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Students develop an understanding of the Distributive Property by studying equivalent Circles of Evaluation. They then apply their understanding of the Distributive Property to make efficient mental computations.

Lesson Goals

Students will be able to…​

  • Recognize how distribution results in equivalent expressions.

  • Apply the Distributive Property to expressions with numbers and variables.

  • Use distribution to efficiently complete mental computations.

Student-facing Lesson Goals

  • Let’s explore the Distributive Property and Circles of Evaluation.

Materials

Key Points For The Facilitator

  • We introduce the Distributive Property by inviting students to look at patterns in structure, and then later apply what they know (to make mental computations or think about area).

🔗The Distributive Property

Overview

Students develop an understanding of the Distributive Property by examining equivalent Circles of Evaluation.

Launch

Display the two Circles of Evaluation, which offer an example of multiplication being distributed over addition.

(​* 9 (+ 40 6))

(+ (​* 9 40) (​* 9 6))

  • What do you Notice and what do you Wonder about the two Circles of Evaluation, above?

  • Possible Noticings: The Circles of Evaluation have different structures. One has a single nested Circle, while the other has two nested Circles. Both include the numbers 9, 40, and 6. Both include addition and multiplication.

  • Possible Wonderings: Are these equivalent? What do they evaluate to? Why is there one 9 in the first Circle of Evaluation, but two 9s in the second Circle?

  • Are these two Circles of Evaluation equivalent? Why or why not?

The Circles of Evaluation are equivalent because of the Distributive Property!

The Distributive Property tells us that a × (b + c) = ab + ac:

(​* a (+ b c))

(+ (​* a b) (​* a c))

Facilitate a discussion where students think deeply about what is happening in the scenarios above, namely, that distribution allows us to rewrite a product as a sum or a sum as a product.

Multiplying the sum of two addends by a number produces the same result as multiplying each addend by that number before finding the sum.

In the next activity, students will use Circles of Evaluation to explore how the mathematical structure of an expression shifts when we apply the Distributive Property.

  • On From Sum to Product, fill in the blanks so that the Circle of Evaluation on the right is equivalent to the Circle of Evaluation on the left.

  • Use computation to confirm that the Circles of Evaluation in each pair are equivalent.

  • After you finish evaluating each Circle of Evaluation, place a checkmark next to the Circle of Evaluation that you think results in a simpler computation.

  • When was it easier to compute the sum? When was it easier to compute the product?

  • There are no right or wrong answers, here. We want students to begin to consider how a nuanced understanding of the Distributive Property can result in more flexible and efficient mental math.

  • On From Product to Sum, the Circle of Evaluation representing a product is provided, and you will complete the Circle of Evaluation that represents a sum.

  • Complete Distribution Challenge, where you will practice applying the Distributive Property in both directions - and with expressions that include fractions and decimals!

The last two questions on each of the above pages ask students to use a made-up function at the top of the Circle of Evaluation, rather than an operator. These functions - which are descriptively named triple, double, times9, and times7 - take in just one input. Students should observe that these functions distribute over addition and subtraction in the same manner as multiplication does.

Investigate

Elise and Dani are discussing a pair of Circles of Evaluation.

(​* 5 43)

(+ 200 8)

Dani thinks the Circles of Evaluation are equivalent. She says, "You can think of 43 as the sum of 40 and 3. Then, you multiply 5 by 40 to get 200, and you add 5 and 3 to get 8."

Elise says, "The second Circle should say 200 + 15, not 200 + 8."

  • Who is correct - Elise or Dani?

  • Elise is correct. We need to multiply each value by 5. It may be helpful for some students to see the Circle for (5 × 40) + (5 × 3).

The activities in the Launch focused on students' internalizing the structure of expressions both before and after applying the Distributive Property. During this segment of the lesson, students apply what they have learned.

Synthesize

  • Summarize the Distributive Property in your own words.

  • Elise and Dani are looking at this problem: 4 × (100 - 1.25). Dani tells Elise that we must always evaluate what’s inside the parentheses first. Do you agree with this idea - or can Dani be more flexible? Explain.

Some students may suggest that distributing is a way of addressing the grouping symbol. Others may agree with Dani. We encourage you to discuss with your students the pros and cons of a flexible approach to solving. In this particular instance, applying the Distributive Property results in a much simpler computation.

🔗The Distributive Property and Mental Math

Overview

Students discover how some computations are simpler and more efficient when we apply the distributive property.

Launch

  • Can you represent the expression 45 × 81 - 45 × 79 in a simpler way?

  • Yes. We can apply the Distributive Property, transforming the expression into 45 × (81 - 79).

  • Can you represent your solving process with a chain of Circles of Evaluation?

  • Invite a student to draw on the board. See below for one possible representation. You might need to draw a Circle or two to get students to see what’s going on

(​- (​* 45 81) (​* 45 79))

→

(​* 45 (​- 81 79))

→

(​* 45 2)

→

90

We can use distribution and mental math to make computation simpler.

In expressions such as 45 × (81 - 79), the multiplication sign is unnecessary and implied when not there. Encourage students to try out this notation, especially if it is one that they have not encountered before.

Investigate

If we were asked to compute 70 × 39, we could use calculator or the standard algorithm. Now that we are familiar with the Distributive Property, we have another valuable tool for efficient mental computation.

Turn to The Distributive Property and Mental Math and look at the first problem.

Our goal here is to make the math easier by creating an equivalent expression that we can solve in our heads! So, instead of finding the product of 70 and 39, we are going to multiply 70 by the difference of 40 and 1.

  • Complete the next Circle of Evaluation, which shows that we are going to distribute 70.

  • Observe how we can now compute our solution by finding the difference between two products.

    • Do you find this process more efficient than the standard algorithm for multiplication? Explain.

  • Complete the rest of the page by creating equivalent expressions that we can solve in our heads.

Like any solving strategy, this takes practice! There is a good chance that students will not find this efficient at first.

Synthesize

  • How can you multiply two 2-digit numbers using mental math?

  • What sorts of problems are simpler to compute using the Distributive Property?

  • Can you think of a multiplication problem that would be easier to solve using the Distributive Property?

We recommend inviting many students to share the problems they come up with. Make a list on the board for students to review together.

🔗Programming Exploration: Distribution

Overview

Extending concepts explored earlier in the lesson, students consider whether various functions that we use when coding are distributive.

Launch

We’ve learned a lot about how the distributive property works. We know that multiplication can distribute over addition or subtraction, allowing us to complete otherwise complex computations with efficiency and ease.

But what about functions that deal with images? For instance, does scale distribute over beside?!

(scale 0.5 (beside aqua-star orange-dot))

→

(beside (scale 0.5 orange-dot) (scale 0.5 aqua-star))

  • Does scale distribute over beside? How do you know?

  • Yes, it does. We know because both Circles of Evaluation produced identical images.

  • What did you discover about the Distributive Property and code?

  • Possible responses: scale distributes over above and beside but rotate does not; flip-vertical distributes over beside but not above; flip-horizontal distributes over above but not beside.

  • Why do you think scale distributed over beside, but rotate did not?

Investigate

Let’s look at whether we can distribute two additional functions: flip-vertical and flip-horizontal.

An interesting feature of these functions is that they require just one input: an Image. Earlier in the lesson, we applied the Distributive Property to 1-input functions like triple and times9.

  • Complete Distribution and Code (2), where you will explore the Distributive Property with some programming functions that consume one input.

  • Reflect with a partner: Why do some functions distribute over above and beside, but not all of them?

Synthesize

  • Do you think code is easier to use and read before or after applying the Distributive Property? Explain.

  • What did you discover about the Distributive Property and code? Did this change your understanding of the Distributive Property in a mathematics context?

These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.

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