Lesson 9

The Distributive Property, Part 1

9.1: Number Talk: Ways to Multiply (5 minutes)

Warm-up

Students perform mental calculations by applying strategies involving the distributive property.

Launch

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Find each product mentally.

\(5 \boldcdot 102\)

\(5 \boldcdot 98\)

\(5 \boldcdot 999\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Once students have had a chance to share a few different ways of reasoning about this product, focus on explanations using the distributive property like \(5 \boldcdot 98 = 5 \boldcdot (90 + 8)\) or \(5 \boldcdot (100 - 2)\), then writing out the two products from distributing. Remind students of the distributive property, and let them know they will spend the next few lessons working with it.

Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

9.2: Ways to Represent Area of a Rectangle (15 minutes)

Activity

The purpose of this activity is to remind students of rectangle diagrams they worked with in a previous unit to represent multiplication. It is also to introduce the convention that for example the expression \(6 \boldcdot 3 + 2\) equals 20. If we want the sum to be carried out before the product, we would need to use parentheses like \(6 \boldcdot (3+2)\).

Launch

Allow students 10 minutes of quiet work time, followed by a whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge about finding area. Some students may benefit from a review of the rectangle diagrams they used in a previous unit to represent multiplication.
Supports accessibility for: Memory; Conceptual processing

Student Facing

  1. Select all the expressions that represent the area of the large, outer rectangle in figure A. Explain your reasoning.

    • \(6 + 3 + 2\)
    • \(6 \boldcdot 3 + 6 \boldcdot 2\)
    • \(6 \boldcdot 3 + 2\)
    • \(6 \boldcdot 5\)
    • \(6 (3+2)\)
    • \(6 \boldcdot 3 \boldcdot 2\)
    One rectangle diagram. Labeled A.
  2. Select all the expressions that represent the area of the shaded rectangle on the left side of figure B. Explain your reasoning.

    • \(4 \boldcdot 7 + 4 \boldcdot 2\)
    • \(4 \boldcdot 7 \boldcdot 2\)
    • \(4 \boldcdot 5\)
    • \(4 \boldcdot 7 - 4 \boldcdot 2\)
    • \(4(7-2)\)
    • \(4(7+2)\)
    • \(4 \boldcdot 2 - 4 \boldcdot 7\)
    One rectangle diagram. Labeled B.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Students may conclude that \(6 \boldcdot 3 + 2\) represents the area of the rectangle. This is a good opportunity to introduce a convention. When we have multiplication and addition in the same expression, it is the convention that the multiplication is done first. So \(6 \boldcdot 3 + 2\) equals \(18 + 2\), or 20, so it doesn’t represent the area of the rectangle, which we know to be 30 square units. If you want the addition to be done first, you need to use parentheses. Therefore, \(6 (3+2)\) does represent the area of the large rectangle. Remind students that “next to” implies multiplication.

By using the rectangle, we can tell that \(6 (3+2)=6 \boldcdot 3 + 6 \boldcdot 2\). This is another example of two expressions that are equivalent because of the distributive property.

Speaking: MLR8 Discussion Supports. Display sentence frames to support whole-class discussion. For example, "Expression ______ matches figure A because _____." or "Figure ___ cannot be represented by expression _____ because ________." Invite students to share their responses with a partner, and prompt them to rehearse what they will say when they share with the whole class. Rehearsing provides students with additional opportunities to clarify their thinking.
Design Principle(s): Optimize output (for explanation)

9.3: Distributive Practice (15 minutes)

Activity

This is for practice going back and forth with the distributive property using numbers, but also to make the point that invoking the distributive property can help you do computations in your head. Some expressions that would be difficult to brute force become simpler after using the distributive property to write an equivalent expression.

Note that there is more than one way to rewrite the last row. For example, \(24-16\) could also be written as \(2(12-8)\) where \((12-8)\) is the difference of two terms. A term is a single number or variable, or variables and numbers multiplied together. This is fine, since there is no reason to insist that students use the greatest common factor at this time. Students should recognize that there is more than one factor that would work, and that the resulting expressions are equivalent. In a subsequent unit students will explicitly study the idea of a greatest common factor.

Launch

Give students the following setup: Suppose your business makes 15 items for \$17 each and sells them for \$20 each. How would you find your profit? One way is to write \(15\boldcdot 20 - 15\boldcdot 17\). But that’s a lot of calculations! An easier way to get the answer is to write \(15(20-17)\), which is easier to figure out in your head.

Allow students 10 minutes of quiet work time, followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts. For example, after students have completed the first two rows of the table, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention

Student Facing

Complete the table. If you get stuck, skip an entry and come back to it, or consider drawing a diagram of two rectangles that share a side.

  column 1   column 2 column 3   column 4     value  
\(5 \boldcdot 98\) \(5 (100-2)\)     \(5 \boldcdot 100 - 5 \boldcdot 2\)     \(500 - 10\) 490
\(33 \boldcdot 12\) \(33 (10 + 2)\)
\(3 \boldcdot 10 - 3 \boldcdot 4\) \(30-12\)
    \(100 (0.04 + 0.06)\)    
\(8 \boldcdot \frac 1 2 + 8 \boldcdot \frac 1 4\)
\(9 + 12\)
\(24 - 16\)

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

  1. Use the distributive property to write two expressions that equal 360. (There are many correct ways to do this.) 
  2. Is it possible to write an expression like \(a(b+c)\) that equals 360 where \(a\) is a fraction? Either write such an expression, or explain why it is impossible.
  3. Is it possible to write an expression like \(a(b-c)\) that equals 360? Either write such an expression, or explain why it is impossible.
  4. How many ways do you think there are to make 360 using the distributive property?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students might understand how to expand an expression with parentheses but struggle with how to approach a sum. Encourage students to think about the rectangle diagrams they have seen and draw a diagram of a partitioned rectangle. Ask students what the sum represents and help them to see that it can represent the sum of the areas of the two smaller rectangles. Remind students that the rectangles have the same width, and ask what that width might have been to produce the two areas, what factor the two areas have in common. Then have them consider the other factors (the lengths) that would produce those products for the areas. 

Activity Synthesis

Invite students to share their strategies and reasoning. Include students who used diagrams of partitioned rectangles. Ask if they noticed any interesting patterns, or if they want to share some examples of their own of calculations that can be made simpler by using the distributive property to write an equivalent expression.

Representing, Speaking, Listening: MLR2 Collect and Display. During whole-class discussion, create a visual display to record a list of the strategies students describe, such as diagrams of partitioned rectangles. Amplify use of mathematical words and phrases that students use, such as patterns, equivalent, sum, etc. that describe their process. Ask students which strategies worked best for them, and ask them to same what they have in common. Remind students that they can borrow language and strategies from the display as needed.
Design Principle(s): Support sense-making; Maximize meta-awareness

Lesson Synthesis

Lesson Synthesis

Arrange students in groups of 2. One partner writes a product of the form \(a(b+c)\) or \(a(b-c)\). The other partner writes an equivalent expression using the distributive property, then each student evaluates their expression. The partners compare which computation was simpler, took less time, etc. Then have one partner write a sum and the other see if they can write an equivalent expression. Evaluate, compare, repeat as time allows. (It may be necessary to give students three numbers to use for the first round, instead of asking them to think of 3 numbers to use.)

9.4: Cool-down - Complete the Equation (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

A term is a single number or variable, or variables and numbers multiplied together. Some examples of terms are 10, \(8x\), \(ab\), and \(7yz\).

When we need to do mental calculations, we often come up with ways to make the calculation easier to do mentally.

Suppose we are grocery shopping and need to know how much it will cost to buy 5 cans of beans at 79 cents a can. We may calculate mentally in this way:

\(5\boldcdot {79}\)
\(5\boldcdot {70}+5\boldcdot {9}\)
\(350+45\)
\(395\)

In general, when we multiply two terms (or factors), we can break up one of the factors into parts, multiply each part by the other factor, and then add the products. The result will be the same as the product of the two original factors. When we break up one of the factors and multiply the parts we are using the distributive property.

The distributive property also works with subtraction. Here is another way to find \(5\boldcdot 79\):

\(5\boldcdot 79\)
\(5\boldcdot {(80-1)}\)
\(400-5\)
\(395\)