Lesson 8

Areas and Equivalent Expressions

These materials, when encountered before Algebra 1, Unit 6, Lesson 8 support success in that lesson.

8.1: Ways to Express the Area (10 minutes)

Warm-up

This activity is to prepare students for the warm-up in the associated Algebra 1 lesson, where they will need to explain why a diagram can be represented by two different expressions.

Student Facing

  1. Here are two rectangles with their side lengths labeled. Write the sum of the areas of the two rectangles.

    on left, 6 by 7 rectangle. on right x by 7 rectangle. 
  2. The two rectangles can be composed into a larger rectangle as shown.

    rectangle partitioned into 2 parts. width = 7. lengths = 6 and x. 
    1. Write the length and width of the new, large rectangle.
    2. Write an expression for the area of the new rectangle.
  3. How are the two expressions for area alike? How are they different?

Student Response

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

Activity Synthesis

  • “How would the values of the two expressions compare if \(x\) is 2?” (They would each equal 56.)
  • “If \(x\) is 10?” (They would each equal 112.)
  • “If \(x\) is 1,000?” (They would each equal 7,042.)
  • “We call the two expressions ‘equivalent expressions.’ What does that mean? How would you define it?” (Equivalent expressions have the same value for all values of the variables.)

Highlight that the two expressions \(42+7x\) and \(7(6+x)\) are equivalent expressions. This means they have the same value for any value of \(x\). Even though we can’t check all possible values of \(x\), we know they are equivalent because they represent the same area, and because they are equivalent by the distributive property.

8.2: Multiplying Two-Digit Numbers and the Distributive Property (15 minutes)

Activity

The work in this activity is similar to the work in the associated Algebra 1 lesson, except that it is more scaffolded, and it focuses on operations with numbers instead of with variables.

Launch

Display the two rectangles from the first part of the activity. Ask students, “What do you notice and wonder about the two rectangles?” Students might notice that they appear to be identical, and that the side lengths of B are equal to the side lengths of A. They might wonder why B is partitioned into smaller rectangles with the dashed lines.

Student Facing

  1. Here are two rectangles.

    rectangle A, length = 12, width = 11. rectangle B, partitioned into 4 parts, widths = 10 and 1. lengths = 10 and 2. 
    1. Find the area of Rectangle A.
    2. Find the area of each of the 4 smaller rectangles that make up Rectangle B.
    3. Use the sum of the areas of the small rectangles to find the area of Rectangle B.
    4. How is finding the area of Rectangle B like multiplying \((10+1)(10+2)\)?
  2. Find the area of this rectangle two different ways:
    rectangle partitioned into 4 parts. lengths = 10 and 6. widths = 10 and 1. 

Student Response

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

Activity Synthesis

Ask students to make sense of the different ways to express the area of the last rectangle. In the course of this discussion, ensure they see work that looks like:

\((10+6)(10+1)\)

\(10 \boldcdot 10 + 10 \boldcdot 6 + 10 \boldcdot 1 + 6 \boldcdot 1\)

\(100 + 60 + 10 + 6\)

Notice that the first 10 is multiplied by both 10 and 1, then the 6 is multiplied by both 10 and 1. These four products are added. This is an example of the distributive property.

8.3: Using the Distributive Property to Write Equivalent Expressions (20 minutes)

Activity

These practice exercises offer an opportunity to practice rewriting expressions using the distributive property. There may be more work here than can reasonably completed in the available time. If that is the case, consider assigning a subset of items that would be most beneficial for your students to practice.

Note that one of the questions gives an expression like \(3x+12\) for the area of a rectangle, and asks for a possible length and width. While 3 and \(x+4\) may be the most likely response, another example of a valid response would be \(\frac15\) and \(15x+60\).

Launch

The intention is for students to reason using the distributive property or by drawing diagrams, so restrict access to calculators unless it’s necessary to access the activity.

Arrange students in groups of 2–4. Encourage students to check in with their group as they work.

Student Facing

  1. Express the area of each rectangle in two ways: as a sum of the areas of the sub-rectangles, and a product of length and width of the large rectangle.

    rectangle A partitioned into 4 rectangles. width = 4, lengths = x, x, x, and 7. rectangle B partitioned into 2 rectangles. width = one half, lengths = a and 8. 
  2. Select all the expressions that are equivalent to \(4(2 + 3x)\). Be prepared to explain or show how you know.
    • \(8 +12x\)
    • \(8 + 3x\)
    • \(4(5x)\)
    • \(12x + 8\)
    • \(2(4) + 3x(4)\)
    • \(12x + 2\)
    • \(2(2+3x) + 2(2+3x)\)
  3. Write at least three expressions that can represent the area of a rectangle that is 12 units long by \((10+a)\) units wide. If you get stuck, try drawing a diagram.
  4. Each expression represents the area of a rectangle. Name a possible length and width of each rectangle. Be prepared to explain or show how you know.
    1. \(3x + 21\)
    2. \(4(9) + 4(20)\)
    3. \(8^2 + 8a\)
    4. \((30)(30) + 30(4)+ 30(b)\)
  5. Sort the expressions into three groups, so that all three of the expressions in a group could represent the area of the same rectangle.
  • \(100+20+90+18\)
  • \(100+90+90+81\)
  • \((10+9)(10+9)\)
  • \(10(2\boldcdot10+2)\)
  • \(12 \boldcdot 19\)
  • \(10 \boldcdot 22\)
  • \((10+2)(10+9)\)
  • \(19^2\)
  • \(2 \boldcdot 100+20\)
rectangle partitioned into 4 rectangles
rectangle partitioned into 4 rectangles
rectangle partitioned into 4 rectangles

Student Response

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

Activity Synthesis

The purpose of this lesson was to remind students of a way to understand the distributive property using expressions for the area of a rectangle. Possible questions for discussion:

  • “What are two different ways to think about the area of the rectangles we’ve seen in this lesson?” (You can add the areas of the individual pieces, or you can multiply the entire length by the width.)
  • “What is an example of two expressions that are equivalent because of the distributive property?” (Answers vary.)
  • “How would you use a diagram to explain to someone who was absent today why \(a(b+c)=ab+ac\) for any value of \(a\), \(b\), and \(c\)?” (Use a rectangle with one side labeled \(a\) and the other \(b+c\). The rectangle is subdivided into two regions with areas \(ab\) and \(ac\). These total to the area of the entire rectangle, which is also length times width, or \(a(b+c)\).)