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

Prerequisites

Coordinates

Relevant Standards

Select one or more standards from the menu on the left (⌘-click on Mac, Ctrl-click elsewhere).

Common Core State Statements
HSA.SSE.A.2

Use the structure of an expression to identify ways to rewrite it.

HSA.SSE.B

Write expressions in equivalent forms to solve problems.

MP.1

Make sense of problems and persevere in solving them

MP.2

Reason abstractly and quantitatively

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)

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)

Warmup

Students should be logged into WeScheme and have their workbooks with a pen or pencil.

Numbers 10 minutes

Overview

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

Launch

Have students explore numbers - just numbers! - in the Interactions window. Make sure both students take a turn at the keyboard.

Investigate

Notice & Wonder

In pairs, students will each try entering a variety of numbers, then together they will write down what they Notice and what they Wonder.

  • What type of numbers did you try?

  • What did you notice? What did you wonder?

  • Did anyone try any irrational numbers? (For example: pi, e)

  • Did anyone try fractions? Decimals?

  • Did anyone try negative numbers?

  • Did you get any error messages?

  • Did anyone try to add, subtract, multiply, or divide numbers?

  • What was the error message?

Student Misconceptions

  • 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.

  • 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 seem to behave 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!

Arithmetic 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 PEMDAS as a grammatical device. The Circles of Evaluation are introduced as "sentence diagramming for arithmetic".

Launch

Have students evaluate the expression in the image below and gather a few responses from the students. Have they seen a problem like this on social media before? Why might we be disagreeing on the solution? How do we know how to evaluate this expression?

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 PEMDAS incorrectly. They are also a bridge representation, which naturally connects to function composition and converting arithmetic into code.

Math is a language, and all languages have rules and grammar. We can draw the structure of those rules 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 , the division function (/) is written at the top, with the 6 on the left and the 3 on the right.

(/ 6 3)

What if we want to use multiple functions? How would we draw the Circle of Evaluation for ? Drawing the Circle of Evaluation for the 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))

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 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

→

Circle of Evaluation

→

(+ 3 8)

Code

→

(+ 3 8)

Have students practice creating Circles of Evaluation using the 6 functions(+, -, *, /, sqr, sqrt).

  • 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

→

Circle of Evaluation

→

(​* 2 (+ 3 8))

Code

→

(* 2 (+ 3 8))

  • Why are there two closing parentheses in a row, at the end of the code?

  • If an expression has three sets of parentheses, how many Circles of Evaluation do you expect to need?

Investigate

Students will complete Practicing Circles (Page 10) page in their workbook. They should draw all of the Circles first and check their work, before converting to code.

Option 1: Have students practice moving between Circles of Evaluation and code with a station review.

  • Convert the expression into a Circle of Evaluation.

  • Convert the Circle of Evaluation model into code.

Once the students feel confident in their work, they can enter the code into their code editor to test it out.

Option 2: Have students complete the Practicing Circles 2 (Page 11) and / or the Practicing Circles with Square Roots (Page 12) with their partners and test their code in their coding environment.

Students will create a Circle of Evaluation based on an expression they’ve created using at least 4 of the 6 functions: +, -, *, /, sqr, sqrt.

Using this graphic organizer, (1) create the code that represents this Circle of Evaluation, (2) translate this into code, (3) evaluate the expression using the order of operations, and (4) then compare and contrast the three methods.

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.

Additional Exercises

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, and 1738598). CCbadge BS:Games by Jen Poole, Emmanuel Schanzer, Ed Campos Jr, and Dorai Sitaram is licensed under a Creative Commons 4.0 Unported License. Based on a work at www.BootstrapWorld.org. Permissions beyond the scope of this license may be available by contacting schanzer@BootstrapWorld.org.