Section C Practice Problems

Problem 1

Area diagram. Length, 4. Width, 1 third.

How are the diagrams alike? How are they different?
How is finding the area of the shaded region in each diagram alike? How is it different?

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Solution

Sample responses:

They are each 4 units wide. They are both rectangles. The diagram on the top has whole number side lengths. The one on the bottom has a fractional side length.
I can use multiplication for both. The top one is multiplication of whole numbers and the bottom one is multiplication of a fraction and a whole number.

Problem 2

What is the area of this rectangle? Explain or show your reasoning.
What is the area of the shaded region? Explain or show your reasoning.
How are these two area calculations alike? How are they different?

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Solution

120 square units. Sample response: $10 \times 12 = 120$
$\frac{12}{5}$ square units. Sample response: There are $3 \times 4$ rectangles shaded and each is $\frac{1}{5}$ of a square unit.
Sample response: In both problems, I multiply the length and width to get the area. In the first problem, the numbers are all whole numbers. In the second problem, I found the number of little rectangles and they are each $\frac{1}{5}$ of a full square.

Problem 3

The shaded part of this diagram represents the top of a stove. What is the area of the stovetop? Explain or show your reasoning.

Area diagram. Length, 3 and 1 half feet. Width, 2 feet.

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Solution

7 square feet. Sample response: Looking at the diagram there are 6 full square feet and then two

\frac{1}{2}

square feet so that makes 7 square feet altogether.

Problem 4

Find the area of the shaded region. Explain or show your reasoning.

Area diagram. Length, 3 feet. Width, 4 and 1 fourth feet.

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Solution

12 \frac{3}{4}

(or equivalent). Sample response: There are 12 full squares (

3 \times 4

) and then 3 rectangles that are each

\frac{1}{4}

square are also shaded.

Problem 5

Select all of the expressions that represent the area of the shaded region in square feet.

$3 + 5 \frac{3}{4}$
$3 \times 5 \frac{3}{4}$
$3 \times \left(5 + \frac{3}{4}\right)$
$(3 \times 5) + \frac{3}{4}$
$3 \times 6 - \left(3 \times \frac{1}{4}\right)$

Area diagram. Length, 5 and 3 fourths feet. Width, 3 feet.

Write one more expression that represents the area of the shaded region in square feet.

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Solution

B, C, E

Sample expression: $3 \times 5 + 3\times \frac{3}{4}$

Problem 6

Tyler says that $9 \frac{11}{12} \times 5$ is a little less than 50.

Do you agree with Tyler? Explain or show your reasoning.
What is the value of $9 \frac{11}{12} \times 5$ ?

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Solution

Yes. Sample response: $10 \times 5$ is 50 and $9 \frac{11}{12}$ is close to but less than 10. So, $9 \frac{11}{12} \times 5$ is a little less than 50.
$49 \frac{7}{12}$

Problem 7

A banner at a sporting event is 8 feet long and $2 \frac{1}{3}$ feet wide.

Sketch and label a diagram of the banner.
Find the area of the banner.

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Solution

$18 \frac{2}{3}$ or $\frac{56}{3}$ square feet (or equivalent).

Problem 8

Find the value of each expression. Explain or show your reasoning.

$3\frac{2}{5} \times 10$
$8 \times \frac{14}{3}$
$3 \frac{41}{100} \times 5$

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Solution

34 (or equivalent). Sample response: First, I found $3 \times 10$ which is 30 and then $\frac{2}{5} \times 10$ which is $\frac{20}{5}$ or 4.
$\frac{112}{3}$ (or equivalent). Sample response: I multiplied 8 and 14 and have that many $\frac{1}{3}$ ’s.
$17 \frac{5}{100}$ (or equivalent). Sample response: I found $3 \times 5$ which is 15 and $\frac{41}{100} \times 5$ is $\frac{205}{100}$ . I knew $\frac{205}{100}$ is $2 \frac{5}{100}$ .

Problem 9

A standard rectangular sheet of paper is $8\frac{1}{2}$ inches wide and 11 inches long. How many times would you need to fold the sheet of paper in half before the area is less than 1 square inch? Explain or show your reasoning.
A rectangular piece of chart paper is 23 inches wide by 33 inches long. How many times would you need to fold it in half before its area is less than 1 square inch?

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Solution

7 times. Sample response: The area of the sheet of paper is $\frac{187}{2}$ or $93\frac{1}{2}$ square inches. The table shows what to multiply the area by after folding the paper each time.
10 times. Sample response: $23 \times 33 = 759$ so I need to continue the table until the denominator in the fraction is more than 759, which is 10 times.

number of folds	multiple
1	$\frac{1}{2}$
2	$\frac{1}{4}$
3	$\frac{1}{8}$
4	$\frac{1}{16}$
5	$\frac{1}{32}$
6	$\frac{1}{64}$
7	$\frac{1}{128}$

Problem 10

Part of the rectangle is shaded.

Area diagram. Length, 6. Width, 3 and 2 fifths.

Write a multiplication expression that represents the shaded area.
Write a division expression that represents the shaded area.
Write some other expressions that represent the shaded area.

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Solution

Sample response:

\frac{17}{5} \times 6

Sample response:

(17 \times 6) \div 5

Sample responses:

3 \frac{2}{5} \times 6

18 + \frac{2}{5} \times 6

24 - \frac{3}{5} \times 6

Problem 11

Here is an image of the Empire State Building in New York City.

The base of the Empire State Building is shaped like a rectangle. What do you think the area of the rectangle is in square meters?

(Hint: A typical bathtub covers about 1 square meter. A typical car parking space is about 10 square meters.)

Make an estimate that is too low.
Make an estimate that is too high.
The length of the rectangle is about 129 $\frac{1}{5}$ meters. The width is about 57 meters. What is the area of the base of the Empire State Building?

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Solution

Sample response: 1,000
Sample response: 1,000,000
$7,364 \frac{2}{5}$ square meters