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Integer Operation

Numbers to Twenty on the MathRack

Each month we will be featuring one of our FREE Teacher Tools for interactive whiteboards. Ready for use in elementary classrooms with any type of interactive whiteboard, these high-quality virtual manipulative tools help teachers connect students with great ways to make sense of math. Use these tools to create an environment for students to explain, discuss, and defend their mathematical thinking.

This month we are featuring “Multiplication: Open Arrays” for Grade 4.

This DreamBox lesson ensures students connect partial products to a concrete area context on the open array. Students learn to efficiently choose and solve partial products that enable better mental multiplication with multi-digit numbers. These lessons teach students to use multiples of 10 and the distributive property as powerful strategies for multiplying and efficient mental computation.

Sample Lesson

Students compose arrays and use partial products to solve multi-digit multiplication problems, compute the area of a rectangle and learn the distributive property. 


The open array model and the DreamBox lesson progression are designed so students learn the distributive property. Specifically, students learn to create smaller arrays using multiples of ten to compose the larger array. Prior to this lesson, students should know multiplication relationships and facts, and have used basic arrays to solve problems and explain multiplication strategies.


1. The teacher begins by listing the math string below one at a time. She asks students to find the sum in their head (mental math). She calls on individual students to discuss their strategies before the subsequent problem is presented.

6 x 10
60 x 10
12 x 2
12 x 20
120 x 20

2. The teacher then draws a rectangle on the board and labels it as shown below.

The teacher explains, “We can represent multiplication problems using arrays; however, with large arrays, we can divide the rectangle into smaller arrays using landmark numbers to find the answer. Draw this rectangle on the paper at your seat and divide it.  Share your drawing with a partner and discuss how you divided your rectangle.” 

After partners share their responses, ask a student to come up to the board and demonstrate how he divided the rectangle and explain his reasoning.   

Possible answers:

a)  “I would decompose the 23 into 20 and 3 because I know the multiplication fact of 9 x 2 and can find 9 x 20 like we did in the number string above.”  

b) “23 because I don’t know my multiplication tables to 23. I would choose 20 because I can multiply 2 x 9.”

c) “I would choose 23 and break it into 2 tens and a 3. I can multiply 9 x 10 easily.”

3. The teacher summarizes, “We can add the products of the smaller arrays together to get the answer to the larger array. The “helper” equations enable us to find the final product of the more difficult equation.”

9 x 23 = (9 x 10) + (9 x 10) + (9 x 3)
           =       90     +     90     +    27
           =                      207

4. Bring up the DreamBox interactive white board lesson.  

5. The teacher explains, “We can use the same strategy to solve two-digit by two-digit multiplication problems as well.” Ask a student to come to the white board and pull the zipper from the corner of the rectangle. Ask students to describe what is happening as the student pulls the zipper. 

6. Ask the student to sit down and request another volunteer to create the first array. After the student has ‘unzipped’ the first array, ask him why he chose to create that array? Ask the student to click done and return to his seat.

7. When the first helper equation appears, ask for another volunteer to solve and type the product of that equation and click done. If the answer is correct, ask the student to explain how he solved that problem in his head and then return to his seat. If incorrect, guide the student to the correct response using questioning.

8. Ask for another volunteer to create the next array and repeat steps 6 and 7 until the large rectangle is covered and all helper equations have been solved. 

9. Ask the class to think about what should be done next and how the helper equations could be used to find the larger product. Then discuss their strategy with a partner. After students have discussed their strategies, ask for a volunteer to explain their strategy and come up to the board to type the final product. 

10. Repeat the strategy discussion with the new problem and repeat steps 6-9. 


Common Core State Standard
Grade CCSS ID Domain Cluster Standard
4 4.NBT.5 Number and Operations in Base Ten Use place value understanding and properties of operations to perform multi-digit arithmetic 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.
4 4.SMP.7 Standards for Mathematical Practice   Look for and make use of structure.

Try the lesson in your classroom today!