Can we break the order of operations?

Oct 25

The order of operations, they are such a straight-forward math topic, right? Or are they?

Here is how I used to approach order of operations:

First, we solve everything inside grouping symbols. Then, we evaluate exponents. Then we complete multiplication and division, moving left to right. Finally we do all the addition and subtraction, moving left to right.

I gave students some mnemonics to help them remember the order. I insisted they complete all operations in that order so they would get the correct answer. Simple as that.

But… I realized there are several problems with this approach:

Students aren’t encouraged to think flexibly (or think at all, really).
We completely ignore properties while learning order of operations.
Order of operations are just one way to simplify an expression (not the only way)!

Here are a couple of common examples of when we do not have to use order of operations:

The distributive property allows students to multiply before evaluating what is inside a grouping symbol.
The commutative property of addition allows students to add in any order they choose, not left to right. (And the same for the commutative property of multiplication).

How about an example that we rarely talk about: We can complete all subtraction before addition. Try it!

But wait… if I show students this, won’t it confuse them?

I used to avoid confusion, but we shouldn’t. First of all, confusion is part of learning. First you are confused, but then you figure it out and learn. We don’t need to avoid confusion. Secondly, avoiding topics like these may lead to misconceptions further down the road.

Let’s let students grapple with breaking order of operations and make sense of when it is okay to do so.

How do we explain why it is okay to subtract before adding?

To explain this with a model, imagine a jar with 10 jellybeans. We want to subtract 7 jellybeans, add 8 jellybeans, and subtract 1 jellybean. Will the order affect the final quantity? No.

For older students who understand negatives, we can rewrite the expression, changing all subtraction to adding a negative. Then we can use the commutative property to rearrange how we choose.

Students are capable of thinking about math. What if we show them multiple paths to simplifying expressions and let them discuss if they agree or disagree? This practice will help them think flexibly when simplifying expressions.

The example problem is from a resource that gets students thinking about the order or operations and when we can break them. See more details here.