# Lesson 10

The Distributive Property, Part 2

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

### Problem 1

Here is a rectangle.

- Explain why the area of the large rectangle is \(2a + 3a + 4a\).
- 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.

### Problem 3

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

\(5t + 4t\)

\(t + 5 + 4\)

\(9t\)

\(4 \boldcdot 5 \boldcdot t\)

\(t(5+4)\)

### Problem 4

Evaluate each expression mentally.

- \(35\boldcdot 91-35\boldcdot 89\)
- \(22\boldcdot 87+22\boldcdot 13\)
- \(\frac{9}{11}\boldcdot \frac{7}{10}-\frac{9}{11}\boldcdot \frac{3}{10}\)

### Problem 5

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

\(b+b+b+b\)

\(b+4\)

\(2b+2b\)

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

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

### Problem 6

Solve each equation. Show your reasoning.

\(111=14a\)

\(13.65 = b + 4.88\)

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

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

\(5.16 = 4e\)

### 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.

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

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

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

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

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

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