(Also available in Pyret)
Students discover the Commutative Property by way of Circles of Evaluation.
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Student-facing Lesson Goals |
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🔗The Commutative Property
Overview
Students examine pairs of Circles of Evaluation to see whether they represent equivalent expressions. The Circles of Evaluation reveal the structural changes that occur when we apply the Commutative Property.
Launch
Build on prior knowledge by reminding students of the activity that they completed in the previous lesson. You might even refer to the list of category names that the class created, and then draw students’ attention to any category names that allude to or suggest commutativity. (Any mention of “changing the order” is a strong hint that students’ have clued into the Commutative Property!)
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What do you Notice about the two Circles of Evaluation in the first problem? What do you Wonder?
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Students might notice that both Circles evaluate to 360, and that the order of the numbers changes from one Circle of Evaluation to the next.
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Work through the remaining problems with your partner. When you finish the page, move on to Discover the Commutative Property (2).
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What did you observe about the Commutative Property?
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Possible responses: In each of the examples, the values were the same but the order of the values was different. Sometimes the structures of the Circles of Evaluation changed, like when I moved a nested circle from the right to the left. The Commutative Property holds for multiplication and addition, but not for subtraction or division.
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For any expression involving only addition or only multiplication, changing the order of the numbers will not change the result.
Ensure comprehension of this key idea, as students will apply their understanding of the Commutative Property throughout the remainder of the lesson.
Investigate
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Turn to Commutative Property Table.
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How many different ways can you apply the Commutative Property? Draw as many different Circles of Evaluation as you can.
This activity is similar to one students’ completed during the previous lesson. In both activities, students develop equivalent expressions. That said, in this lesson, students cannot write down any equivalent Circle of Evaluation; they must develop equivalent arithmetic expressions that illustrate commutativity.
As students work, draw their attention to the operation as they rearrange the values inside the Circles of Evaluation, helping to reinforce that the Commutative Property does not apply for division and subtraction.
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How come some of the expressions generated fewer equivalent Circles of Evaluation than others?
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Possible response: When expressions included addition and multiplication, there were many equivalent expressions, whereas division and subtraction resulted in fewer (or no) equivalent expressions.
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What did you notice about the final problem on the page?
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Possible response: It was impossible to apply the Commutative Property because there were no instances of multiplication or addition, only division and subtraction.
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Common Misconception
Students may attempt to apply other properties when developing equivalent Circles. In such instances, help students to see that their Circle of Evaluation is indeed equivalent, but is not an example of the Commutative Property at work.
Synthesize
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Which Circles were equivalent most often?
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Which Circles were equivalent least often? (Or maybe even not at all!)
🔗Tying it All Together
Overview
Students connect and apply what they have learned so far about translation, computation, and the Commutative Property.
Launch
Claire, Walker, and Soraya are translating verbal expressions into Circles of Evaluation. Their teacher asks them to represent the following expression: "one half the product of five and six."
Claire draws this: |
Walker draws this: |
Soraya draws this: |
(* 1/2 (* 5 6)) |
(* (* 5 6) 1/2) |
(/ (* 5 6) 2) |
All three students compute that this expression evaluates to 15.
The teacher looks at their work. She definitely likes Claire’s Circle of Evaluation, but she can’t decide if Soraya’s and Walker’s are good translations or not.
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What do you think? Are Soraya and Walker correct - or just Claire?
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Arguments against Walker’s Circle of Evaluation: The verbal expression starts with one half, and then multiplies that by 5x6. Walker’s Circle is not a direct translation.
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Arguments against Soraya’s Circle of Evaluation: The expression divides by two, whereas the verbal expression seems to imply multiplication by one half.
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Arguments for Walker’s and Soraya’s Circles of Evaluation: When we use computation, these Circles evaluate to 15. The Commutative Property indicates that we can multiply factors in any order. Similarly, dividing by two produces the same result as multiplying by one half.
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Challenge students to consider a variety of perspectives: although Claire’s Circle is a more direct translation, Walker has demonstrated an understanding of the Commutative Property, while Soraya’s work suggests an understanding of fraction multiplication. All students' Circles of Evaluation highlight how computation can produce equivalent Circles.
Investigate
Walker, Claire and Soraya’s Circles are different but still equivalent. Computation helps us to verify that!
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Look at Which Circle of Evaluation is Correct? with some additional work by Claire and Walker. Their teacher awards credit when her students translate the expression precisely or when they show a deep understanding of computation or commutativity.
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In the column on the right, record if Claire, Walker, or both students correctly translated the words into a Circle.
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Complete Which Circle of Evaluation is Correct? Challenge to analyze Circles of Evaluation and commutativity for a more complex expression in words.
Discuss and debrief with students. Invite students to verbally share their responses to reinforce important vocabulary and concepts that students will use again and again in future lessons.
Synthesize
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Summarize the Commutative Property in your own words.
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How might a strong understanding of the Commutative Property be useful when you’re doing computations in your head?
🔗Programming Exploration: Commutativity 20 minutes
Overview
Extending concepts explored earlier in the lesson, students consider whether various functions that we use when coding are commutative.
Launch
You already know that in math, the Commutative Property allows us to rewrite arithmetic expressions in a variety of different ways. We learned that for any expression involving only addition or only multiplication, changing the order of the numbers will not change the result.
But how about functions in WeScheme?!
As a programmer, you will definitely want (and need!) to know if you can change around the order of a function’s arguments… or if you need to always use one "correct" order.
Investigate
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Turn to Commutativity and Code (Images) and open the Commutativity and Associativity Starter File.
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For each function, draw a second Circle of Evaluation that changes the order of the arguments. Translate the Circles of Evaluation to code, then sketch the image that you think your Circle will return. Finally, test your code in WeScheme.
As students work, encourage them to always make predictions before testing the code. Similarly, the activity will be more valuable if students discuss why the code did or did not produce identical images. Debrief to ensure comprehension.
When everyone is finished, check in with students. Did everyone discover that none of the functions were commutative? There is a good chance your students will wonder if any WeScheme functions are commutative!
In the last activity, we discovered that none of the image-producing functions on the page were commutative! Do you think there are any functions in WeScheme that are commutative? Let’s answer that question.
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Turn to Commutativity and Code (String, Number, Color Blending), where we will test four additional functions for commutativity.
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With your partner, complete Commutativity and Code (Images).
There is a good chance your students will want to play with and explore blend-images. Please note that color blending in WeScheme does not behave exactly as it would in the real world, because computers typically use "RGB" (red, blue, green) to express color. Try blending yellow and blue: on a normal primary color wheel, these two would blend to be green. In RGB…the results might surprise you.
Synthesize
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What did you learn about the Commutative Property in WeScheme? Did anything surprise you?
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How were the programming activities in this lesson similar to the paper-and-pencil activities? How were they different?
These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). 
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