order-of-op

Creating My Dream Lessons…Digitally.

While creating math lessons for middle school, I’ve been reminded of a game I used to play when I was in eighth grade. My math teacher, Mr. W, would start by giving us a target number (such as 27), and four other numbers (such as 2, 3, 3, 6). Our challenge was to use basic operations and all four numbers to create an expression equal to 27 (with extra points for being the first to finish). Inevitably, someone would run to the board and write 2 + 3 × 6 – 3. They were initially surprised to find out they were wrong. It would be correct, if expressions simplified left-to-right. But they don’t. Mr. W would get frustrated that students weren’t using parentheses, and students would get frustrated at losing the first-place bonus points.

One of the Order of Operations digital tools our team at DreamBox developed is similar to this game. But—like all of our DreamBox lessons—there are innovative and transformative representations that help students make sense of concepts in new, deeper ways. In DreamBox, students can choose from a set of integers, perhaps 2 through 7, and they are given an unfinished equation to complete. They might see __ + 6 × __ = 32. Beneath the equation is an array of lights that represents the values and expression each student creates. This array of lights is a unique representation developed by DreamBox that connects students with the order of operations in a way that can’t be achieved with a worksheet or a multiple choice item. Like the first student to run to the board in Mr. W’s class, some students may try 2 + 6 × 4 to get 32, thinking that it will simplify left-to-right. However, the lights will first create a 6 by 4 array, and then add 2 more lights, for a total of 26, not 32. The student can then change the expression to 2 + 6 × 5 to get a correct answer.

In later levels, as students become more fluent with the order of operations, they use equations with division, parentheses, exponents, and negative values. In the parentheses lessons, they may once again see __ + 6 × __ = 32, but this time they’ll have the option to set a pair of parentheses and create (2 + 6) × 4 to produce an 8 by 4 array for a correct answer.

When students use this tool, I know they’re learning more than just the order of operations. These lessons are also DreamBox’s way of introducing exponents. A student may only have the values 2 through 7 on hand, but to get to a large target value they may try squaring 7 (to get 49) or cubing 4 (to get 64). Students get several chances to generate an acceptable equation, and some of the fun is observing what happens with a correct answer. I hope students have the courage to play with exponents, maybe even to try 75 and receive the feedback: “There are not enough lights to represent these values. You created an expression that is equal to 16,807. Woah! Exponents are crazy!” In later levels, the students can try (–3)5, and watch as it multiplies –3 again and again. The lights will change between positive and negative values as the expression scales by –3 each time.

This interactivity, exploration, and sense-making are what I love about the lessons we design at DreamBox. There are multiple ways to get a right answer, but even wrong answers are an opportunity to explore, enjoy, and ultimately learn mathematics. One of the best things about designing software to create innovative digital manipulatives is seeing my dream lessons come to life.

Joe Trahan

Peace Corps Volunteer in Guinea, West Africa | MEd in Secondary Mathematics from GWU, Washington DC | 6-year teacher of Mathematics in Bethesda, MD