# Change Maker: Order of Operations

I came up with this idea a couple weeks ago, then I used the game this week in my 6th grade general classes.  It might not be an original idea or a game of a similar nature might already exist for practicing order of operations; however, I wanted to share the experience of my students.

Basically, I wanted to introduce students to the order of operations (groupings excluded) in a way that did not appeal to some rambling along the lines of, “multiplication is repeated addition, exponents are repeated multiplication, subtraction is the inverse of addition, division is the inverse of multiplication.”  While the conversations and work in class during the following days touched on these ideas, I wanted students to recognize that some operations are more powerful than others because of how much they change a value.  I knew almost of my students probably remembered PEMDAS, but I knew they probably couldn’t give me an explanation why the acronym is true other than saying, “That’s just what it is.”  Furthermore, I wanted to begin the process of nixing this trick.  My thought was if I gave students a simple experience that made the idea of change to a value evident, students would sort the strength of operations themselves.

Long story short, I created the imperfect game known as Change Maker.  In each round, students worked with the same numbers rolled by one member of a duo; however, each student rolled another die that corresponded to different operations (addition, subtraction, multiplication, division, exponents, your choice).  Whoever changed the starting value more would win a round.  I was a tad worried about the game because I needed to clarify how to calculate change in value.  I was also apprehensive because I knew from my trial plays that sometimes the amount of change created by an operation did not mesh with the order of operations perfectly.  For instance, a student who got 8+1 for a round had a change of 1 compared to the student who got 8/1 for a change of 0.  If a pair of students got these type of rolls frequently, their conclusion would be that addition is stronger than division.  My hope was with 24 students each playing about 20 rounds of this game, these kind of rolls would be an irregular event.

In observing two classes and interacting with many students as they played this game, I was pleased to find that my concerns were a bit exaggerated.  Once students had played about 10 rounds, I kept hearing similar answers as I questioned students about their strategy for the game. “Get exponents.”  Why?  “It changes the starting number more.”  I asked students how their strategy would change if exponents were not allowed.  “Get multiplication.”  Which operations change a number the least?  “Addition and subtraction.”

The game was a great chance for students to practice calculation skills, but more than this practice I was happy to find the order of operations began to rise naturally in students as they played this game.  When I discussed the game with the entire class later in the day, students were able to share the correct order of operations based on the results of their game.  I had to interject to reveal the equal strength of multiplication and division, but everything else was mathematically accurate.  Later in the week, I was able to use the game as a reference when students were completing order of operations problems.  For instance, I would use a hint such as, “Think back to the game we played.  Would you want to add or divide to create more change?”

I know it’s an imperfect game, but does anything else exist that helps students construct the order of operations?  Let me know!