.Grade 6
California6.G.1Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.Grade 6
California6.G.2Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas ùòù = ùò≠ ùò∏ ùò© and ùòù = ùò£ ùò© to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.Grade 6
California6.G.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
California6.G.4Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
California6.RP.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Grade 6
California6.RP.2Understand the concept of a unit rate ùò¢/ùò£ associated with a ratio ùò¢:ùò£ with ùò£ ‚â† 0, and use rate language in the context of a ratio relationship.Grade 6
California6.RP.3Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Grade 6
California6.SP.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.Grade 6
California6.SP.2Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.Grade 6
California6.SP.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.Grade 6
California6.SP.4Display numerical data in plots on a number line, including dot plots, histograms, and box plots.Grade 6
California6.SP.5Summarize numerical data sets in relation to their context, such as by:Grade 6
California6.NS.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.Grade 6
California6.NS.2Fluently divide multi-digit numbers using the standard algorithm.Grade 6
California6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Grade 6
California6.NS.4Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1‚Äì100 with a common factor as a multiple of a sum of two whole numbers with no common factor.Grade 6
California6.NS.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.Grade 6
California6.NS.6Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
California6.NS.7Understand ordering and absolute value of rational numbers.Grade 6
California6.NS.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
California7.EE.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Grade 7
California7.EE.2Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.Grade 7
California7.EE.3Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
California7.EE.4Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Grade 7
California7.G.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
California7.G.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
California7.G.3Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.Grade 7
California7.G.4Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.Grade 7
California7.G.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Grade 7
California7.G.6Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Grade 7
California7.RP.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.Grade 7
California7.RP.2Recognize and represent proportional relationships between quantities.Grade 7
California7.RP.3Use proportional relationships to solve multistep ratio and percent problems.Grade 7
California7.SP.1Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.Grade 7
California7.SP.2Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.Grade 7
California7.SP.3Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.Grade 7
California7.SP.4Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.Grade 7
California7.SP.5Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.Grade 7
California7.SP.6Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Grade 7
California7.SP.7Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.Grade 7
California7.SP.8Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Grade 7
California7.NS.1Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.Grade 7
California7.NS.2Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Grade 7
California7.NS.3Solve real-world and mathematical problems involving the four operations with rational numbers.Grade 7
California8.EE.1Know and apply the properties of integer exponents to generate equivalent numerical expressions.Grade 8
California8.EE.2Use square root and cube root symbols to represent solutions to equations of the form ùòπ¬≤ = ùò± and ùòπ¬≥ = ùò±, where ùò± is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that ‚àö2 is irrational.Grade 8
California8.EE.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.Grade 8
California8.EE.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.Grade 8
California8.EE.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.Grade 8
California8.EE.6Use similar triangles to explain why the slope ùòÆ is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation ùò∫ = ùòÆùòπ for a line through the origin and the equation ùò∫ = ùòÆùòπ + ùò£ for a line intercepting the vertical axis at ùò£.Grade 8
California8.EE.7Solve linear equations in one variable.Grade 8
California8.EE.8Analyze and solve pairs of simultaneous linear equations.Grade 8
California8.F.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.Grade 8
California8.F.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Grade 8
California8.F.3Interpret the equation ùò∫ = ùòÆùòπ + ùò£ as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.Grade 8
California8.F.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (ùòπ, ùò∫) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Grade 8
California8.F.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.Grade 8
California8.G.1Verify experimentally the properties of rotations, reflections, and translations:Grade 8
California8.G.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
California8.G.3Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Grade 8
California8.G.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Grade 8
California8.G.5Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.Grade 8
California8.G.6Explain a proof of the Pythagorean Theorem and its converse.Grade 8
California8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
California8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
California8.G.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Grade 8
California8.SP.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
California8.SP.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
California8.SP.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Grade 8
California8.SP.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.Grade 8
California8.NS.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.Grade 8
California8.NS.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., œÄ¬≤).Grade 8
CaliforniaA-SSE.1Interpret expressions that represent a quantity in terms of its context.Algebra I
CaliforniaA-SSE.2Use the structure of an expression to identify ways to rewrite it.Algebra I
CaliforniaA-SSE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Algebra I
CaliforniaA-APR.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Algebra I
CaliforniaA-CED.1Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.Algebra I
CaliforniaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra I
CaliforniaA-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.Algebra I
CaliforniaA-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Algebra I
CaliforniaA-REI.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.Algebra I
CaliforniaA-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Algebra I
CaliforniaA-REI.3.1Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.Algebra I
CaliforniaA-REI.4Solve quadratic equations in one variable.Algebra I
CaliforniaA-REI.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.Algebra I
CaliforniaA-REI.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Algebra I
CaliforniaA-REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.Algebra I
CaliforniaA-REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).Algebra I
CaliforniaA-REI.11Explain why the ùòπ-coordinates of the points where the graphs of the equations ùò∫ = ùòß(ùòπ) and ùò∫ = ùëî(ùòπ) intersect are the solutions of the equation ùòß(ùòπ) = ùëî(ùòπ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ùòß(ùòπ) and/or ùëî(ùòπ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Algebra I
CaliforniaA-REI.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.Algebra I
CaliforniaF-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ùòß is a function and ùòπ is an element of its domain, then ùòß(ùòπ) denotes the output of ùòß corresponding to the input ùòπ. The graph of ùòß is the graph of the equation ùò∫ = ùòß(ùòπ).Algebra I
CaliforniaF-IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Algebra I
CaliforniaF-IF.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.Algebra I
CaliforniaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Algebra I
CaliforniaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Algebra I
CaliforniaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Algebra I
CaliforniaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra I
CaliforniaF-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Algebra I
CaliforniaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Algebra I
CaliforniaF-BF.1Write a function that describes a relationship between two quantities.Algebra I
CaliforniaF-BF.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.Algebra I
CaliforniaF-BF.3Identify the effect on the graph of replacing ùòß(ùòπ) by ùòß(ùòπ) + ùò¨, ùò¨ ùòß(ùòπ), ùòß(ùò¨ùòπ), and ùòß(ùòπ + ùò¨) for specific values of ùò¨ (both positive and negative); find the value of ùò¨ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Algebra I
CaliforniaF-BF.4Find inverse functions.Algebra I
CaliforniaF-LE.1Distinguish between situations that can be modeled with linear functions and with exponential functions.Algebra I
CaliforniaF-LE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).Algebra I
CaliforniaF-LE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.Algebra I
CaliforniaF-LE.5Interpret the parameters in a linear or exponential function in terms of a context.Algebra I
CaliforniaF-LE.6Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.Algebra I
CaliforniaN-RN.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.Algebra I
CaliforniaN-RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.Algebra I
CaliforniaN-RN.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.Algebra I
CaliforniaN-Q.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.Algebra I
CaliforniaN-Q.2Define appropriate quantities for the purpose of descriptive modeling.Algebra I
CaliforniaN-Q.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.Algebra I
CaliforniaS-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).Algebra I
CaliforniaS-ID.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.Algebra I
CaliforniaS-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).Algebra I
CaliforniaS-ID.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.Algebra I
CaliforniaS-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.Algebra I
CaliforniaS-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Algebra I
CaliforniaS-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.Algebra I
CaliforniaS-ID.9Distinguish between correlation and causation.Algebra I
CaliforniaG-CO.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Geometry
CaliforniaG-CO.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).Geometry
CaliforniaG-CO.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.Geometry
CaliforniaG-CO.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.Geometry
CaliforniaG-CO.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Geometry
CaliforniaG-CO.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Geometry
CaliforniaG-CO.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Geometry
CaliforniaG-CO.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.Geometry
CaliforniaG-CO.9Prove theorems about lines and angles.Geometry
CaliforniaG-CO.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Geometry
CaliforniaG-CO.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.Geometry
CaliforniaG-SRT.1Verify experimentally the properties of dilations given by a center and a scale factor:Geometry
CaliforniaG-SRT.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.Geometry
CaliforniaG-SRT.3Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.Geometry
CaliforniaG-SRT.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.Geometry
CaliforniaG-SRT.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.Geometry
CaliforniaG-SRT.7Explain and use the relationship between the sine and cosine of complementary angles.Geometry
CaliforniaG-SRT.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Geometry
CaliforniaG-SRT.8.1Derive and use the trigonometric ratios for special right triangles (30¬∞, 60¬∞, 90¬∞ and 45¬∞, 45¬∞, 90¬∞).Geometry
CaliforniaG-SRT.9Derive the formula ùê¥ = 1/2 ùò¢ùò£ sin(ùê∂) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.Geometry
CaliforniaG-SRT.10Prove the Laws of Sines and Cosines and use them to solve problems.Geometry
CaliforniaG-SRT.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).Geometry
CaliforniaG-C.1Prove that all circles are similar.Geometry
CaliforniaG-C.2Identify and describe relationships among inscribed angles, radii, and chords.Geometry
CaliforniaG-C.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.Geometry
CaliforniaG-C.4Construct a tangent line from a point outside a given circle to the circle.Geometry
CaliforniaG-C.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Convert between degrees and radians.Geometry
CaliforniaG-GPE.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.Geometry
CaliforniaG-GPE.2Derive the equation of a parabola given a focus and directrix.Geometry
CaliforniaG-GPE.4Use coordinates to prove simple geometric theorems algebraically.Geometry
CaliforniaG-GPE.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).Geometry
CaliforniaG-GPE.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.Geometry
CaliforniaG-GPE.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.Geometry
CaliforniaG-GMD.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.Geometry
CaliforniaG-GMD.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.Geometry
CaliforniaG-GMD.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.Geometry
CaliforniaG-GMD.5Know that the effect of a scale factor ùëò greater than zero on length, area, and volume is to multiply each by ùëò, ùëò¬≤, and ùëò¬≥, respectively; determine length, area and volume measures using scale factors.Geometry
CaliforniaG-GMD.6Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.Geometry
CaliforniaG-MG.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).Geometry
CaliforniaG-MG.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).Geometry
CaliforniaG-MG.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).Geometry
CaliforniaS-CP.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (‚Äúor,‚Äù ‚Äúand,‚Äù ‚Äúnot‚Äù).Geometry
CaliforniaS-CP.2Understand that two events ùòà and ùòâ are independent if the probability of ùòà and ùòâ occurring together is the product of their probabilities, and use this characterization to determine if they are independent.Geometry
CaliforniaS-CP.3Understand the conditional probability of ùòà given ùòâ as ùòó(ùòà and ùòâ)/ùòó(ùòâ), and interpret independence of ùòà and ùòâ as saying that the conditional probability of ùòà given ùòâ is the same as the probability of ùòà, and the conditional probability of ùòâ given ùòà is the same as the probability of ùòâ.Geometry
CaliforniaS-CP.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.Geometry
CaliforniaS-CP.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.Geometry
CaliforniaS-CP.6Find the conditional probability of ùòà given ùòâ as the fraction of ùòâ‚Äôs outcomes that also belong to ùòà, and interpret the answer in terms of the model.Geometry
CaliforniaS-CP.7Apply the Addition Rule, ùòó(ùòà or ùòâ) = ùòó(ùòà) + ùòó(ùòâ) ‚Äì ùòó(ùòà and ùòâ), and interpret the answer in terms of the model.Geometry
CaliforniaS-CP.8Apply the general Multiplication Rule in a uniform probability model, ùòó(ùòà and ùòâ) = ùòó(ùòà)ùòó(ùòâ|ùòà) = ùòó(ùòâ)ùòó(ùòà|ùòâ), and interpret the answer in terms of the model.Geometry
CaliforniaS-CP.9Use permutations and combinations to compute probabilities of compound events and solve problems.Geometry
CaliforniaS-MD.6Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Geometry
CaliforniaS-MD.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Geometry
CaliforniaA-SSE.1Interpret expressions that represent a quantity in terms of its context.Algebra II
CaliforniaA-SSE.2Use the structure of an expression to identify ways to rewrite it.Algebra II
CaliforniaA-SSE.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.Algebra II
CaliforniaA-APR.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Algebra II
CaliforniaA-APR.2Know and apply the Remainder Theorem: For a polynomial ùò±(ùòπ) and a number ùò¢, the remainder on division by ùòπ ‚Äì ùò¢ is ùò±(ùò¢), so ùò±(ùò¢) = 0 if and only if (ùòπ ‚Äì ùò¢) is a factor of ùò±(ùòπ).Algebra II
CaliforniaA-APR.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Algebra II
CaliforniaA-APR.4Prove polynomial identities and use them to describe numerical relationships.Algebra II
CaliforniaA-APR.5Know and apply the Binomial Theorem for the expansion of (ùòπ + ùò∫)‚Åø in powers of ùòπ and y for a positive integer ùòØ, where ùòπ and ùò∫ are any numbers, with coefficients determined for example by Pascal‚Äôs Triangle.Algebra II
CaliforniaA-APR.6Rewrite simple rational expressions in different forms; write ùò¢(ùòπ)/ùò£(ùòπ) in the form ùò≤(ùòπ) + ùò≥(ùòπ)/ùò£(ùòπ), where ùò¢(ùòπ), ùò£(ùòπ), ùò≤(ùòπ), and ùò≥(ùòπ) are polynomials with the degree of ùò≥(ùòπ) less than the degree of ùò£(ùòπ), using inspection, long division, or, for the more complicated examples, a computer algebra system.Algebra II
CaliforniaA-APR.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.Algebra II
CaliforniaA-CED.1Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.Algebra II
CaliforniaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra II
CaliforniaA-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.Algebra II
CaliforniaA-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Algebra II
CaliforniaA-REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.Algebra II
CaliforniaA-REI.3.1Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.Algebra II
CaliforniaA-REI.11Explain why the ùòπ-coordinates of the points where the graphs of the equations ùò∫ = ùòß(ùòπ) and ùò∫ = ùëî(ùòπ) intersect are the solutions of the equation ùòß(ùòπ) = ùëî(ùòπ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ùòß(ùòπ) and/or ùëî(ùòπ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Algebra II
CaliforniaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Algebra II
CaliforniaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Algebra II
CaliforniaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Algebra II
CaliforniaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra II
CaliforniaF-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Algebra II
CaliforniaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Algebra II
CaliforniaF-BF.1Write a function that describes a relationship between two quantities.Algebra II
CaliforniaF-BF.3Identify the effect on the graph of replacing ùòß(ùòπ) by ùòß(ùòπ) + ùò¨, ùò¨ ùòß(ùòπ), ùòß(ùò¨ùòπ), and ùòß(ùòπ + ùò¨) for specific values of ùò¨ (both positive and negative); find the value of ùò¨ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Algebra II
CaliforniaF-BF.4Find inverse functions.Algebra II
CaliforniaF-LE.4For exponential models, express as a logarithm the solution to ùò¢ùò£ to the ùò§ùòµ power = ùò• where ùò¢, ùò§, and ùò• are numbers and the base ùò£ is 2, 10, or ùò¶; evaluate the logarithm using technology.Algebra II
CaliforniaF-LE.4.1Prove simple laws of logarithms.Algebra II
CaliforniaF-LE.4.2Use the definition of logarithms to translate between logarithms in any base.Algebra II
CaliforniaF-LE.4.3Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.Algebra II
CaliforniaF-TF.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.Algebra II
CaliforniaF-TF.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.Algebra II
CaliforniaF-TF.2.1Graph all 6 basic trigonometric functions.Algebra II
CaliforniaF-TF.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.Algebra II
CaliforniaF-TF.8Prove the Pythagorean identity sin¬≤(Œ∏) + cos¬≤(Œ∏) = 1 and use it to find sin(Œ∏), cos(Œ∏), or tan(Œ∏) given sin(Œ∏), cos(Œ∏), or tan(Œ∏) and the quadrant of the angle.Algebra II
CaliforniaG-GPE.3.1Given a quadratic equation of the form ax¬≤ + by2 + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola, and graph the equation.Algebra II
CaliforniaN-CN.1Know there is a complex number ùò™ such that ùò™¬≤ = ‚Äì1, and every complex number has the form ùò¢ + ùò£ùò™ with ùò¢ and ùò£ real.Algebra II
CaliforniaN-CN.2Use the relation ùò™¬≤ = ‚Äì1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.Algebra II
CaliforniaN-CN.7Solve quadratic equations with real coefficients that have complex solutions.Algebra II
CaliforniaN-CN.8Extend polynomial identities to the complex numbers.Algebra II
CaliforniaN-CN.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.Algebra II
CaliforniaS-ID.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.Algebra II
CaliforniaS-IC.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.Algebra II
CaliforniaS-IC.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.Algebra II
CaliforniaS-IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.Algebra II
CaliforniaS-IC.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.Algebra II
CaliforniaS-IC.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.Algebra II
CaliforniaS-IC.6Evaluate reports based on data.Algebra II
CaliforniaS-MD.6Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Algebra II
CaliforniaS-MD.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Algebra II
CaliforniaA-SSE.1Interpret expressions that represent a quantity in terms of its context.Mathematics I
CaliforniaA-CED.1Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.Mathematics I
CaliforniaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Mathematics I
CaliforniaA-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.Mathematics I
CaliforniaA-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Mathematics I
CaliforniaA-REI.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.Mathematics I
CaliforniaA-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.Mathematics I
CaliforniaA-REI.3.1Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.Mathematics I
CaliforniaA-REI.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.Mathematics I
CaliforniaA-REI.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.Mathematics I
CaliforniaA-REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).Mathematics I
CaliforniaA-REI.11Explain why the ùòπ-coordinates of the points where the graphs of the equations ùò∫ = ùòß(ùòπ) and ùò∫ = ùëî(ùòπ) intersect are the solutions of the equation ùòß(ùòπ) = ùëî(ùòπ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ùòß(ùòπ) and/or ùëî(ùòπ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Mathematics I
CaliforniaA-REI.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.Mathematics I
CaliforniaF-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ùòß is a function and ùòπ is an element of its domain, then ùòß(ùòπ) denotes the output of ùòß corresponding to the input ùòπ. The graph of ùòß is the graph of the equation ùò∫ = ùòß(ùòπ).Mathematics I
CaliforniaF-IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Mathematics I
CaliforniaF-IF.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.Mathematics I
CaliforniaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Mathematics I
CaliforniaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Mathematics I
CaliforniaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Mathematics I
CaliforniaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Mathematics I
CaliforniaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Mathematics I
CaliforniaF-BF.1Write a function that describes a relationship between two quantities.Mathematics I
CaliforniaF-BF.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.Mathematics I
CaliforniaF-BF.3Identify the effect on the graph of replacing ùòß(ùòπ) by ùòß(ùòπ) + ùò¨, ùò¨ ùòß(ùòπ), ùòß(ùò¨ùòπ), and ùòß(ùòπ + ùò¨) for specific values of ùò¨ (both positive and negative); find the value of ùò¨ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Mathematics I
CaliforniaF-LE.1Distinguish between situations that can be modeled with linear functions and with exponential functions.Mathematics I
CaliforniaF-LE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).Mathematics I
CaliforniaF-LE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.Mathematics I
CaliforniaF-LE.5Interpret the parameters in a linear or exponential function in terms of a context.Mathematics I
CaliforniaG-CO.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.Mathematics I
CaliforniaG-CO.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).Mathematics I
CaliforniaG-CO.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.Mathematics I
CaliforniaG-CO.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.Mathematics I
CaliforniaG-CO.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.Mathematics I
CaliforniaG-CO.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.Mathematics I
CaliforniaG-CO.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.Mathematics I
CaliforniaG-CO.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.Mathematics I
CaliforniaG-CO.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).Mathematics I
CaliforniaG-CO.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.Mathematics I
CaliforniaG-GPE.4Use coordinates to prove simple geometric theorems algebraically.Mathematics I
CaliforniaG-GPE.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).Mathematics I
CaliforniaG-GPE.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.Mathematics I
CaliforniaN-Q.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.Mathematics I
CaliforniaN-Q.2Define appropriate quantities for the purpose of descriptive modeling.Mathematics I
CaliforniaN-Q.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.Mathematics I
CaliforniaS-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).Mathematics I
CaliforniaS-ID.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.Mathematics I
CaliforniaS-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).Mathematics I
CaliforniaS-ID.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.Mathematics I
CaliforniaS-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.Mathematics I
CaliforniaS-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.Mathematics I
CaliforniaS-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.Mathematics I
CaliforniaS-ID.9Distinguish between correlation and causation.Mathematics I
CaliforniaA-SSE.1Interpret expressions that represent a quantity in terms of its context.Mathematics II
CaliforniaA-SSE.2Use the structure of an expression to identify ways to rewrite it.Mathematics II
CaliforniaA-SSE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Mathematics II
CaliforniaA-APR.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Mathematics II
CaliforniaA-CED.1Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.Mathematics II
CaliforniaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Mathematics II
CaliforniaA-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Mathematics II
CaliforniaA-REI.4Solve quadratic equations in one variable.Mathematics II
CaliforniaA-REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.Mathematics II
CaliforniaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Mathematics II
CaliforniaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Mathematics II
CaliforniaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Mathematics II
CaliforniaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Mathematics II
CaliforniaF-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Mathematics II
CaliforniaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Mathematics II
CaliforniaF-BF.1Write a function that describes a relationship between two quantities.Mathematics II
CaliforniaF-BF.3Identify the effect on the graph of replacing ùòß(ùòπ) by ùòß(ùòπ) + ùò¨, ùò¨ ùòß(ùòπ), ùòß(ùò¨ùòπ), and ùòß(ùòπ + ùò¨) for specific values of ùò¨ (both positive and negative); find the value of ùò¨ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Mathematics II
CaliforniaF-BF.4Find inverse functions.Mathematics II
CaliforniaF-LE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.Mathematics II
CaliforniaF-LE.6Apply quadratic functions to physical problems, such as the motion of an object under the force of gravity.Mathematics II
CaliforniaF-TF.8Prove the Pythagorean identity sin¬≤(Œ∏) + cos¬≤(Œ∏) = 1 and use it to find sin(Œ∏), cos(Œ∏), or tan(Œ∏) given sin(Œ∏), cos(Œ∏), or tan(Œ∏) and the quadrant of the angle.Mathematics II
CaliforniaG-CO.9Prove theorems about lines and angles.Mathematics II
CaliforniaG-SRT.1Verify experimentally the properties of dilations given by a center and a scale factor:Mathematics II
CaliforniaG-SRT.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.Mathematics II
CaliforniaG-SRT.3Use the properties of similarity transformations to establish the Angle-Angle (AA) criterion for two triangles to be similar.Mathematics II
CaliforniaG-SRT.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.Mathematics II
CaliforniaG-SRT.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.Mathematics II
CaliforniaG-SRT.7Explain and use the relationship between the sine and cosine of complementary angles.Mathematics II
CaliforniaG-SRT.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.Mathematics II
CaliforniaG-SRT.8.1Derive and use the trigonometric ratios for special right triangles (30¬∞, 60¬∞, 90¬∞ and 45¬∞, 45¬∞, 90¬∞).Mathematics II
CaliforniaG-C.1Prove that all circles are similar.Mathematics II
CaliforniaG-C.2Identify and describe relationships among inscribed angles, radii, and chords.Mathematics II
CaliforniaG-C.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.Mathematics II
CaliforniaG-C.4Construct a tangent line from a point outside a given circle to the circle.Mathematics II
CaliforniaG-C.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Convert between degrees and radians.Mathematics II
CaliforniaG-GPE.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.Mathematics II
CaliforniaG-GPE.2Derive the equation of a parabola given a focus and directrix.Mathematics II
CaliforniaG-GPE.4Use coordinates to prove simple geometric theorems algebraically.Mathematics II
CaliforniaG-GPE.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.Mathematics II
CaliforniaG-GMD.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.Mathematics II
CaliforniaG-GMD.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.Mathematics II
CaliforniaG-GMD.5Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k¬≤, and k¬≥, respectively; determine length, area and volume measures using scale factors.Mathematics II
CaliforniaG-GMD.6Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.Mathematics II
CaliforniaN-RN.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.Mathematics II
CaliforniaN-RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.Mathematics II
CaliforniaN-RN.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.Mathematics II
CaliforniaN-CN.1Know there is a complex number ùò™ such that ùò™¬≤ = ‚Äì1, and every complex number has the form ùò¢ + ùò£ùò™ with ùò¢ and ùò£ real.Mathematics II
CaliforniaN-CN.2Use the relation ùò™¬≤ = ‚Äì1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.Mathematics II
CaliforniaN-CN.7Solve quadratic equations with real coefficients that have complex solutions.Mathematics II
CaliforniaN-CN.8Extend polynomial identities to the complex numbers.Mathematics II
CaliforniaN-CN.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.Mathematics II
CaliforniaS-CP.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (‚Äúor,‚Äù ‚Äúand,‚Äù ‚Äúnot‚Äù).Mathematics II
CaliforniaS-CP.2Understand that two events ùòà and ùòâ are independent if the probability of ùòà and ùòâ occurring together is the product of their probabilities, and use this characterization to determine if they are independent.Mathematics II
CaliforniaS-CP.3Understand the conditional probability of ùòà given ùòâ as ùòó(ùòà and ùòâ)/ùòó(ùòâ), and interpret independence of ùòà and ùòâ as saying that the conditional probability of ùòà given ùòâ is the same as the probability of ùòà, and the conditional probability of ùòâ given ùòà is the same as the probability of ùòâ.Mathematics II
CaliforniaS-CP.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.Mathematics II
CaliforniaS-CP.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.Mathematics II
CaliforniaS-CP.6Find the conditional probability of ùòà given ùòâ as the fraction of ùòâ‚Äôs outcomes that also belong to ùòà, and interpret the answer in terms of the model.Mathematics II
CaliforniaS-CP.7Apply the Addition Rule, ùòó(ùòà or ùòâ) = ùòó(ùòà) + ùòó(ùòâ) ‚Äì ùòó(ùòà and ùòâ), and interpret the answer in terms of the model.Mathematics II
CaliforniaS-CP.8Apply the general Multiplication Rule in a uniform probability model, ùòó(ùòà and ùòâ) = ùòó(ùòà)ùòó(ùòâ|ùòà) = ùòó(ùòâ)ùòó(ùòà|ùòâ), and interpret the answer in terms of the model.Mathematics II
CaliforniaS-CP.9Use permutations and combinations to compute probabilities of compound events and solve problems.Mathematics II
CaliforniaS-MD.6Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Mathematics II
CaliforniaS-MD.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Mathematics II
CaliforniaA-SSE.1Interpret expressions that represent a quantity in terms of its context.Mathematics III
CaliforniaA-SSE.2Use the structure of an expression to identify ways to rewrite it.Mathematics III
CaliforniaA-SSE.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.Mathematics III
CaliforniaA-APR.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.Mathematics III
CaliforniaA-APR.2Know and apply the Remainder Theorem: For a polynomial ùò±(ùòπ) and a number ùò¢, the remainder on division by ùòπ ‚Äì ùò¢ is ùò±(ùò¢), so ùò±(ùò¢) = 0 if and only if (ùòπ ‚Äì ùò¢) is a factor of ùò±(ùòπ).Mathematics III
CaliforniaA-APR.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Mathematics III
CaliforniaA-APR.4Prove polynomial identities and use them to describe numerical relationships.Mathematics III
CaliforniaA-APR.5Know and apply the Binomial Theorem for the expansion of (ùòπ + ùò∫)‚Åø in powers of ùòπ and y for a positive integer ùòØ, where ùòπ and ùò∫ are any numbers, with coefficients determined for example by Pascal‚Äôs Triangle.Mathematics III
CaliforniaA-APR.6Rewrite simple rational expressions in different forms; write ùò¢(ùòπ)/ùò£(ùòπ) in the form ùò≤(ùòπ) + ùò≥(ùòπ)/ùò£(ùòπ), where ùò¢(ùòπ), ùò£(ùòπ), ùò≤(ùòπ), and ùò≥(ùòπ) are polynomials with the degree of ùò≥(ùòπ) less than the degree of ùò£(ùòπ), using inspection, long division, or, for the more complicated examples, a computer algebra system.Mathematics III
CaliforniaA-APR.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.Mathematics III
CaliforniaA-CED.1Create equations and inequalities in one variable including ones with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.Mathematics III
CaliforniaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Mathematics III
CaliforniaA-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.Mathematics III
CaliforniaA-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Mathematics III
CaliforniaA-REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.Mathematics III
CaliforniaA-REI.11Explain why the ùòπ-coordinates of the points where the graphs of the equations ùò∫ = ùòß(ùòπ) and ùò∫ = ùëî(ùòπ) intersect are the solutions of the equation ùòß(ùòπ) = ùëî(ùòπ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ùòß(ùòπ) and/or ùëî(ùòπ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.Mathematics III
CaliforniaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Mathematics III
CaliforniaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.Mathematics III
CaliforniaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.Mathematics III
CaliforniaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Mathematics III
CaliforniaF-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Mathematics III
CaliforniaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Mathematics III
CaliforniaF-BF.1Write a function that describes a relationship between two quantities.Mathematics III
CaliforniaF-BF.3Identify the effect on the graph of replacing ùòß(ùòπ) by ùòß(ùòπ) + ùò¨, ùò¨ ùòß(ùòπ), ùòß(ùò¨ùòπ), and ùòß(ùòπ + ùò¨) for specific values of ùò¨ (both positive and negative); find the value of ùò¨ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.Mathematics III
CaliforniaF-BF.4Find inverse functions.Mathematics III
CaliforniaF-LE.4For exponential models, express as a logarithm the solution to ùò¢ùò£ to the ùò§ùòµ power = ùò• where ùò¢, ùò§, and ùò• are numbers and the base ùò£ is 2, 10, or ùò¶; evaluate the logarithm using technology.Mathematics III
CaliforniaF-LE.4.1Prove simple laws of logarithms.Mathematics III
CaliforniaF-LE.4.2Use the definition of logarithms to translate between logarithms in any base.Mathematics III
CaliforniaF-LE.4.3Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.Mathematics III
CaliforniaF-TF.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.Mathematics III
CaliforniaF-TF.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.Mathematics III
CaliforniaF-TF.2.1Graph all 6 basic trigonometric functions.Mathematics III
CaliforniaF-TF.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.Mathematics III
CaliforniaG-SRT.9Derive the formula ùê¥ = 1/2 ùò¢ùò£ sin(ùê∂) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.Mathematics III
CaliforniaG-SRT.10Prove the Laws of Sines and Cosines and use them to solve problems.Mathematics III
CaliforniaG-SRT.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).Mathematics III
CaliforniaG-GPE.3.1Given a quadratic equation of the form ax¬≤ + by¬≤ + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola, and graph the equationMathematics III
CaliforniaG-GMD.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.Mathematics III
CaliforniaG-MG.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).Mathematics III
CaliforniaG-MG.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).Mathematics III
CaliforniaG-MG.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).Mathematics III
CaliforniaN-CN.8Extend polynomial identities to the complex numbers.Mathematics III
CaliforniaN-CN.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.Mathematics III
CaliforniaS-ID.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.Mathematics III
CaliforniaS-IC.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.Mathematics III
CaliforniaS-IC.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.Mathematics III
CaliforniaS-IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.Mathematics III
CaliforniaS-IC.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.Mathematics III
CaliforniaS-IC.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.Mathematics III
CaliforniaS-IC.6Evaluate reports based on data.Mathematics III
CaliforniaS-MD.6Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).Mathematics III
CaliforniaS-MD.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).Mathematics III
CaliforniaN-Q.1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.High School - Number and Quantity
CaliforniaN-Q.2Define appropriate quantities for the purpose of descriptive modeling.High School - Number and Quantity
CaliforniaN-Q.3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.High School - Number and Quantity
CaliforniaN-CN.1Know there is a complex number ùò™ such that ùò™¬≤ = ‚Äì1, and every complex number has the form ùò¢ + ùò£ùò™ with ùò¢ and ùò£ real.High School - Number and Quantity
CaliforniaN-CN.2Use the relation ùò™¬≤ = ‚Äì1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.High School - Number and Quantity
CaliforniaN-CN.3Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.High School - Number and Quantity
CaliforniaN-CN.4Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.High School - Number and Quantity
CaliforniaN-CN.5Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.High School - Number and Quantity
CaliforniaN-CN.6Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.High School - Number and Quantity
CaliforniaN-CN.7Solve quadratic equations with real coefficients that have complex solutions.High School - Number and Quantity
CaliforniaN-CN.8Extend polynomial identities to the complex numbers.High School - Number and Quantity
CaliforniaN-CN.9Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.High School - Number and Quantity
CaliforniaN-RN.1Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.High School - Number and Quantity
CaliforniaN-RN.2Rewrite expressions involving radicals and rational exponents using the properties of exponents.High School - Number and Quantity
CaliforniaN-RN.3Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.High School - Number and Quantity
CaliforniaN-VM.1Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., ùô´, |ùô´|, ||ùô´||, ùô´).High School - Number and Quantity
CaliforniaN-VM.2Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.High School - Number and Quantity
CaliforniaN-VM.3Solve problems involving velocity and other quantities that can be represented by vectors.High School - Number and Quantity
CaliforniaN-VM.4Add and subtract vectors.High School - Number and Quantity
CaliforniaN-VM.5Multiply a vector by a scalar.High School - Number and Quantity
CaliforniaN-VM.6Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.High School - Number and Quantity
CaliforniaN-VM.7Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.High School - Number and Quantity
CaliforniaN-VM.8Add, subtract, and multiply matrices of appropriate dimensions.High School - Number and Quantity
CaliforniaN-VM.9Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.High School - Number and Quantity
CaliforniaN-VM.10Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.High School - Number and Quantity
CaliforniaN-VM.11Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.High School - Number and Quantity
CaliforniaN-VM.12Work with 2 √ó 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.High School - Number and Quantity
CaliforniaA-APR.1Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.High School - Algebra
CaliforniaA-APR.2Know and apply the Remainder Theorem: For a polynomial ùò±(ùòπ) and a number ùò¢, the remainder on division by ùòπ ‚Äì ùò¢ is ùò±(ùò¢), so ùò±(ùò¢) = 0 if and only if (ùòπ ‚Äì ùò¢) is a factor of ùò±(ùòπ).High School - Algebra
CaliforniaA-APR.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.High School - Algebra
CaliforniaA-APR.4Prove polynomial identities and use them to describe numerical relationships.High School - Algebra
CaliforniaA-APR.5Know and apply the Binomial Theorem for the expansion of (ùòπ + ùò∫)‚Åø in powers of ùòπ and y for a positive integer ùòØ, where ùòπ and ùò∫ are any numbers, with coefficients determined for example by Pascal‚Äôs Triangle.High School - Algebra
CaliforniaA-APR.6Rewrite simple rational expressions in different forms; write ùò¢(ùòπ)/ùò£(ùòπ) in the form ùò≤(ùòπ) + ùò≥(ùòπ)/ùò£(ùòπ), where ùò¢(ùòπ), ùò£(ùòπ), ùò≤(ùòπ), and ùò≥(ùòπ) are polynomials with the degree of ùò≥(ùòπ) less than the degree of ùò£(ùòπ), using inspection, long division, or, for the more complicated examples, a computer algebra system.High School - Algebra
CaliforniaA-APR.7Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.High School - Algebra
CaliforniaA-CED.1Create equations and inequalities in one variable including ones with absolute value and use them to solve problems.High School - Algebra
CaliforniaA-CED.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.High School - Algebra
CaliforniaA-CED.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context.High School - Algebra
CaliforniaA-CED.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.High School - Algebra
CaliforniaA-REI.1Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.High School - Algebra
CaliforniaA-REI.2Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.High School - Algebra
CaliforniaA-REI.3Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.High School - Algebra
CaliforniaA-REI.3.1Solve one-variable equations and inequalities involving absolute value, graphing the solutions and interpreting them in context.High School - Algebra
CaliforniaA-REI.4Solve quadratic equations in one variable.High School - Algebra
CaliforniaA-REI.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.High School - Algebra
CaliforniaA-REI.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.High School - Algebra
CaliforniaA-REI.7Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.High School - Algebra
CaliforniaA-REI.8Represent a system of linear equations as a single matrix equation in a vector variable.High School - Algebra
CaliforniaA-REI.9Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 √ó 3 or greater).High School - Algebra
CaliforniaA-REI.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).High School - Algebra
CaliforniaA-REI.11Explain why the ùòπ-coordinates of the points where the graphs of the equations ùò∫ = ùòß(ùòπ) and ùò∫ = ùëî(ùòπ) intersect are the solutions of the equation ùòß(ùòπ) = ùëî(ùòπ); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ùòß(ùòπ) and/or ùëî(ùòπ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.High School - Algebra
CaliforniaA-REI.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.High School - Algebra
CaliforniaA-SSE.1Interpret expressions that represent a quantity in terms of its context.High School - Algebra
CaliforniaA-SSE.2Use the structure of an expression to identify ways to rewrite it.High School - Algebra
CaliforniaA-SSE.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.High School - Algebra
CaliforniaA-SSE.4Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.High School - Algebra
CaliforniaF-BF.1Write a function that describes a relationship between two quantities.High School - Functions
CaliforniaF-BF.2Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.High School - Functions
CaliforniaF-BF.3Identify the effect on the graph of replacing ùòß(ùòπ) by ùòß(ùòπ) + ùò¨, ùò¨ ùòß(ùòπ), ùòß(ùò¨ùòπ), and ùòß(ùòπ + ùò¨) for specific values of ùò¨ (both positive and negative); find the value of ùò¨ given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology.High School - Functions
CaliforniaF-BF.4Find inverse functions.High School - Functions
CaliforniaF-BF.5Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.High School - Functions
CaliforniaF-IF.1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If ùòß is a function and ùòπ is an element of its domain, then ùòß(ùòπ) denotes the output of ùòß corresponding to the input ùòπ. The graph of ùòß is the graph of the equation ùò∫ = ùòß(ùòπ).High School - Functions
CaliforniaF-IF.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.High School - Functions
CaliforniaF-IF.3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.High School - Functions
CaliforniaF-IF.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.High School - Functions
CaliforniaF-IF.5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.High School - Functions
CaliforniaF-IF.6Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.High School - Functions
CaliforniaF-IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.High School - Functions
CaliforniaF-IF.8Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.High School - Functions
CaliforniaF-IF.9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).High School - Functions
CaliforniaF-IF.10Demonstrate an understanding of functions and equations defined parametrically and graph them.High School - Functions
CaliforniaF-IF.11Graph polar coordinates and curves. Convert between polar and rectangular coordinate systems.High School - Functions
CaliforniaF-LE.1Distinguish between situations that can be modeled with linear functions and with exponential functions.High School - Functions
CaliforniaF-LE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).High School - Functions
CaliforniaF-LE.3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.High School - Functions
CaliforniaF-LE.4For exponential models, express as a logarithm the solution to ùò¢ùò£ to the ùò§ùòµ power = ùò• where ùò¢, ùò§, and ùò• are numbers and the base ùò£ is 2, 10, or ùò¶; evaluate the logarithm using technology.High School - Functions
CaliforniaF-LE.4.1Prove simple laws of logarithms.High School - Functions
CaliforniaF-LE.4.2Use the definition of logarithms to translate between logarithms in any base.High School - Functions
CaliforniaF-LE.4.3Understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.High School - Functions
CaliforniaF-LE.5Interpret the parameters in a linear or exponential function in terms of a context.High School - Functions
CaliforniaF-LE.6Apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.High School - Functions
CaliforniaF-TF.1Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.High School - Functions
CaliforniaF-TF.2Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.High School - Functions
CaliforniaF-TF.2.1Graph all 6 basic trigonometric functions.High School - Functions
CaliforniaF-TF.3Use special triangles to determine geometrically the values of sine, cosine, tangent for œÄ/3, œÄ/4 and œÄ/6, and use the unit circle to express the values of sine, cosine, and tangent for œÄ‚Äìùòπ, œÄ+ùòπ, and 2œÄ‚Äìùòπ in terms of their values for ùòπ, where ùòπ is any real number.High School - Functions
CaliforniaF-TF.4Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.High School - Functions
CaliforniaF-TF.5Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.High School - Functions
CaliforniaF-TF.6Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.High School - Functions
CaliforniaF-TF.7Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.High School - Functions
CaliforniaF-TF.8Prove the Pythagorean identity sin¬≤(Œ∏) + cos¬≤(Œ∏) = 1 and use it to find sin(Œ∏), cos(Œ∏), or tan(Œ∏) given sin(Œ∏), cos(Œ∏), or tan(Œ∏) and the quadrant of the angle.High School - Functions
CaliforniaF-TF.9Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.High School - Functions
CaliforniaF-TF.10Prove the half angle and double angle identities for sine and cosine and use them to solve problems.High School - Functions
CaliforniaG-C.1Prove that all circles are similar.High School - Geometry
CaliforniaG-C.2Identify and describe relationships among inscribed angles, radii, and chords.High School - Geometry
CaliforniaG-C.3Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.High School - Geometry
CaliforniaG-C.4Construct a tangent line from a point outside a given circle to the circle.High School - Geometry
CaliforniaG-C.5Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Convert between degrees and radians.High School - Geometry
CaliforniaG-CO.1Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.High School - Geometry
CaliforniaG-CO.2Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).High School - Geometry
CaliforniaG-CO.3Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.High School - Geometry
CaliforniaG-CO.4Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.High School - Geometry
CaliforniaG-CO.5Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.High School - Geometry
CaliforniaG-CO.6Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.High School - Geometry
CaliforniaG-CO.7Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.High School - Geometry
CaliforniaG-CO.8Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.High School - Geometry
CaliforniaG-CO.9Prove theorems about lines and angles.High School - Geometry
CaliforniaG-CO.10Prove theorems about triangles.High School - Geometry
CaliforniaG-CO.11Prove theorems about parallelograms.High School - Geometry
CaliforniaG-CO.12Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).High School - Geometry
CaliforniaG-CO.13Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.High School - Geometry
CaliforniaG-GPE.1Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.High School - Geometry
CaliforniaG-GPE.2Derive the equation of a parabola given a focus and directrix.High School - Geometry
CaliforniaG-GPE.3Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.High School - Geometry
CaliforniaG-GPE.3.1Given a quadratic equation of the form ax¬≤ + by¬≤ + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola, and graph the equation.High School - Geometry
CaliforniaG-GPE.4Use coordinates to prove simple geometric theorems algebraically.High School - Geometry
CaliforniaG-GPE.5Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).High School - Geometry
CaliforniaG-GPE.6Find the point on a directed line segment between two given points that partitions the segment in a given ratio.High School - Geometry
CaliforniaG-GPE.7Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.High School - Geometry
CaliforniaG-GMD.1Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.High School - Geometry
CaliforniaG-GMD.2Give an informal argument using Cavalieri‚Äôs principle for the formulas for the volume of a sphere and other solid figures.High School - Geometry
CaliforniaG-GMD.3Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.High School - Geometry
CaliforniaG-GMD.4Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.High School - Geometry
CaliforniaG-GMD.5Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k¬≤, and k¬≥, respectively; determine length, area and volume measures using scale factors.High School - Geometry
CaliforniaG-GMD.6Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve real-world and mathematical problems.High School - Geometry
CaliforniaG-MG.1Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).High School - Geometry
CaliforniaG-MG.2Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).High School - Geometry
CaliforniaG-MG.3Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).High School - Geometry
CaliforniaG-SRT.1Verify experimentally the properties of dilations given by a center and a scale factor:High School - Geometry
CaliforniaG-SRT.2Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.High School - Geometry
CaliforniaG-SRT.3Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.High School - Geometry
CaliforniaG-SRT.4Prove theorems about triangles.High School - Geometry
CaliforniaG-SRT.5Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.High School - Geometry
CaliforniaG-SRT.6Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.High School - Geometry
CaliforniaG-SRT.7Explain and use the relationship between the sine and cosine of complementary angles.High School - Geometry
CaliforniaG-SRT.8Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.High School - Geometry
CaliforniaG-SRT.8.1Derive and use the trigonometric ratios for special right triangles (30¬∞, 60¬∞, 90¬∞ and 45¬∞, 45¬∞, 90¬∞).High School - Geometry
CaliforniaG-SRT.9Derive the formula ùê¥ = 1/2 ùò¢ùò£ sin(ùê∂) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.High School - Geometry
CaliforniaG-SRT.10Prove the Laws of Sines and Cosines and use them to solve problems.High School - Geometry
CaliforniaG-SRT.11Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).High School - Geometry
CaliforniaS-CP.1Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (‚Äúor,‚Äù ‚Äúand,‚Äù ‚Äúnot‚Äù).High School - Statistics and Probability
CaliforniaS-CP.2Understand that two events ùòà and ùòâ are independent if the probability of ùòà and ùòâ occurring together is the product of their probabilities, and use this characterization to determine if they are independent.High School - Statistics and Probability
CaliforniaS-CP.3Understand the conditional probability of ùòà given ùòâ as ùòó(ùòà and ùòâ)/ùòó(ùòâ), and interpret independence of ùòà and ùòâ as saying that the conditional probability of ùòà given ùòâ is the same as the probability of ùòà, and the conditional probability of ùòâ given ùòà is the same as the probability of ùòâ.High School - Statistics and Probability
CaliforniaS-CP.4Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.High School - Statistics and Probability
CaliforniaS-CP.5Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.High School - Statistics and Probability
CaliforniaS-CP.6Find the conditional probability of ùòà given ùòâ as the fraction of ùòâ‚Äôs outcomes that also belong to ùòà, and interpret the answer in terms of the model.High School - Statistics and Probability
CaliforniaS-CP.7Apply the Addition Rule, ùòó(ùòà or ùòâ) = ùòó(ùòà) + ùòó(ùòâ) ‚Äì ùòó(ùòà and ùòâ), and interpret the answer in terms of the model.High School - Statistics and Probability
CaliforniaS-CP.8Apply the general Multiplication Rule in a uniform probability model, ùòó(ùòà and ùòâ) = ùòó(ùòà)ùòó(ùòâ|ùòà) = ùòó(ùòâ)ùòó(ùòà|ùòâ), and interpret the answer in terms of the model.High School - Statistics and Probability
CaliforniaS-CP.9Use permutations and combinations to compute probabilities of compound events and solve problems.High School - Statistics and Probability
CaliforniaS-ID.1Represent data with plots on the real number line (dot plots, histograms, and box plots).High School - Statistics and Probability
CaliforniaS-ID.2Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.High School - Statistics and Probability
CaliforniaS-ID.3Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).High School - Statistics and Probability
CaliforniaS-ID.4Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.High School - Statistics and Probability
CaliforniaS-ID.5Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.High School - Statistics and Probability
CaliforniaS-ID.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.High School - Statistics and Probability
CaliforniaS-ID.7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.High School - Statistics and Probability
CaliforniaS-ID.8Compute (using technology) and interpret the correlation coefficient of a linear fit.High School - Statistics and Probability
CaliforniaS-ID.9Distinguish between correlation and causation.High School - Statistics and Probability
CaliforniaS-IC.1Understand statistics as a process for making inferences about population parameters based on a random sample from that population.High School - Statistics and Probability
CaliforniaS-IC.2Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.High School - Statistics and Probability
CaliforniaS-IC.3Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.High School - Statistics and Probability
CaliforniaS-IC.4Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.High School - Statistics and Probability
CaliforniaS-IC.5Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.High School - Statistics and Probability
CaliforniaS-IC.6Evaluate reports based on data.High School - Statistics and Probability
CaliforniaS-MD.1Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.High School - Statistics and Probability
CaliforniaS-MD.2Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.High School - Statistics and Probability
CaliforniaS-MD.3Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.High School - Statistics and Probability
CaliforniaS-MD.4Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.High School - Statistics and Probability
CaliforniaS-MD.5Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.High School - Statistics and Probability
CaliforniaS-MD.6Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).High School - Statistics and Probability
CaliforniaS-MD.7Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).High School - Statistics and Probability
CCSSMA-APR.B.3Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.Algebra
CCSSMA-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Algebra
CCSSMA-SSE.A.2Use the structure of an expression to identify ways to rewrite it.Algebra
CCSSMA-SSE.B.3Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Algebra
CCSSMF-BF.A.1Write a function that describes a relationship between two quantities.Algebra
CCSSMF-IF.A.2Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.Algebra
CCSSMF-IF.B.4For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Algebra
CCSSMF-IF.C.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Algebra
CCSSMS-ID.B.6Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.Algebra
CCSSM1.MD.B.3Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
CCSSM1.MD.C.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
CCSSM1.NBT.A.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
CCSSM1.NBT.B.2Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases: 10 can be thought of as a bundle of ten ones - called a 'ten.'. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).Grade 1
CCSSM1.NBT.B.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with symbols.Grade 1
CCSSM1.NBT.C.4Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
CCSSM1.NBT.C.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1
CCSSM1.NBT.C.6Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 1
CCSSM1.OA.B.3Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)Grade 1
CCSSM1.OA.B.4Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8. Add and subtract within 20.Grade 1
CCSSM1.OA.C.6Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Grade 1
CCSSM1.OA.D.7Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.Grade 1
CCSSM1.OA.D.8Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.Grade 1
CCSSM2.G.A.1Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Grade 2
CCSSM2.MD.C.7Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Grade 2
CCSSM2.MD.D.10Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.Grade 2
CCSSM2.MD.D.9Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.Grade 2
CCSSM2.NBT.A.1Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: 100 can be thought of as a bundle of ten tens - called a 'hundred.'. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).Grade 2
CCSSM2.NBT.A.2Count within 1000; skip-count by 5s, 10s, and 100s.Grade 2
CCSSM2.NBT.A.3Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.Grade 2
CCSSM2.NBT.A.4Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using symbols to record the results of comparisons.Grade 2
CCSSM2.NBT.B.5Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 2
CCSSM2.NBT.B.6Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
CCSSM2.NBT.B.7Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Grade 2
CCSSM2.NBT.B.8Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.Grade 2
CCSSM2.OA.A.1Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 2
CCSSM2.OA.B.2Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.Grade 2
CCSSM3.G.A.1Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Grade 3
CCSSM3.MD.A.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.Grade 3
CCSSM3.MD.B.3Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step √±how many more√Æ and √±how many less√Æ problems using information presented in scaled bar graphs.Grade 3
CCSSM3.MD.B.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.Grade 3
CCSSM3.MD.C.5Recognize area as an attribute of plane figures and understand concepts of area measurement.Grade 3
CCSSM3.MD.C.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).Grade 3
CCSSM3.NBT.A.1Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
CCSSM3.NBT.A.2Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
CCSSM3.NBT.A.3Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 _ 80, 5 _ 60) using strategies based on place value and properties of operations.Grade 3
CCSSM3.NF.A.1Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.Grade 3
CCSSM3.NF.A.2Understand a fraction as a number on the number line; represent fractions on a number line diagram. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.Grade 3
CCSSM3.NF.A.3Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, orGrade 3
CCSSM3.OA.A.1Interpret products of whole numbers, e.g., interpret 5 _ 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 _ 7.Grade 3
CCSSM3.OA.A.2Interpret whole-number quotients of whole numbers, e.g., interpret 56 ¬Ö 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ¬Ö 8.Grade 3
CCSSM3.OA.A.3Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 3
CCSSM3.OA.A.4Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 _ ? = 48, 5 = _ ¬Ö 3, 6 _ 6 = ?Grade 3
CCSSM3.OA.B.5Apply properties of operations as strategies to multiply and divide. Examples: If 6 _ 4 = 24 is known, then 4 _ 6 = 24 is also known. (Commutative property of multiplication.) 3 _ 5 _ 2 can be found by 3 _ 5 = 15, then 15 _ 2 = 30, or by 5 _ 2 = 10, then 3 _ 10 = 30. (Associative property of multiplication.) Knowing that 8 _ 5 = 40 and 8 _ 2 = 16, one can find 8 _ 7 as 8 _ (5 + 2) = (8 _ 5) + (8 _ 2) = 40 + 16 = 56. (Distributive property.)Grade 3
CCSSM3.OA.B.6Understand division as an unknown-factor problem. For example, find 32 ¬Ö 8 by finding the number that makes 32 when multiplied by 8.Grade 3
CCSSM3.OA.C.7Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 _ 5 = 40, one knows 40 ¬Ö 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
CCSSM4.G.A.1Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Grade 4
CCSSM4.G.A.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.Grade 4
CCSSM4.MD.A.1Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
CCSSM4.MD.A.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
CCSSM4.MD.B.4Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
CCSSM4.MD.C.5Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.Grade 4
CCSSM4.MD.C.6Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Grade 4
CCSSM4.MD.C.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Grade 4
CCSSM4.NBT.A.1Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 / 70 = 10 by applying concepts of place value and division.Grade 4
CCSSM4.NBT.A.2Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
CCSSM4.NBT.A.3Use place value understanding to round multi-digit whole numbers to any place.Grade 4
CCSSM4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm.Grade 4
CCSSM4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
CCSSM4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
CCSSM4.NF.A.1Explain why a fraction a/b is equivalent to a fraction (n x a) / (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
CCSSM4.NF.A.2Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, orGrade 4
CCSSM4.NF.B.3Understand a fraction a/b with a > 1 as a sum of fractions 1/b. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.Grade 4
CCSSM4.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 x (1/4), recording the conclusion by the equation 5/4 = 5 x (1/4). Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 x(2/5) as 6 x (1/5), recognizing this product as 6/5. (In general, n x (a/b) = (n x a) / b.) Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?Grade 4
CCSSM4.NF.C.5Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.Grade 4
CCSSM4.NF.C.6Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.Grade 4
CCSSM4.NF.C.7Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, orGrade 4
CCSSM4.OA.A.1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
CCSSM4.OA.A.2Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Grade 4
CCSSM4.OA.B.4Find all factor pairs for a whole number in the range 1 - 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1 ¬Ñ 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1 - 100 is prime or composite.Grade 4
CCSSM4.OA.C.5Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule 'Add 3' and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.Grade 4
CCSSM5.G.A.1Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and the given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate).Grade 5
CCSSM5.G.A.2Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
CCSSM5.G.B.3Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.Grade 5
CCSSM5.G.B.4Classify two-dimensional figures in a hierarchy based on properties.Grade 5
CCSSM5.MD.B.2Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.Grade 5
CCSSM5.NBT.A.1Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
CCSSM5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.Grade 5
CCSSM5.NBT.A.4Use place value understanding to round decimals to any place.Grade 5
CCSSM5.NBT.B.5Fluently multiply multi-digit whole numbers using the standard algorithm.Grade 5
CCSSM5.NBT.B.6Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 5
CCSSM5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 5
CCSSM5.NF.B.3Interpret a fraction as division of the numerator by the denominator (a/b = a / b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?Grade 5
CCSSM5.NF.B.4Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
CCSSM5.NF.B.5Interpret multiplication as scaling (resizing), by comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.Grade 5
CCSSM5.NF.B.6Solve real world problems involving multiplication of fractions and mixed numbers by using visual fraction models or equations to represent the problem.Grade 5
CCSSM5.NF.B.7Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.Grade 5
CCSSM5.OA.A.1Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.Grade 5
CCSSM5.OA.B.3Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule √±Add 3√Æ and the starting number 0, and given the rule √±Add 6√Æ and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.Grade 5
CCSSM6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents.Grade 6
CCSSM6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers.Grade 6
CCSSM6.EE.A.3Apply the properties of operations to generate equivalent expressions.Grade 6
CCSSM6.EE.B.5Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
CCSSM6.EE.B.7Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.Grade 6
CCSSM6.EE.B.8Write an inequality of the form x > c or x < c to represent a constraint or condition in a real world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
CCSSM6.G.A.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.Grade 6
CCSSM6.NS.A.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions.Grade 6
CCSSM6.NS.B.2Fluently divide multi-digit numbers using the standard algorithm.Grade 6
CCSSM6.NS.B.3Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Grade 6
CCSSM6.NS.C.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.Grade 6
CCSSM6.NS.C.6Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
CCSSM6.NS.C.7Understand ordering and absolute value of rational numbers.Grade 6
CCSSM6.NS.C.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
CCSSM6.RP.A.1Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Grade 6
CCSSM6.RP.A.2Understand the concept of a unit rate a/b associated with a ratio a:b with b ? 0, and use rate language in the context of a ratio relationship. For example, This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar. We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.Grade 6
CCSSM6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams or equations.Grade 6
CCSSM7.EE.A.1Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Grade 7
CCSSM7.EE.B.3Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
CCSSM7.G.A.1Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
CCSSM7.G.A.2Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
CCSSM7.G.B.5Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.Grade 7
CCSSM7.NS.A.1Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.Grade 7
CCSSM7.NS.A.2Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.Grade 7
CCSSM7.NS.A.3Solve real-world and mathematical problems involving the four operations with rational numbers.Grade 7
CCSSM7.RP.A.1Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/3 hour, compute the unit rate as the complex fraction 1/2 divided by 1/4 per hour, equivalently 2 miles per hour.Grade 7
CCSSM7.RP.A.2Recognize and represent proportional relationships between quantities.Grade 7
CCSSM7.RP.A.3Use proportional relationships to solve multistep ratio and percent problems.Grade 7
CCSSM8.EE.A.3Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much ones is than the other.Grade 8
CCSSM8.EE.A.4Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology.Grade 8
CCSSM8.EE.B.5Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.Grade 8
CCSSM8.EE.B.6Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.Grade 8
CCSSM8.EE.C.7Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).Grade 8
CCSSM8.EE.C.8Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.Grade 8
CCSSM8.F.A.1Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.Grade 8
CCSSM8.F.A.2Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.Grade 8
CCSSM8.F.A.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.Grade 8
CCSSM8.F.B.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Grade 8
CCSSM8.F.B.5Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.Grade 8
CCSSM8.G.A.1Verify experimentally the properties of rotations, reflections, and translations.Grade 8
CCSSM8.G.A.2Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
CCSSM8.G.A.4Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Grade 8
CCSSM8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
CCSSM8.G.B.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
CCSSM8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
CCSSM8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
CCSSMK.CC.A.1Count to 100 by ones and by tensKindergarten
CCSSMK.CC.A.2Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Kindergarten
CCSSMK.CC.A.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).Kindergarten
CCSSMK.CC.B.4Understand the relationship between numbers and quantities; connect counting to cardinality. When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. Understand that each successive number name refers to a quantity that is one larger.Kindergarten
CCSSMK.CC.B.5Count to answer 'how many' questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
CCSSMK.CC.C.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.Kindergarten
CCSSMK.CC.C.7Compare two numbers between 1 and 10 presented as written numerals.Kindergarten
CCSSMK.NBT.A.1Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Kindergarten
CCSSMK.OA.A.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
CCSSMK.OA.A.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.Kindergarten
CCSSMK.OA.A.3Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).Kindergarten
CCSSMK.OA.A.4For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.Kindergarten
CCSSMK.OA.A.5Fluently add and subtract within 5.Kindergarten
ColoradoP.OA.A.1By the end of the preschool experience (approximately 60 months/5 years old), students may: Represent addition and subtraction in different ways, such as with fingers, objects, and drawings.Prekindergarten
ColoradoP.OA.A.2By the end of the preschool experience (approximately 60 months/5 years old), students may: Solve addition and subtraction problems set in simple contexts. Add and subtract up to at least five to or from a given number to find a sum or difference up to 10.Prekindergarten
ColoradoP.OA.A.3By the end of the preschool experience (approximately 60 months/5 years old), students may: With adult assistance, begin to use counting on (adding 1 or 2, for example) from the larger number for addition.Prekindergarten
Colorado1Children may: Add a group of three and a group of two, counting ‚ÄúOne, two three ‚Ä¶‚Äù and then counting on ‚ÄúFour, five!‚Äù while keeping track using their fingers.Prekindergarten
Colorado2Children may: Take three away from five, counting ‚ÄúFive, four, three ‚Ä¶ two!‚Äù while keeping track using their fingers.Prekindergarten
Colorado3Children may: Say after receiving more crackers at snack time, ‚ÄúI had two and now I have four.‚ÄùPrekindergarten
Colorado4Children may: Predict what will happen when one more object is taken away from a group of five or fewer objects, and then verify their prediction by taking the object away and counting the remaining objects.Prekindergarten
ColoradoP.OA.B.4By the end of the preschool experience (approximately 60 months/5 years old), students may: Fill in missing elements of simple patterns.Prekindergarten
ColoradoP.OA.B.5By the end of the preschool experience (approximately 60 months/5 years old), students may: Duplicate simple patterns in a different location than demonstrated, such as making the same alternating color pattern with blocks at a table that was demonstrated on the rug. Extend patterns, such as making an eight-block tower of the same pattern that was demonstrated with four blocks.Prekindergarten
ColoradoP.OA.B.6By the end of the preschool experience (approximately 60 months/5 years old), students may: Identify the core unit of sequentially repeating patterns, such as color in a sequence of alternating red and blue blocks.Prekindergarten
ColoradoP.MD.A.1By the end of the preschool experience (approximately 60 months/5 years old), students may: Use comparative language, such as shortest, heavier, biggest, or later.Prekindergarten
ColoradoP.MD.A.2By the end of the preschool experience (approximately 60 months/5 years old), students may: Compare or order up to five objects based on their measurable attributes, such as height or weight.Prekindergarten
ColoradoP.MD.A.3By the end of the preschool experience (approximately 60 months/5 years old), students may: Measure using the same unit, such as putting together snap cubes to see how tall a book is.Prekindergarten
ColoradoP.G.A.1By the end of the preschool experience (approximately 60 months/5 years old), students may: Name and describe shapes in terms of length of sides, number of sides, and number of angles/corners.Prekindergarten
ColoradoP.G.A.2By the end of the preschool experience (approximately 60 months/5 years old), students may: Correctly name basic shapes (circle, square, rectangle, triangle) regardless of size and orientation.Prekindergarten
ColoradoP.G.A.3By the end of the preschool experience (approximately 60 months/5 years old), students may: Analyze, compare, and sort two-and three-dimensional shapes and objects in different sizes. Describe their similarities, differences, and other attributes, such as size and shape.Prekindergarten
ColoradoP.G.A.4By the end of the preschool experience (approximately 60 months/5 years old), students may: Compose simple shapes to form larger shapes.Prekindergarten
ColoradoP.G.B.5By the end of the preschool experience (approximately 60 months/5 years old), students may: Understand and use language related to directionality, order, and the position of objects, including up/down and in front/behind.Prekindergarten
ColoradoP.G.B.6By the end of the preschool experience (approximately 60 months/5 years old), students may: Correctly follow directions involving their own position in space, such as ‚ÄúStand up‚Äù and ‚ÄúMove forward.‚ÄùPrekindergarten
ColoradoP.CC.A.1By the end of the preschool experience (approximately 60 months/5 years old), students may: Count verbally or sign to at least 20 by ones.Prekindergarten
ColoradoP.CC.B.2By the end of the preschool experience (approximately 60 months/5 years old), students may: Instantly recognize, without counting, small quantities of up to five objects and say or sign the number.Prekindergarten
ColoradoP.CC.C.3By the end of the preschool experience (approximately 60 months/5 years old), students may: Say or sign the number names in order when counting, pairing one number word that corresponds with one object, up to at least 10.Prekindergarten
ColoradoP.CC.C.4By the end of the preschool experience (approximately 60 months/5 years old), students may: Use the number name of the last object counted to answer ‚ÄúHow many?‚Äù questions for up to approximately 10 objects.Prekindergarten
ColoradoP.CC.C.5By the end of the preschool experience (approximately 60 months/5 years old), students may: Accurately count as many as five objects in a scattered configuration or out of a collection of more than five objects.Prekindergarten
ColoradoP.CC.C.6By the end of the preschool experience (approximately 60 months/5 years old), students may: Understand that each successive number name refers to a quantity that is one larger.Prekindergarten
ColoradoP.CC.D.7By the end of the preschool experience (approximately 60 months/5 years old), students may: Identify whether the number of objects in one group is more than, less than or the same as objects in another group for up to at least five objects.Prekindergarten
ColoradoP.CC.D.8By the end of the preschool experience (approximately 60 months/5 years old), students may: Identify and use numbers related to order or position from first to fifth.Prekindergarten
ColoradoP.CC.E.9By the end of the preschool experience (approximately 60 months/5 years old), students may: Associate a number of objects with a written numeral 0-5.Prekindergarten
ColoradoP.CC.E.10By the end of the preschool experience (approximately 60 months/5 years old), students may: Recognize and, with support, write some numerals up to 10.Prekindergarten
ColoradoK.OA.A.1Students can: Represent addition and subtraction with objects, fingers, mental images, drawings (drawings need not show details, but should show the mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
ColoradoK.OA.A.2Students can: Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.Kindergarten
ColoradoK.OA.A.3Students can: Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).Kindergarten
ColoradoK.OA.A.4Students can: For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.Kindergarten
ColoradoK.MD.A.1Students can: Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.Kindergarten
ColoradoK.MD.A.2Students can: Directly compare two objects with a measurable attribute in common, to see which object has ‚Äúmore of‚Äù/‚Äúless of‚Äù the attribute, and describe the difference.Kindergarten
ColoradoK.MD.B.3Students can: Classify objects into given categories; count the numbers of objects in each category and sort the categories by count. (Limit category counts to be less than or equal to 10.)Kindergarten
ColoradoK.G.A.1Students can: Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.Kindergarten
ColoradoK.G.A.2Students can: Correctly name shapes regardless of their orientations or overall size.Kindergarten
ColoradoK.G.A.3Students can: Identify shapes as two-dimensional (lying in a plane, ‚Äúflat‚Äù) or three-dimensional (‚Äúsolid‚Äù).Kindergarten
ColoradoK.G.B.4Students can: Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/‚Äúcorners‚Äù) and other attributes (e.g., having sides of equal length).Kindergarten
ColoradoK.G.B.5Students can: Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.Kindergarten
ColoradoK.G.B.6Students can: Compose simple shapes to form larger shapes.Kindergarten
ColoradoK.CC.A.1Students can: Count to 100 by ones and by tens.Kindergarten
ColoradoK.CC.A.2Students can: Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Kindergarten
ColoradoK.CC.A.3Students can: Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).Kindergarten
ColoradoK.CC.B.4Students can: Apply the relationship between numbers and quantities and connect counting to cardinality.Kindergarten
ColoradoK.CC.B.5Students can: Count to answer ‚Äúhow many?‚Äù questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
ColoradoK.CC.C.6Students can: Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. (Include groups with up to 10 objects.)Kindergarten
ColoradoK.CC.C.7Students can: Compare two numbers between 1 and 10 presented as written numerals.Kindergarten
ColoradoK.NBT.A.1Students can: Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10+8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Kindergarten
Colorado1.OA.A.1Students can: Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Grade 1
Colorado1.OA.A.2Students can: Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.Grade 1
Colorado1.OA.B.3Students can: Apply properties of operations as strategies to add and subtract. (Students need not use formal terms for these properties.)Grade 1
Colorado1.OA.C.6Students can: Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 ‚àí 4 = 13 ‚àí 3 ‚àí 1 = 10‚àí1=9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 ‚àí 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).Grade 1
Colorado1.OA.D.7Students can: Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.Grade 1
Colorado1.OA.D.8Students can: Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.Grade 1
Colorado1.MD.A.1Students can: Order three objects by length; compare the lengths of two objects indirectly by using a third object.Grade 1
Colorado1.MD.A.2Students can: Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.Grade 1
Colorado1.MD.B.3Students can: Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
Colorado1.MD.C.4Students can: Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
Colorado1.G.A.1Students can: Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.Grade 1
Colorado1.G.A.2Students can: Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names, such as ‚Äúright rectangular prisms.‚Äù)Grade 1
Colorado1.G.A.3Students can: Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.Grade 1
Colorado1.NBT.A.1Students can: Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
Colorado1.NBT.B.2Students can: Understand that the two digits of a two-digit number represent amounts of tens and ones.Grade 1
Colorado1.NBT.B.3Students can: Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.Grade 1
Colorado1.NBT.C.4Students can: Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
Colorado1.NBT.C.5Students can: Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1
Colorado1.NBT.C.6Students can: Subtract multiples of 10 in the range 10‚Äì90 from multiples of 10 in the range 10‚Äì90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 1
Colorado2.OA.A.1Students can: Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 2
Colorado2.OA.B.2Students can: Fluently add and subtract within 20 using mental strategies. (See 1.OA.C.6 for a list of strategies.) By end of Grade 2, know from memory all sums of two one-digit numbers.Grade 2
Colorado2.OA.C.3Students can: Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.Grade 2
Colorado2.OA.C.4Students can: Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.Grade 2
Colorado2.MD.A.1Students can: Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.Grade 2
Colorado2.MD.A.2Students can: Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.Grade 2
Colorado2.MD.A.3Students can: Estimate lengths using units of inches, feet, centimeters, and meters.Grade 2
Colorado2.MD.A.4Students can: Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.Grade 2
Colorado2.MD.B.5Students can: Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.Grade 2
Colorado2.MD.B.6Students can: Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2,‚Ä¶, and represent whole-number sums and differences within 100 on a number line diagram.Grade 2
Colorado2.MD.C.7Students can: Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.Grade 2
Colorado2.MD.C.8Students can: Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using \$ and ¬¢ symbols appropriately.Grade 2
Colorado2.MD.D.9Students can: Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.Grade 2
Colorado2.MD.D.10Students can: Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.Grade 2
Colorado2.G.A.1Students can: Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. (Sizes are compared directly or visually, not compared by measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.Grade 2
Colorado2.G.A.2Students can: Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.Grade 2
Colorado2.G.A.3Students can: Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.Grade 2
Colorado2.NBT.A.1Students can: Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones.Grade 2
Colorado2.NBT.A.2Students can: Count within 1000; skip-count by 5s, 10s, and 100s.Grade 2
Colorado2.NBT.A.3Students can: Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.Grade 2
Colorado2.NBT.A.4Students can: Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.Grade 2
Colorado2.NBT.B.5Students can: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 2
Colorado2.NBT.B.6Students can: Add up to four two-digit numbers using strategies based on place value and properties of operations.Grade 2
Colorado2.NBT.B.7Students can: Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.Grade 2
Colorado2.NBT.B.8Students can: Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.Grade 2
Colorado2.NBT.B.9Students can: Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)Grade 2
Colorado3.OA.A.1Students can: Interpret products of whole numbers, e.g., interpret 5 √ó 7 as the total number of objects in 5 groups of 7 objects each.Grade 3
Colorado3.OA.A.2Students can: Interpret whole-number quotients of whole numbers, e.g., interpret 56 √∑ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each.Grade 3
Colorado3.OA.A.3Students can: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.Grade 3
Colorado3.OA.A.4Students can: Determine the unknown whole number in a multiplication or division equation relating three whole numbers.Grade 3
Colorado3.OA.B.5Students can: Apply properties of operations as strategies to multiply and divide. (Students need not use formal terms for these properties.)Grade 3
Colorado3.OA.C.7Students can: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 √ó 5 = 40, one knows 40 √∑ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.Grade 3
Colorado3.OA.D.8Students can: Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This evidence outcome is limited to problems posed with whole numbers and having whole-number answers; students should know how to perform operations in the conventional order of operations when there are no parentheses to specify a particular order.)Grade 3
Colorado3.OA.D.9Students can: Identify arithmetic patterns (including patterns in the addition table or multiplication table) and explain them using properties of operations.Grade 3
Colorado3.MD.A.1Students can: Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram.Grade 3
Colorado3.MD.A.2Students can: Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). (This excludes compound units such as cm¬≥ and finding the geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.Grade 3
Colorado3.MD.B.3Students can: Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step ‚Äúhow many more‚Äù and ‚Äúhow many less‚Äù problems using information presented in scaled bar graphs.Grade 3
Colorado3.MD.B.4Students can: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-whole numbers, halves, or quarters.Grade 3
Colorado3.MD.C.5Students can: Recognize area as an attribute of plane figures and understand concepts of area measurement.Grade 3
Colorado3.MD.C.6Students can: Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).Grade 3
Colorado3.MD.C.7Students can: Use concepts of area and relate area to the operations of multiplication and addition.Grade 3
Colorado3.MD.D.8Students can: Solve real-world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters.Grade 3
Colorado3.G.A.1Students can: Explain that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.Grade 3
Colorado3.G.A.2Students can: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.Grade 3
Colorado3.NBT.A.1Students can: Use place value understanding to round whole numbers to the nearest 10 or 100.Grade 3
Colorado3.NBT.A.2Students can: Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.Grade 3
Colorado3.NBT.A.3Students can: Multiply one-digit whole numbers by multiples of 10 in the range 10 - 90 (e.g., 9 √ó 80, 5 √ó 60) using strategies based on place value and properties of operations.Grade 3
Colorado3.NF.A.1Students can: Describe a fraction 1/ùëè as the quantity formed by 1 part when a whole is partitioned into ùëè equal parts; understand a fraction ùëé/ùëè as the quantity formed by ùëé parts of size 1/ùëè.Grade 3
Colorado3.NF.A.2Students can: Describe a fraction as a number on the number line; represent fractions on a number line diagram.Grade 3
Colorado3.NF.A.3Students can: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.Grade 3
Colorado4.OA.A.1Students can: Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 √ó 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.Grade 4
Colorado4.OA.A.2Students can: Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Grade 4
Colorado4.OA.A.3Students can: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Grade 4
Colorado4.OA.B.4Students can: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.Grade 4
Colorado4.OA.C.5Students can: Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.Grade 4
Colorado4.MD.A.1Students can: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.Grade 4
Colorado4.MD.A.2Students can: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.Grade 4
Colorado4.MD.A.3Students can: Apply the area and perimeter formulas for rectangles in real-world and mathematical problems.Grade 4
Colorado4.MD.B.4Students Can: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots.Grade 4
Colorado4.MD.C.5Students can: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement.Grade 4
Colorado4.MD.C.6Students can: Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure.Grade 4
Colorado4.MD.C.7Students can: Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.Grade 4
Colorado4.G.A.1Students can: Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.Grade 4
Colorado4.G.A.2Students can: Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.Grade 4
Colorado4.G.A.3Students can: Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry.Grade 4
Colorado4.NBT.A.1Students can: Explain that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.Grade 4
Colorado4.NBT.A.2Students can: Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.Grade 4
Colorado4.NBT.A.3Students can: Use place value understanding to round multi-digit whole numbers to any place.Grade 4
Colorado4.NBT.B.5Students can: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Colorado4.NBT.B.6Students can: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 4
Colorado4.NF.A.1Students can: Explain why a fraction ùëé/ùëè is equivalent to a fraction (ùëõ √ó ùëé)/(ùëõ √ó ùëè) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.Grade 4
Colorado4.NF.A.2Students can: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.Grade 4
Colorado4.NF.B.3Students can: Understand a fraction ùëé/ùëè with ùëé > 1 as a sum of fractions 1/ùëè.Grade 4
Colorado4.NF.B.4Students can: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.Grade 4
Colorado4.NF.C.5Students can: Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. (Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade.)Grade 4
Colorado4.NF.C.6Students can: Use decimal notation for fractions with denominators 10 or 100.Grade 4
Colorado4.NF.C.7Students can: Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.Grade 4
Colorado5.OA.A.1Students can: Use grouping symbols (parentheses, brackets, or braces) in numerical expressions, and evaluate expressions with these symbols.Grade 5
Colorado5.OA.A.2Students can: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.Grade 5
Colorado5.OA.B.3Students can: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.Grade 5
Colorado5.MD.A.1Students can: Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real-world problems.Grade 5
Colorado5.MD.B.2Students can: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots.Grade 5
Colorado5.MD.C.3Students can: Recognize volume as an attribute of solid figures and understand concepts of volume measurement.Grade 5
Colorado5.MD.C.4Students can: Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.Grade 5
Colorado5.MD.C.5Students can: Relate volume to the operations of multiplication and addition and solve real-world and mathematical problems involving volume.Grade 5
Colorado5.G.A.1Students can: Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., ùë•-axis and ùë•-coordinate, ùë¶-axis and ùë¶-coordinate).Grade 5
Colorado5.G.A.2Students can: Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.Grade 5
Colorado5.G.B.3Students can: Explain that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.Grade 5
Colorado5.G.B.4Students can: Classify two-dimensional figures in a hierarchy based on properties.Grade 5
Colorado5.NBT.A.1Students can: Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.Grade 5
Colorado5.NBT.A.2Students can: Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.Grade 5
Colorado5.NBT.A.4Students can: Use place value understanding to round decimals to any place.Grade 5
Colorado5.NBT.B.5Students can: Fluently multiply multi-digit whole numbers using the standard algorithm.Grade 5
Colorado5.NBT.B.6Students can: Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.Grade 5
Colorado5.NBT.B.7Students can: Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.Grade 5
Colorado5.NF.A.1Students can: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.Grade 5
Colorado5.NF.A.2Students can: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.Grade 5
Colorado5.NF.B.3Students can: Interpret a fraction as division of the numerator by the denominator (ùëé/ùëè = ùëé √∑ ùëè). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem.Grade 5
Colorado5.NF.B.4Students can: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.Grade 5
Colorado5.NF.B.6Students can: Solve real-world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.Grade 5
Colorado5.NF.B.7Students can: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)Grade 5
Colorado6.EE.A.3Students can: Apply the properties of operations to generate equivalent expressions.Grade 6
Colorado6.EE.A.4Students can: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).Grade 6
Colorado6.EE.B.5Students can: Describe solving an equation or inequality as a process of answering a question: Which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.Grade 6
Colorado6.EE.B.6Students can: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; recognize that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.Grade 6
Colorado6.EE.B.7Students can: Solve real-world and mathematical problems by writing and solving equations of the form ùë• ¬± ùëù = ùëû and ùëùùë• = ùëû for cases in which ùëù, ùëû and ùë• are all nonnegative rational numbers.Grade 6
Colorado6.EE.B.8Students can: Write an inequality of the form ùë• > ùëê, ùë• ‚â• ùëê, ùë• < ùëê, or ùë• ‚â§ ùëê to represent a constraint or condition in a real-world or mathematical problem. Show that inequalities of the form ùë• > ùëê, ùë• ‚â• ùëê, ùë• < ùëê, or ùë• ‚â§ ùëê have infinitely many solutions; represent solutions of such inequalities on number line diagrams.Grade 6
Colorado6.EE.C.9Students can: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.Grade 6
Colorado6.SP.A.1Students can: Identify a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.Grade 6
Colorado6.SP.A.2Students can: Demonstrate that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread, and overall shape.Grade 6
Colorado6.SP.A.3Students can: Explain that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.Grade 6
Colorado6.SP.B.4Students can: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.Grade 6
Colorado6.SP.B.5Students can: Summarize numerical data sets in relation to their context, such as by:Grade 6
Colorado6.G.A.1Students can: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Colorado6.G.A.2Students can: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas ùëâ = ùëôùë§‚Ñé and ùëâ = ùëè‚Ñé to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.Grade 6
Colorado6.G.A.3Students can: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Colorado6.G.A.4Students can: Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.Grade 6
Colorado6.RP.A.1Students can: Apply the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.Grade 6
Colorado6.RP.A.2Students can: Apply the concept of a unit rate ùëé/ùëè associated with a ratio ùëé:ùëè with ùëè ‚â† 0, and use rate language in the context of a ratio relationship.Grade 6
Colorado6.RP.A.3Students can: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.Grade 6
Colorado6.NS.A.1Students can: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem.Grade 6
Colorado6.NS.B.3Students can: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.Grade 6
Colorado6.NS.B.4Students can: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.Grade 6
Colorado6.NS.C.5Students can: Explain why positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.Grade 6
Colorado6.NS.C.6Students can: Describe a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.Grade 6
Colorado6.NS.C.8Students can: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.Grade 6
Colorado7.EE.A.1Students can: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.Grade 7
Colorado7.EE.A.2Students can: Demonstrate that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related.Grade 7
Colorado7.EE.B.3Students can: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.Grade 7
Colorado7.EE.B.4Students can: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.Grade 7
Colorado7.SP.A.1Students can: Understand that statistics can be used to gain information about a population by examining a sample of the population; explain that generalizations about a population from a sample are valid only if the sample is representative of that population. Explain that random sampling tends to produce representative samples and support valid inferences.Grade 7
Colorado7.SP.A.2Students can: Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions.Grade 7
Colorado7.SP.B.3Students can: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability.Grade 7
Colorado7.SP.B.4Students can: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.Grade 7
Colorado7.SP.C.5Students can: Explain that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.Grade 7
Colorado7.SP.C.6Students can: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability.Grade 7
Colorado7.SP.C.7Students can: Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.Grade 7
Colorado7.SP.C.8Students can: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.Grade 7
Colorado7.G.A.1Students can: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.Grade 7
Colorado7.G.A.2Students can: Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.Grade 7
Colorado7.G.A.3Students can: Describe the two-dimensional figures that result from slicing three-dimensional figures, as in cross sections of right rectangular prisms and right rectangular pyramids.Grade 7
Colorado7.G.B.4Students can: State the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.Grade 7
Colorado7.G.B.5Students can: Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.Grade 7
Colorado7.G.B.6Students can: Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.Grade 7
Colorado7.RP.A.1Students can: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas, and other quantities measured in like or different units.Grade 7
Colorado7.RP.A.3Students can: Use proportional relationships to solve multistep ratio and percent problems.Grade 7
Colorado7.NS.A.1Students can: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.Grade 7
Colorado7.NS.A.2Students can: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.Grade 7
Colorado7.NS.A.3Students can: Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)Grade 7
Colorado8.EE.A.1Students can: Know and apply the properties of integer exponents to generate equivalent numerical expressions.Grade 8
Colorado8.EE.A.2Students can: Use square root and cube root symbols to represent solutions to equations of the form ùë•¬≤ = ùëù and ùë•¬≥ = ùëù, where ùëù is a positive rational number. Evaluate square roots of small perfect squares (up to 100) and cube roots of small perfect cubes (up to 64). Know that ‚àö2 is irrational.Grade 8
Colorado8.EE.A.3Students can: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other.Grade 8
Colorado8.EE.A.4Students can: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.Grade 8
Colorado8.EE.B.5Students can: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.Grade 8
Colorado8.EE.B.6Students can: Use similar triangles to explain why the slope ùëö is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation ùë¶ = ùëöùë• for a line through the origin and the equation ùë¶ = ùëöùë• + ùëè for a line intercepting the vertical axis at ùëè.Grade 8
Colorado8.F.A.1Students can: Define a function as a rule that assigns to each input exactly one output. Show that the graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required for Grade 8.)Grade 8
Colorado8.F.A.2Students can: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).Grade 8
Colorado8.F.A.3Students can: Interpret the equation ùë¶ = ùëöx + ùëè as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.Grade 8
Colorado8.F.B.4Students can: Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (ùë•, ùë¶) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.Grade 8
Colorado8.F.B.5Students can: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.Grade 8
Colorado8.SP.A.1Students can: Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.Grade 8
Colorado8.SP.A.2Students can: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.Grade 8
Colorado8.SP.A.3Students can: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Grade 8
Colorado8.SP.A.4Students can: Explain that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.Grade 8
Colorado8.G.A.1Students can: Verify experimentally the properties of rotations, reflections, and translations.Grade 8
Colorado8.G.A.2Students can: Demonstrate that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.Grade 8
Colorado8.G.A.3Students can: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.Grade 8
Colorado8.G.A.4Students can: Demonstrate that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.Grade 8
Colorado8.G.A.5Students can: Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.Grade 8
Colorado8.G.B.6Students can: Explain a proof of the Pythagorean Theorem and its converse.Grade 8
Colorado8.G.B.7Students can: Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.Grade 8
Colorado8.G.B.8Students can: Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.Grade 8
Colorado8.G.C.9Students can: State the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.Grade 8
Colorado8.NS.A.1Students can: Demonstrate informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Define irrational numbers as numbers that are not rational.Grade 8
Colorado8.NS.A.2Students can: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., ùúã¬≤).Grade 8
ColoradoHS.A-SSE.A.1Students can: Interpret expressions that represent a quantity in terms of its context.High School
ColoradoHS.A-SSE.A.2Students can: Use the structure of an expression to identify ways to rewrite it.High School
ColoradoHS.A-SSE.B.3Students can: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.High School
ColoradoHS.A-SSE.B.4Students can: Use the formula for the sum of a finite geometric series (when the common ratio is not 1) to solve problems.High School
ColoradoHS.A-APR.A.1Students can: Explain that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.High School
ColoradoHS.A-APR.B.2Students can: Know and apply the Remainder Theorem. For a polynomial ùëù(ùë•) and a number ùëé, the remainder on division by ùë• ‚Äì ùëé is ùëù(ùëé), so ùëù(ùëé) = 0 if and only if (ùë• ‚Äì ùëé) is a factor of ùëù(ùë•). (Students need not apply the Remainder Theorem to polynomials of degree greater than 4.)High School
ColoradoHS.A-APR.B.3Students can: Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.High School
ColoradoHS.A-APR.C.4Students can: Prove polynomial identities and use them to describe numerical relationships.High School
ColoradoHS.A-APR.C.5Students can: Know and apply the Binomial Theorem for the expansion of in powers of ùë• and ùë¶ for a positive integer ùëõ, where ùë• and ùë¶ are any numbers, with coefficients determined for example by Pascal‚Äôs Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.)High School
ColoradoHS.A-APR.D.6Students can: Rewrite simple rational expressions in different forms; write ùëé(ùë•)/ùëè(ùë•) in the form ùëû(ùë•) + ùëü(ùë•)/ùëè(ùë•), where ùëé(ùë•), ùëè(ùë•), ùëû(ùë•), and ùëü(ùë•) are polynomials with the degree of ùëü(ùë•) less than the degree of ùëè(ùë•), using inspection, long division, or, for the more complicated examples, a computer algebra system.High School
ColoradoHS.A-APR.D.7Students can: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expressions; add, subtract, multiply, and divide rational expressions.High School
ColoradoHS.A-CED.A.1Students can: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.High School
ColoradoHS.A-CED.A.2Students can: Create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales.High School
ColoradoHS.A-CED.A.3Students can: Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.High School
ColoradoHS.A-CED.A.4Students can: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.High School
ColoradoHS.A-REI.A.1Students can: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.High School
ColoradoHS.A-REI.A.2Students can: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.High School
ColoradoHS.A-REI.B.3Students can: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.High School
ColoradoHS.A-REI.C.5Students can: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.High School
ColoradoHS.A-REI.C.6Students can: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.High School
ColoradoHS.A-REI.C.7Students can: Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.High School
ColoradoHS.A-REI.C.8Students can: Represent a system of linear equations as a single matrix equation in a vector variable.High School
ColoradoHS.A-REI.C.9Students can: Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 √ó 3 or greater).High School
ColoradoHS.A-REI.D.10Students can: Explain that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).High School
ColoradoHS.A-REI.D.11Students can: Explain why the ùë•-coordinates of the points where the graphs of the equations ùë¶ =ùëì(ùë•) and ùë¶ = ùëî(ùë•) intersect are the solutions of the equation ùëì(ùë•) = ùëî(ùë•); find the solutions approximately e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where ùëì(ùë•) and/or ùëî(ùë•) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.High School
ColoradoHS.A-REI.D.12Students can: Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.High School
ColoradoHS.F-IF.A.1Students can: Explain that a function is a correspondence from one set (called the domain) to another set (called the range) that assigns to each element of the domain exactly one element of the range. If ùëì is a function and ùë• is an element of its domain, then ùëì(ùë•) denotes the output of ùëì corresponding to the input ùë•. The graph of ùëì is the graph of the equation ùë¶ = ùëì(ùë•).High School
ColoradoHS.F-IF.A.2Students can: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.High School
ColoradoHS.F-IF.A.3Students can: Demonstrate that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.High School
ColoradoHS.F-IF.B.4Students can: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.High School
ColoradoHS.F-IF.B.5Students can: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.High School
ColoradoHS.F-IF.B.6Students can: Calculate and interpret the average rate of change presented symbolically or as a table, of a function over a specified interval. Estimate the rate of change from a graph.High School
ColoradoHS.F-IF.C.7Students can: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.High School
ColoradoHS.F-IF.C.8Students can: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.High School
ColoradoHS.F-IF.C.9Students can: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).High School
ColoradoHS.F-BF.A.1Students can: Write a function that describes a relationship between two quantities.High School
ColoradoHS.F-BF.A.2Students can: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.High School
ColoradoHS.F-BF.B.3Students can: Identify the effect on the graph of replacing ùëì(ùë•) by ùëì(ùë•) + ùëò, ùëòùëì(ùë•), ùëì(ùëòx), and ùëì(ùë• + ùëò) for specific values of ùëò both positive and negative; find the value of ùëò given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.High School
ColoradoHS.F-BF.B.4Students can: Find inverse functions.High School
ColoradoHS.F-BF.B.5Students can: Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.High School
ColoradoHS.F-LE.A.1Students can: Distinguish between situations that can be modeled with linear functions and with exponential functions.High School
ColoradoHS.F-LE.A.2Students can: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).High School
ColoradoHS.F-LE.A.3Students can: Use graphs and tables to describe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.High School
ColoradoHS.F-LE.A.4Students can: For exponential models, express as a logarithm the solution to ùëéùëè·∂ú·µó = ùëë where ùëé, ùëê, and ùëë are numbers and the base ùëè is 2, 10, or ùëí; evaluate the logarithm using technology.High School
ColoradoHS.F-LE.B.5Students can: Interpret the parameters in a linear or exponential function in terms of a context.High School
ColoradoHS.F-TF.A.1Students can: Use radian measure of an angle as the length of the arc on the unit circle subtended by the angle.High School
ColoradoHS.F-TF.A.2Students can: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.High School
ColoradoHS.F-TF.A.3Students can: Use special triangles to determine geometrically the values to sine, cosine, tangent for ùúã/3, ùúã/4, and ùúã/6 and use the unit circle to express the values sine, cosine, and tangent for ùë•, ùúã + ùë•, and 2ùúã - ùë• and in terms of their values for ùë• where ùë• is any real number.High School
ColoradoHS.F-TF.A.4Students can: Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.High School
ColoradoHS.F-TF.B.5Students can: Model periodic phenomena with trigonometric functions with specified amplitude, frequency, and midline.High School
ColoradoHS.F-TF.B.6Students can: Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.High School
ColoradoHS.F-TF.B.7Students can: Use inverse function to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.High School
ColoradoHS.F-TF.C.8Students can: Prove the Pythagorean identity sin¬≤ (ùúÉ) + cos¬≤ (ùúÉ) = 1 and use it to find sin(ùúÉ), cos(ùúÉ), or tan(ùúÉ) given sin(ùúÉ), cos(ùúÉ), or tan(ùúÉ) and the quadrant of the angle.High School
ColoradoHS.F-TF.C.9Students can: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.High School
ColoradoHS.S-ID.A.1Students can: Model data in context with plots on the real number line (dot plots, histograms, and box plots).High School
ColoradoHS.S-ID.A.2Students can: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.High School
ColoradoHS.S-ID.A.3Students can: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).High School
ColoradoHS.S-ID.A.4Students can: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages and identify data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.High School
ColoradoHS.S-ID.B.5Students can: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.High School
ColoradoHS.S-ID.B.6Students can: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.High School
ColoradoHS.S-ID.B.7Students can: Distinguish between correlation and causation.High School
ColoradoHS.S-ID.C.7Students can: Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.High School
ColoradoHS.S-ID.C.8Students can: Using technology, compute and interpret the correlation coefficient of a linear fit.High School
ColoradoHS.S-IC.A.1Students can: Describe statistics as a process for making inferences about population parameters based on a random sample from that population.High School
ColoradoHS.S-IC.A.2Students can: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.High School
ColoradoHS.S-IC.B.3Students can: Identify the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.High School
ColoradoHS.S-IC.B.4Students can: Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.High School
ColoradoHS.S-IC.B.5Students can: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.High School
ColoradoHS.S-IC.B.6Students can: Evaluate reports based on data. Define and explain the meaning of significance, both statistical (using p-values) and practical (using effect size).High School
ColoradoHS.S-CP.A.1Students can: Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (‚Äúor,‚Äù ‚Äúand,‚Äù ‚Äúnot‚Äù).High School
ColoradoHS.S-CP.A.2Students can: Explain that two events ùê¥ and ùêµ are independent if the probability of ùê¥ and ùêµ occurring together is the product of their probabilities, and use this characterization to determine if they are independent.High School
ColoradoHS.S-CP.A.3Students can: Using the conditional probability of ùê¥ given ùêµ as ùëÉ(ùê¥ and ùêµ)/ùëÉ(ùêµ), interpret the independence of ùê¥ and ùêµ as saying that the conditional probability of ùê¥ given ùêµ is the same as the probability of ùê¥, and the conditional probability of ùêµ given ùê¥ is the same as the probability of ùêµ.High School
ColoradoHS.S-CP.A.4Students can: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.High School
ColoradoHS.S-CP.A.5Students can: Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.High School
ColoradoHS.S-CP.B.6Students can: Find the conditional probability of ùê¥ given ùêµ as the fraction of ùêµ‚Äôs outcomes that also belong to ùê¥, and interpret the answer in terms of the model.High School
ColoradoHS.S-CP.B.7Students can: Apply the Addition Rule, ùëÉ(ùê¥ or ùêµ) = ùëÉ(ùê¥) + ùëÉ(ùêµ) ‚Äì ùëÉ (ùê¥ and ùêµ), and interpret the answer in terms of the model.High School
ColoradoHS.S-CP.B.8Students can: Apply the general Multiplication Rule in a uniform probability model, ùëÉ(ùê¥ and ùêµ) = ùëÉ(ùê¥) ùëÉ(ùêµ ‚à£ ùê¥) = ùëÉ(ùêµ) ùëÉ(ùê¥ ‚à£ ùêµ), and interpret the answer in terms of the model.High School
ColoradoHS.S-CP.B.9Students can: Use permutations and combinations to compute probabilities of compound events and solve problems.High School
ColoradoHS.S-MD.A.1Students can: Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.High School
ColoradoHS.S-MD.A.2Students can: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.High School
ColoradoHS.S-MD.A.3Students can: Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.High School
ColoradoHS.S-MD.A.4Students can: Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.High School
ColoradoHS.S-MD.B.5Students can: Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.High School
ColoradoHS.S-MD.B.6Students can: Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).High School
ColoradoHS.S-MD.B.7Students can: Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).High School
ColoradoHS.G-CO.A.1Students can: State precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.High School
ColoradoHS.G-CO.A.2Students can: Represent transformations in the plane using e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).High School
ColoradoHS.G-CO.A.3Students can: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.High School
ColoradoHS.G-CO.A.4Students can: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.High School
ColoradoHS.G-CO.A.5Students can: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using appropriate tools (e.g., graph paper, tracing paper, or geometry software). Specify a sequence of transformations that will carry a given figure onto another.High School
ColoradoHS.G-CO.B.6Students can: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.High School
ColoradoHS.G-CO.B.7Students can: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.High School
ColoradoHS.G-CO.B.8Students can: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.High School
ColoradoHS.G-CO.C.9Students can: Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment‚Äôs endpoints.High School
ColoradoHS.G-CO.C.10Students can: Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180¬∞; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.High School
ColoradoHS.G-CO.C.11Students can: Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.High School
ColoradoHS.G-CO.D.12Students can: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.High School
ColoradoHS.G-CO.D.13Students can: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.High School
ColoradoHS.G-SRT.A.1Students can: Verify experimentally the properties of dilations given by a center and a scale factor.High School
ColoradoHS.G-SRT.A.2Students can: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.High School
ColoradoHS.G-SRT.A.3Students can: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.High School
ColoradoHS.G-SRT.B.4Students can: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.High School
ColoradoHS.G-SRT.B.5Students can: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.High School
ColoradoHS.G-SRT.C.6Students can: Explain that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.High School
ColoradoHS.G-SRT.C.7Students can: Explain and use the relationship between the sine and cosine of complementary angles.High School
ColoradoHS.G-SRT.C.8Students can: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.High School
ColoradoHS.G-SRT.D.9Students can: Derive the formula ùê¥ = ¬Ω ùëéùëè sin(ùê∂) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.High School
ColoradoHS.G-SRT.D.10Students can: Prove the Laws of Sines and Cosines and use them to solve problems.High School
ColoradoHS.G-SRT.D.11Students can: Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).High School
ColoradoHS.G-C.A.1Students can: Prove that all circles are similar.High School
ColoradoHS.G-C.A.2Students can: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.High School
ColoradoHS.G-C.A.3Students can: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.High School
ColoradoHS.G-C.A.4Students can: Construct a tangent line from a point outside a given circle to the circle.High School
ColoradoHS.G-C.B.5Students can: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.High School
ColoradoHS.G-GPE.A.1Students can: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.High School
ColoradoHS.G-GPE.A.2Students can: Derive the equation of a parabola given a focus and directrix.High School
ColoradoHS.G-GPE.A.3Students can: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.High School
ColoradoHS.G-GPE.B.4Students can: Use coordinates to prove simple geometric theorems algebraically.High School
ColoradoHS.G-GPE.B.5Students can: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).High School
ColoradoHS.G-GPE.B.6Students can: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.High School
ColoradoHS.G-GPE.B.7Students can: Use coordinates and the distance formula to compute perimeters of polygons and areas of triangles and rectangles.High School
ColoradoHS.G-GMD.A.1Students can: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri‚Äôs principle, and informal limit arguments.High School
ColoradoHS.G-GMD.A.2Students can: Give an informal argument using Cavalieri‚Äôs principle for the formulas for the volume of a sphere and other solid figures.High School
ColoradoHS.G-GMD.A.3Students can: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.High School
ColoradoHS.G-GMD.B.4Students can: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.High School
ColoradoHS.G-MG.A.1Students can: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).High School
ColoradoHS.G-MG.A.2Students can: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).High School
ColoradoHS.G-MG.A.3Students can: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).High School
ColoradoHS.N-RN.A.1Students can: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.High School
ColoradoHS.N-RN.A.2Students can: Rewrite expressions involving radicals and rational exponents using the properties of exponents.High School
ColoradoHS.N-RN.B.3Students can: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.High School
ColoradoHS.N-Q.A.1Students can: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.High School
ColoradoHS.N-Q.A.2Students can: Define appropriate quantities for the purpose of descriptive modeling.High School
ColoradoHS.N-Q.A.3Students can: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.High School
ColoradoHS.N-CN.A.1Students can: Define complex number ùëñ such that ùëñ¬≤ = ‚Äì1, and show that every complex number has the form ùëé + ùëèùëñ where ùëé and ùëè are real numbers.High School
ColoradoHS.N-CN.A.2Students can: Use the relation ùëñ¬≤ = ‚Äì1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.High School
ColoradoHS.N-CN.A.3Students can: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.High School
ColoradoHS.N-CN.B.4Students can: Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.High School
ColoradoHS.N-CN.B.5Students can: Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.High School
ColoradoHS.N-CN.B.6Students can: Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.High School
ColoradoHS.N-CN.C.7Students can: Solve quadratic equations with real coefficients that have complex solutions.High School
ColoradoHS.N-CN.C.8Students can: Extend polynomial identities to the complex numbers.High School
ColoradoHS.N-CN.C.9Students can: Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.High School
ColoradoHS.N-VM.A.1Students can: Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., ùíó, |ùíó|, ||ùíó||, ùë£).High School
ColoradoHS.N-VM.A.2Students can: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.High School
ColoradoHS.N-VM.A.3Students can: Solve problems involving velocity and other quantities that can be represented by vectors.High School
ColoradoHS.N-VM.B.5Students can: Multiply a vector by a scalar.High School
ColoradoHS.N-VM.C.6Students can: Use matrices to represent and manipulate data, e.g., as when all of the payoffs or incidence relationships in a network.High School
ColoradoHS.N-VM.C.7Students can: Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.High School
ColoradoHS.N-VM.C.9Students can: Understand that, unlike the multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.High School
ColoradoHS.N-VM.C.10Students can: Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.High School
ColoradoHS.N-VM.C.11Students can: Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimension to produce another vector. Work with matrices as transformations of vectors.High School
ColoradoHS.N-VM.C.12Students can: Work with 2 √ó 2 matrices as transformations of the plane and interpret the absolute value of the determinant in terms of area.High School
ConnecticutK.CC.A.1Count to 100 by ones and by tens.Kindergarten
ConnecticutK.CC.A.2Count forward beginning from a given number within the known sequence (instead of having to begin at 1).Kindergarten
ConnecticutK.CC.A.3Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).Kindergarten
ConnecticutK.CC.B.4Understand the relationship between numbers and quantities; connect counting to cardinality.Kindergarten
ConnecticutK.CC.B.5Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.Kindergarten
ConnecticutK.CC.C.6Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.Kindergarten
ConnecticutK.CC.C.7Compare two numbers between 1 and 10 presented as written numerals.Kindergarten
ConnecticutK.G.A.1Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.Kindergarten
ConnecticutK.G.A.2Correctly name shapes regardless of their orientations or overall size.Kindergarten
ConnecticutK.G.A.3Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).Kindergarten
ConnecticutK.G.B.4Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length).Kindergarten
ConnecticutK.G.B.5Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.Kindergarten
ConnecticutK.G.B.6Compose simple shapes to form larger shapes.Kindergarten
ConnecticutK.MD.A.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.Kindergarten
ConnecticutK.MD.A.2Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference.Kindergarten
ConnecticutK.MD.B.3Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.Kindergarten
ConnecticutK.NBT.A.1Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.Kindergarten
ConnecticutK.OA.A.1Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.Kindergarten
ConnecticutK.OA.A.2Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.Kindergarten
ConnecticutK.OA.A.3Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).Kindergarten
ConnecticutK.OA.A.4For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.Kindergarten
ConnecticutK.OA.A.5Fluently add and subtract within 5.Kindergarten
Connecticut1.G.A.1Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.Grade 1
Connecticut1.G.A.2Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.Grade 1
Connecticut1.G.A.3Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.Grade 1
Connecticut1.MD.A.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.Grade 1
Connecticut1.MD.A.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.Grade 1
Connecticut1.MD.B.3Tell and write time in hours and half-hours using analog and digital clocks.Grade 1
Connecticut1.MD.C.4Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.Grade 1
Connecticut1.NBT.A.1Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.Grade 1
Connecticut1.NBT.B.2Understand that the two digits of a two-digit number represent amounts of tens and ones.Grade 1
Connecticut1.NBT.B.3Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.Grade 1
Connecticut1.NBT.C.4Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.Grade 1
Connecticut1.NBT.C.5Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.Grade 1