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Lesson 11The Distributive Property, Part 3

Let's practice writing equivalent expressions by using the distributive property.

Learning Targets:

  • I can use the distributive property to write equivalent expressions with variables.

11.1 The Shaded Region

A rectangle with dimensions 6 cm and w cm is partitioned into two smaller rectangles.

Explain why each of these expressions represents the area, in cm2, of the shaded portion.

  • 6w-24  
  • 6(w-4)  
a rectangle is shown with a height of 6 and the length of w.

11.2 Matching to Practice Distributive Property

Match each expression in Column 1 to an equivalent expression in Column 2. If you get stuck, consider drawing a diagram.

Column 1

  1. a(1+2+3)
  2. 2(12-4)
  3. 12a+3b
  4. \frac23(15a-18)
  5. 6a+10b
  6. 0.4(5-2.5a)
  7. 2a+3a

Column 2

  1. 3(4a+b)
  2. 12 \boldcdot 2 - 4 \boldcdot 2
  3. 2(3a+5b)
  4. (2+3)a
  5. a+2a+3a
  6. 10a-12
  7. 2-a

11.3 Writing Equivalent Expressions Using the Distributive Property

The distributive property can be used to write equivalent expressions. In each row, use the distributive property to write an equivalent expression. If you get stuck, draw a diagram.

product sum or difference
3(3+x)
4x-20
(9-5)x
4x+7x
3(2x+1)
10x-5
x+2x+3x
\frac12 (x-6)
y(3x+4z)
2xyz-3yz+4xz

Are you ready for more?

This rectangle has been cut up into squares of varying sizes. Both small squares have side length 1 unit. The square in the middle has side length x units.

a rectangle is formed by 9 different sized squares
  1. Suppose that x is 3. Find the area of each square in the diagram. Then find the area of the large rectangle.
  2. Find the side lengths of the large rectangle assuming that x is 3. Find the area of the large rectangle by multiplying the length times the width. Check that this is the same area you found before.
  3. Now suppose that we do not know the value of x . Write an expression for the side lengths of the large rectangle that involves x .

Lesson 11 Summary

The distributive property can be used to write a sum as a product, or write a product as a sum. You can always draw a partitioned rectangle to help reason about it, but with enough practice, you should be able to apply the distributive property without making a drawing.

Here are some examples of expressions that are equivalent due to the distributive property. 

\begin {align} 9+18&=9(1+2)\\[10pt] 2(3x+4)&=6x+8\\[10pt] 2n+3n+n&=n(2+3+1)\\[10pt] 11b-99a&=11(b-9a)\\[10pt] k(c+d-e)&=kc+kd-ke\\ \end {align}

Lesson 11 Practice Problems

  1. For each expression, use the distributive property to write an equivalent expression.

    1. 4(x+2)
    2. (6+8)\boldcdot x
    1. 4(2x+3)
    2. 6(x+y+z)

  2. Priya rewrites the expression 8y - 24 as 8(y-3) . Han rewrites 8y-24 as 2(4y-12) . Are Priya's and Han's expressions each equivalent to 8y-24 ? Explain your reasoning.

  3. Select all the expressions that are equivalent to 16x + 36 .

    1. 16(x+20)
    2. x(16+36)
    3. 4(4x+9)
    4. 2(8x+18)
    5. 2(8x+36)
  4. The area of a rectangle is 30 + 12x . List at least 3 possibilities for the length and width of the rectangle. 

  5. Select all the expressions that are equivalent to \frac{1}{2}z .

    1. z + z
    2. z \div 2
    3. z \boldcdot z
    4. \frac{1}{4}z + \frac{1}{4} z
    5. 2z

    1. What is the perimeter of a square with side length:

      3 cm

      7 cm

      s cm

    2. If the perimeter of a square is 360 cm, what is its side length?

    3. What is the area of a square with side length:

      3 cm

      7 cm

      s cm

    4. If the area of a square is 121 cm2, what is its side length?

  7. Solve each equation.

    1. 10=4y
    2. 5y = 17.5
    3. 1.036=10y
    4. 0.6y = 1.8
    5. 15=0.1y