Lesson 10The Distributive Property, Part 2

Let's use rectangles to understand the distributive property with variables.

Learning Targets:

  • I can use a diagram of a split rectangle to write different expressions with variables representing its area.

10.1 Possible Areas

  1. A rectangle has a width of 4 units and a length of m units. Write an expression for the area of this rectangle.
  1. What is the area of the rectangle if m is 3 units? 2.2 units? \frac15 unit?
  1. Could the area of this rectangle be 11 square units? Why or why not?

10.2 Partitioned Rectangles When Lengths are Unknown

  1. Here are two rectangles. The length and width of one rectangle are 8 and 5. The width of the other rectangle is 5, but its length is unknown so we labeled it x . Write an expression for the sum of the areas of the two rectangles.

  2. The two rectangles can be composed into one larger rectangle as shown. What are the width and length of the new, large rectangle?

    A figure of a small and large rectangle. The small rectangle is drawn to the left and adjacent to the large rectangle forming one large rectangle. The small rectangle has length of x and width 5. The large rectangle has length 8 and width 5.
  3. Write an expression for the total area of the large rectangle as the product of its width and its length.

10.3 Areas of Partitioned Rectangles

For each rectangle, write expressions for the length and width and two expressions for the total area. Record them in the table. Check your expressions in each row with your group and discuss any disagreements.

Six different sized rectangles labeled A, B, C, D, E, and F.  Rectangle A is partioned by a vertical line segment into two smaller rectangles. The vertical side is labeled 3 and the top horizontal side lengths are labeled "a" and 5.  Rectangle B is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled one third and the top horizontal side lengths are labeled 6 and x. Rectangle C is partitioned by 2 vertical line segments into three equally sized rectangles. The vertical side is labeled r and the top horizontal side lengths are each labeled 1. Rectangle D is partitioned by 3 vertical line segments into 4 equally sized rectangles. The vertical side is labeled 6, and the top horizontal side lengths are each labeled 4. Rectangle E is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled m and the top horizontal side lengths are labeled 6 and 8. Rectangle F is partitioned by a vertical line segment into two smaller rectangles. The vertical side is labeled 5 and the top horizontal side lengths are labeled 3 x and 8.
width length area as a product of
width times length
area as a sum of the areas
of the smaller rectangles

Are you ready for more?

Here is an area diagram of a rectangle.

  1. Find the lengths w , x , y , and z , and the area A . All values are whole numbers.
  2. Can you find another set of lengths that will work? How many possibilities are there?

Lesson 10 Summary

Here is a rectangle composed of two smaller rectangles A and B. 

A rectangle is partitioned by a vertical line segment creating two smaller rectangles, A and B. Rectangle A has a vertical side length of 3 and horizontal side length of 2. Rectangle B has a horizontal side length of x.

Based on the drawing, we can make several observations about the area of the rectangle:

  • One side length of the large rectangle is 3 and the other is 2+x , so its area is 3(2+x) .
  • Since the large rectangle can be decomposed into two smaller rectangles, A and B, with no overlap, the area of the large rectangle is also the sum of the areas of rectangles A and B: 3(2) + 3(x) or 6+3x .
  • Since both expressions represent the area of the large rectangle, they are equivalent to each other. 3(2+x) is equivalent to 6 + 3x .

We can see that multiplying 3 by the sum 2+x is equivalent to multiplying 3 by 2 and then 3 by x  and adding the two products. This relationship is an example of the distributive property.

3(2+x) = 3 \boldcdot 2 + 3 \boldcdot x

Lesson 10 Practice Problems

  1. Here is a rectangle.

    1. Explain why the area of the large rectangle is 2a + 3a + 4a .
    2. Explain why the area of the large rectangle is (2+3+4)a .
  2. Is the area of the shaded rectangle  6(2-m) or 6(m-2) ?

    Explain how you know.

  3. Choose the expressions that do not represent the total area of the rectangle. Select all that apply.

    1. 5t + 4t
    2. t + 5 + 4
    3. 9t
    4. 4 \boldcdot 5 \boldcdot t
    5. t(5+4)
    A rectangle partitioned by a vertical line segment into two smaller rectangles. the vertical side is labeled t and the top horizontal side lengths are labeled 5 and 4.
  4. Evaluate each expression mentally.

    1. 35\boldcdot 91-35\boldcdot 89
    1. 22\boldcdot 87+22\boldcdot 13
    1. \frac{9}{11}\boldcdot \frac{7}{10}-\frac{9}{11}\boldcdot \frac{3}{10}
  5. Select all the expressions that are equivalent to 4b .

    1. b+b+b+b
    2. b+4
    3. 2b+2b
    4. b \boldcdot b \boldcdot b \boldcdot b
    5. b \div \frac{1}{4}
  6. Solve each equation. Show your reasoning.

    1. 111=14g
    2. 13.65 = h + 4.88
    1. k+ \frac{1}{3} = 5\frac{1}{8}
    2. \frac{2}{5} m = \frac{17}{4}
    1. 5.16 = 4n
  7. Andre ran 5\frac{1}{2} laps of a track in 8 minutes at a constant speed. It took Andre x minutes to run each lap. Select all the equations that represent this situation.

    1. \left(5\frac{1}{2}\right)x = 8
    2. 5 \frac{1}{2} + x = 8
    3. 5 \frac{1}{2} - x = 8
    4. 5 \frac{1}{2} \div x = 8
    5. x = 8 \div \left(5\frac{1}{2}\right)
    6. x = \left(5\frac{1}{2}\right) \div 8