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.](https://staging-cms-im.s3.amazonaws.com/GcxYkZ7ktZ36EhfEHLwaZxcQ?response-content-disposition=inline%3B%20filename%3D%226-6.6.B5.PP.Image.New.0505.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.B5.PP.Image.New.0505.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T132407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4819d3b1d8ed7e8de4628376680283c9abb8ea4e87a47ba4d943c04ec52b1183)
- 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.
![Area diagram partitioned into two attached rectangles one with a shaded area.](https://staging-cms-im.s3.amazonaws.com/aW1ZyhB9s4wkPu3aynTfYrmd?response-content-disposition=inline%3B%20filename%3D%226-6.6.B5.PP.Image.New.0606.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.B5.PP.Image.New.0606.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T132407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=791831d79478c6fe0adcbc436ec3a30dc01762623d4f2e9d5c7fc3d4bba54e8e)
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.](https://staging-cms-im.s3.amazonaws.com/i8C7gQR8V35gG1KMJQNDsiKg?response-content-disposition=inline%3B%20filename%3D%226-6.6.B.PP.Image.03.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.B.PP.Image.03.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T132407Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=907f605dc185daea171d55eec0afcdd33af808c9c3ed9fd8524527f97001b436)
\(5t + 4t\)
\(t + 5 + 4\)
\(9t\)
\(4 \boldcdot 5 \boldcdot t\)
\(t(5+4)\)
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.
\(\left(5\frac{1}{2}\right)x = 8\)
\(5 \frac{1}{2} + x = 8\)
\(5 \frac{1}{2} - x = 8\)
\(5 \frac{1}{2} \div x = 8\)
\(x = 8 \div \left(5\frac{1}{2}\right)\)
\(x = \left(5\frac{1}{2}\right) \div 8\)