Lesson 10

The Distributive Property, Part 2

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

Problem 1

Here is a rectangle.

Area diagram. A rectangle partioned vertically into 3 smaller rectangles. 
  1. Explain why the area of the large rectangle is \(2a + 3a + 4a\).
  2. Explain why the area of the large rectangle is \((2+3+4)a\).

Problem 2

Is the area of the shaded rectangle \(6(2-m)\) or \(6(m-2)\)?

Explain how you know.

Area diagram partitioned into two attached rectangles one with a shaded area.

Problem 3

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

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.
A:

\(5t + 4t\)

B:

\(t + 5 + 4\)

C:

\(9t\)

D:

\(4 \boldcdot 5 \boldcdot t\)

E:

\(t(5+4)\)

Problem 4

Evaluate each expression mentally.

  1. \(35\boldcdot 91-35\boldcdot 89\)
  2. \(22\boldcdot 87+22\boldcdot 13\)
  3. \(\frac{9}{11}\boldcdot \frac{7}{10}-\frac{9}{11}\boldcdot \frac{3}{10}\)
(From Unit 4, Lesson 9.)

Problem 5

Select all the expressions that are equivalent to \(4b\).

A:

\(b+b+b+b\)

B:

\(b+4\)

C:

\(2b+2b\)

D:

\(b \boldcdot b \boldcdot b \boldcdot b\)

E:

\(b \div \frac{1}{4}\)

(From Unit 4, Lesson 8.)

Problem 6

Solve each equation. Show your reasoning.

\(111=14a\)

 

\(13.65 = b + 4.88\)

\(c+ \frac{1}{3} = 5\frac{1}{8}\)

\(\frac{2}{5} d = \frac{17}{4}\)

 

\(5.16 = 4e\)

(From Unit 4, Lesson 4.)

Problem 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.

A:

\(\left(5\frac{1}{2}\right)x = 8\)

B:

\(5 \frac{1}{2} + x = 8\)

C:

\(5 \frac{1}{2} - x = 8\)

D:

\(5 \frac{1}{2} \div x = 8\)

E:

\(x = 8 \div \left(5\frac{1}{2}\right)\)

F:

\(x = \left(5\frac{1}{2}\right) \div 8\)

(From Unit 4, Lesson 2.)