Order of Operations

Students are given a challenging expression that exposes common misconceptions about order of operations. The goal is to demonstrate that a brittle, fixed notion of order of operations is insufficient, and lead students to a deeper understanding of order of operations as a grammatical device. The Circles of Evaluation are introduced as "sentence diagramming for arithmetic".

Humans use verbs like "throw", "run", "build" and "jump" to describe operations on nouns like "ball", "puppy", and "blocks". Mathematics has "operations" and functions, like addition and subtraction. Just as you can "throw a ball", a person can also "add four and five".

A mathematical expression is an instruction for doing something, which specifies the verbs and nouns involved. The expression 4 + 5$\displaystyle 4 + 5$ tells us to add 4 and 5. To evaluate an expression, we follow the instructions. The expression 4 + 5$\displaystyle 4 + 5$ evaluates to 9.

If you were to write instructions for getting ready for school, it would matter very much which instruction came first: putting on your socks, putting on your shoes, etc. Sometimes we need multiple expressions in mathematics, and the order matters there, too!

🖼Show image Mathematicians didn’t always agree on the order of operations, but at some point it became important to develop rules to help them work together.

The pyramid on the right is a model for summarizing the order of operations. When evaluating an expression, we begin by applying the operations written at the top of the pyramid (operations in parentheses and other grouping symbols). Only after we have completed all of those operations can we move down to the lower level. If both operations from the same level are present (as in 4 + 2 - 1$\displaystyle 4 + 2 - 1$), we read the expression from left to right, applying the operations in the order in which they appear. This set of rules is brittle, and doesn’t always make it clear what we need to do. Mnemonic devices for the order of operations like PEMDAS, GEMDAS, etc focus on how to get the answer. What we need is a better way to read math.

Instead of using a rule for computing answers, let’s start by diagramming the math itself!

That means that Numbers (e.g. - 3$\displaystyle 3$, -29$\displaystyle -29$, 77.01$\displaystyle 77.01$…​) are still written by themselves. It’s only when we want to do something like add, subtract, etc. that we need to draw a Circle.

If we want to draw the Circle of Evaluation for 6 ÷ 3$\displaystyle 6 \div 3$, the division function (/) is written at the top, with the 6 on the left and the 3 on the right.

What if we want to use multiple functions? How would we draw the Circle of Evaluation for 6 ÷ ( 1 + 2 )$\displaystyle 6 \div ( 1 + 2 )$? Drawing the Circle of Evaluation for the 1 + 2$\displaystyle 1 + 2$ is easy. But how do we take the result of that circle, and divide 6 by it?

We basically replace the 3 from our earlier Circle of Evaluation with another Circle, which adds 1 and 2!

Aside from helping us catch mistakes before they happen, Circles of Evaluation are also a useful way to think about transformation in mathematics. For example, you may have heard that "addition is commutative, so a + b$\displaystyle a + b$ can always be written as b + a$\displaystyle b + a$." For example, 1 + 2$\displaystyle 1 + 2$ can be transformed to 2 + 1$\displaystyle 2 + 1$.

Suppose another student tells you that 1 + 2 × 3$\displaystyle 1 + 2 \;\times\; 3$ can be rewritten as 2 + 1 × 3$\displaystyle 2 + 1 \;\times\; 3$. This is obviously wrong, but why?

Take a moment to think: what’s the problem? We can use the Circles of Evaluation to figure it out!

The first Circle is just the original expression. The second expression represents what the (incorrect) commutativity transformation gives us:

In this case, the student failed to see the structure, viewing the term to the right of the +$\displaystyle +$ sign as 2$\displaystyle 2$ instead of 2 × 3$\displaystyle 2 \;\times\; 3$. The Circles of Evaluation help us see the structure of the expression, rather than forcing us to construct it and keep it in our heads.

Spend some time ensuring that students have drawn their circles correctly. You may want to have them compare their circles with a partner and another pair of partners or you may want to post an answer key. Students will use their circles to write code in the next segment of the lesson - so this step is crucial.

You may also want to have students complete Completing Circles of Evaluation from Arithmetic Expressions, Matching Circles of Evaluation and Arithmetic Expressions and/or Matching Circles of Evaluation to Expressions (Desmos).

Students learn how to use the Circles of Evaluation to translate arithmetic expressions into code.

When converting a Circle of Evaluation to code, it’s useful to imagine a spider crawling through the circle from the left and exiting on the right. The first thing the spider does is cross over a curved line (an open parenthesis!), then visit the operation - also called the function - at the top. After that, she crawls from left to right, visiting each of the inputs to the function. Finally, she has to leave the circle by crossing another curved line (a close parenthesis).

All of the expressions that follow the function name are called arguments to the function. The following diagram summarizes the shape of an expression that uses a function. 🖼Show image

Arithmetic expressions involving more than one operation, will end up with more than one circle, and more than one pair of parentheses.

If you have time, start with the two pages in the student workbook that scaffold translating circles to code: Completing Partial Code from Circles of Evaluation and Matching Circles of Evaluation & Code.

Note: Translate Arithmetic to Circles of Evaluation & Code (Intro) offers students the scaffold of extra parentheses. Those scaffolds drop away on Translate Arithmetic to Circles of Evaluation & Code 2.

There is one page of more complex problems - Arithmetic Expressions to Circles of Evaluation & Code - Challenge - so that you’re ready to challenge students who fly. Make sure these students know that we use sqrt as the name of the square root function, and sqr as the function that squares its input.

Have students share back what they learned from the Circles of Evaluation.

Circles of Evaluation are a powerful tool that can be used without ever getting students on computers. If you have time to introduce students to the WeScheme, typing their code into the Interactions Area gives students a chance to get feedback on their use of parentheses as well as the satisfaction of seeing their code successfully evaluate the expressions they’ve generated.

Now that we understand the structure of Circles of Evaluation, we can use them to write code for any function!