Students learn to model arithmetic expressions with a visual tool for order of operations, known as "Circles of Evaluation".

 Prerequisites Relevant StandardsOK Select one or more standards from the menu on the left (⌘-click on Mac, Ctrl-click elsewhere). Oklahoma Standards OK.A1.A.3 Generate equivalent algebraic expressions and use algebraic properties to evaluate expressions and arithmetic and geometric sequences. OK.PA.A.3 Generate equivalent numerical and algebraic expressions and use algebraic properties to evaluate expressions. Lesson Goals Students will be able to: Model an arithmetic expression using Circles of Evaluation. Translate Circles of Evaluation into code. Student-facing Goals I can write Circles of Evaluation for a given arithmetic expression. I can translate a Circle of Evaluation model into code. I can use numbers and operations in a programming environment. Materials Preparation Make sure all materials have been gathered Supplemental Resources Key Points For The Facilitator Error messages are the computer trying to give us a clue that something is wrong. Model reacting to error messages with interest to demonstrate to students that the messages are a helpful tool. After the first few exercises in creating Circles of Evaluation, ask students whether they create them from the 'inside-out' (drawing the inner circles first) or from the 'outside-in.' After they’ve given their responses, have them try using the OTHER way! Up until now, we didn’t have a visual spatial model for explaining the order of operations. Ask students to compare Circles of Evaluation to previous methods they’ve learned (PEMDAS, GEMAS, etc) For a memory hook, model the "bug that crawls through the circle" explanation. Students may benefit from using multiple colors to distinguish between the different smaller expressions and parentheses.

Glossary
circle of evaluation

a diagram of the structure of an expression (arithmetic or code)

definitions area

the left-most text box in the Editor where definitions for values and functions are written

editor

software in which you can write and evaluate code

error message

information from the computer about errors in code

expression

a computation written in the rules of some language (such as arithmetic, code, or a Circle of Evaluation)

function

a mathematical object that consumes inputs and produces an output

interactions area

the right-most text box in the Editor, where expressions are entered to evaluate

value

a specific piece of data, like 5 or "hello"

🔗Numbers 10 minutes

Overview

Students experiment with the Editor, exploring the different kinds of numbers and how they behave in this programming language.

Launch

This screen is called the Editor, and it looks something like the diagram you see here. There are a few buttons at the top, but most of the screen is taken up by two large boxes: the Definitions Area on the left and the Interactions Area on the right.

The Definitions Area is where programmers define values and functions that they want to keep, while the Interactions Area allows them to experiment with those values and functions. This is like writing function definitions on a blackboard, and having students use those functions to compute answers on scrap paper.

For now, we will only be writing programs in the Interactions Area on the right.

Investigate

Math is a language, just like English, Spanish, or any other language. We use nouns, like "bread", "tomato", "mustard" and "cheese" to describe physical objects. Math has values, like the numbers 1, 2 or 3, to describe quantities.

Try typing the number 42 on the right, and then hitting "Enter" or "Return". What did this number evaluate to? (Hint: Numbers should evaluate to themselves - if you didn’t get back the same number you put in, something is very wrong!)

If working in pairs, make sure you each take a turn at the keyboard. Suggestions:

• How large of a number can you enter?

• How small of a number can you enter?

• What happens if you type two numbers on the same line?

• Do fractions work? Decimals?

• Do negative numbers work?

Remember, we’re only trying numbers for now, not operations like 3 - 6$\displaystyle 3 - 6$, √16$\displaystyle \sqrt{16}$ or 4^2$\displaystyle 4^2$

 Notice & Wonder In pairs, students will each try entering a variety of numbers in the Interactions Area, hitting "Enter" each time to see what the computer does. Then they will write down what they Notice and Wonder on Notice and Wonder (Page 2).
• What did you Notice? What do you Wonder?

• Did you get any error messages? If so, read it carefully - what do you think it means?

Student Misconceptions

• Students who write decimals as .5 (without the leading zero) may get an error message, causing them to think that Pyret doesn’t have decimals! They just need to add the zero.

• Students who try division by writing 3/2 and get an answer may falsely assume that they’ve performed division. In fact, what they’ve done is entered a rational number. ("Two-thirds" is equivalent to the expression "two divided by three", but only insofar as they result in the same value. "2" is equivelent to expression "10 minus 8", for the same reason!)

• Rational numbers can be converted back and forth between fraction and decimal forms by clicking on them.

Synthesize

Our programming language knows about many types of numbers, and they behave pretty much the way they do in math. Our Editor is also pretty smart, and can automatically switch between showing a rational number as a fraction or a decimal, just by clicking on it!

🔗Order of Operations 30 minutes

Overview

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 not good enough, 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".

Launch

Humans also use verbs like "throw", "run", "build" and "jump" to describe operations on these nouns. Mathematics has functions - or "operations" - like addition and subtraction, which are operations performed on values. Just as you can "spread mustard on bread", a person can also "add four and five".

A mathematical expression is like a sentence: it’s an instruction for doing something. The expression 4+5 tells us to add 4 and 5. To evaluate an expression, we follow the instructions in the expression. The expression 4 + 5$\displaystyle 4 + 5$ evaluates to 9.

Sometimes, we need multiple expressions to accomplish a task, and it will matter in which order they come. For exmple, if you were to write instructions for making a sandwich, it would matter very much which instruction came first: melting the cheese, slicing the bread, spreading the mustard, etc. The order of functions matters in mathematics, too.

Mathematicians didn’t always agree on the order of operations, but now we have a common set of rules for how to evaluate expressions. The pyramid on the right summarizes the order. When evaluating an expression, we begin by applying the operations written at the top of the pyramid (multiplication and division). Only after we have completed all of those operations can we move down to the lower level. If both operations 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.

But this set of rules is brittle, and doesn’t always make it clear what we need to do. Check out the expression below. What do you think the answer is? This math problem went viral on social media recently, with math teachers arguing about what the answer was! Why might they disagree on the solution?

6 ÷ 2(1 + 2)$\displaystyle 6 \div 2(1 + 2)$

Order of Operations mneumonic devices like PEMDAS, GEMDAS, etc focus on how to get the answer. What we need is a better way to read math.

Instead of a rule for computing answers, let’s start by diagramming the math itself! We can draw the structure of this grammer in mathematics using something called the Circles of Evaluation. The rules are simple:

1) Every Circle must have one - and only one! - function, written at the top

That means that Numbers (e.g. - 3, -29, 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.

2) The inputs to the function are written left-to-right, in the middle of the 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.

/
 6 3
(/ 6 3)

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 divide 6 by that circle?

Circles can contain other Circles

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

/
6
+
 1 2
(/ 6 (+ 1 2))

If you’d like to have students practice connecting expressions with Circles of Evaluation before you move on to talking about code, turn to Completing Circles of Evaluation from Arithmetic Expressions (2) (Page 3), Creating Circles of Evaluation from Arithmetic Expressions (3) (Page 4), and/or Matching Circles of Evaluation and Arithmetic Expressions (Page 5) in the workbook.

Circles of Evaluation help us write 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).

Expression

3 + 8$\displaystyle 3 + 8$

Circle of Evaluation

+
 3 8
(+ 3 8)

Code

3 + 8

Practice creating Circles of Evaluation using the common operators (+, -, *, /).

• Do spaces matter when typing in functions?

• Does the order of the numbers matter in the functions? Which functions?

• What do the error messages tell us?

• What connections do you see between the expression, circle, and code?

Expression

2 × (3 + 8)$\displaystyle 2 \;\times\; (3 + 8)$

Circle of Evaluation

​*
2
+
 3 8
(​* 2 (+ 3 8))

Code

2 ​* (​3 + 8​)

Circles of Evaluation help us get the correct answer

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 "any subtraction can be transformed to a negative addition." For example, 1 - 2$\displaystyle 1 - 2$ can be transformed to 1 + -2$\displaystyle 1 + (-2)$.

Suppose someone tells you that 1 - 2 * 3 + 4$\displaystyle 1 - 2 * 3 + 4$ can be rewritten as 1 + -2 * 3 + 4$\displaystyle 1 + (-2) * 3 + 4$. These two expressions will definitely give us the same answer, but this transformation is actually incorrect! It doesn’t use the negative addition rule at all! 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 multiplication happens first, so let’s see how multiplication changes this circle:

+
​-
1
​*
 2 3
4
(+ (​- 1 (​* 2 3)) 4)

multiplication

+
​-
 1 6
4
(+ (​- 1 6) 4)

As you can see, replacing the subtraction with a negative addition happens to the result of the multiplication. We can’t actually change the 2 into a -2, because it isn’t actually being subtracted from 1!

Sure, we got the same answer - but that doesn’t mean the way we got it was correct. If all that mattered was getting the right answer, we could just as easily have replaced the whole expression with 5 - 6$\displaystyle 5 - 6$. And that is definitely not a correct transformation!

Any time you make a transformation in math (replacing 10 - 2$\displaystyle 10 - 2$ with 8$\displaystyle 8$ because of subtraction, or replacing 2 + 6$\displaystyle 2 + 6$ with 6 + 2$\displaystyle 6 + 2$ because of commutativity), you need to make sure the transformation is correct. The Circles of Evaluation help us see these transformation visually, rather than forcing us to keep them in our heads.

 Circles of Evaluation The Circles of Evaluation are a critical pedagogical tool in this course. They place the focus on the structure of mathematical expressions, as a means of combating the harmful student belief that the only thing that matters is the answer. They can be used to diagram arithmetic sentences to expose common misconceptions about Order of Operations, and make an excellent scaffold for tracing mistakes when a student applies the Order of Operations incorrectly. They are also a bridge representation, which naturally connects to function composition and converting arithmetic into code.

Investigate

The Circles of Evaluation are a great way to visualize other functions you already know, such as square and square root!

Note: In Pyret, we treat operators like +, -, *, and / differently - they are written in between their inputs, just like in math. We also use letters instead of symbols for function names, so taking the square root is written as num-sqrt and squaring is written as num-sqr.

 Strategies For English Language Learners MLR 7 - Compare and Connect: Gather students' graphic organizers to highlight and analyze a few of them as a class, asking students to compare and connect different representations.

🔗Closing

Have students share back what they learned from the Circles of Evaluation. You may want to assign traditional Order of Operations problems from your math book, but instead of asking them simply to compute the answer - or even list the steps - have them draw the circle.