Section A Practice Problems

Problem 1

Here is a diagram of the floor in a room.

What is the area of the floor? Explain or show your reasoning.

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Solution
18 square yards. Sample response: I multiplied the length and the width.

Problem 2

What are the unknown side lengths? Explain or show your reasoning.

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Solution

The missing top side length is 4 cm. The missing left side length is 12 cm. Sample response: The sum of the length of the two top sides is equal to the length of the bottom, so  6+46 + 4 is 1010 cm. The length of the left slide is the sum of the lengths of the two sides on the right, so  8+48+4 is 1212 cm.

Problem 3

Which of these units would you use to measure the length of a pencil? Select all that apply.

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Solution
A,D

Problem 4

Find the area of the figure shown here. Explain or show your reasoning.

6-sided shape. 
6-sided shape. Straight sides. All side lengths meet at right angles. Side lengths. Bottom, 15 ft. Right side rises 12 ft, then goes left 4 ft, and goes down 2 ft. Top side length, blank. Left side length, blank.   

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Solution

158 square feet. Sample response: The length of the vertical side that is not labeled is 10 feet because 12−2=1012−2=10. If the piece at the upper right is cut off, it is 2 feet by 4 feet or 8 square feet. The rest is a 15-foot-by-10-foot square and that’s 150 square feet, so 150+8=158150 + 8 = 158.

Problem 5

Which has the greater volume? Explain or show your reasoning.

A
Figure made of 3 rows of cubes. First row, 4 cubes. Second row, 2 cubes. Third row, 1 cube.

B
Rectangular prism. 2 cubes by 2 cubes by 2 cubes.

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Solution
Shape B. Sample response: There are 7 cubes in Shape A and 8 cubes in Shape B, assuming there is one cube hidden behind, so Shape B has a greater volume.

Problem 6

What is the volume of the figure? Explain or show your reasoning.

Figure made of cubes. Base: 3 cubes by 3 cubes by 3 cubes. Tower of 3 cubes vertically stacked on top of back right cube. 

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Solution

12 unit cubes. Sample response: The figure is made of 12 cubes, so its volume is 12 unit cubes.

Problem 7

  1. What is the volume of this rectangular prism? Explain or show your reasoning.

    Prism. 3 by 2 by 1 cubes.

  2. What is the volume of this rectangular prism? Explain or show your reasoning. 

    Rectangular prism. 3 cubes by 2 cubes by 4 cubes. 

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Solution
  1. 6 unit cubes. Sample response: 3×23\times2 is 6.
  2. 24 unit cubes. Sample response: There are 4 layers of 6 cubes.

Problem 8

Find the volume of each rectangular prism. Explain or show your reasoning.

A
Prism. 3 by 1 by 4 cubes.

B
Rectangular prism. Base, 2 cubes by 3 cubes. Height shown with 3 cubes. 

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Solution
  1. 12 unit cubes. Sample response: 3× 43 \times 4 is 12.
  2. 18 unit cubes. Sample response: There are 3× 23 \times 2 or 6 cubes in each layer and 3 layers, so that’s 3× 63 \times 6 or 18 cubes.

Problem 9

Find the volume of some objects around you.

  1. List the objects.

  2. Which has the least volume? Which has the greatest?

  3. Pick 2 of your objects and compare their volumes.

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Solution

Sample responses:

  1. pencil, teacher’s desk, book, paper clip

  2. The paper clip takes up the least space and the teacher’s desk takes up the most.

  3. The volume of the pencil is less than the volume of the book. All of the objects have less volume than the desk. They all fit inside the desk.

Problem 10

  1. How many different rectangular prisms can you make with 18 cubes? Explain or show your reasoning.

  2. How many different rectangular prisms can you make with 24 cubes? Explain or show your reasoning.

  3. How do the side lengths of the prisms compare to each other? What patterns do you notice? Is this pattern true for the rectangular prisms you can make with 36 cubes?

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Solution

1. 4. Sample response:

  • 1 unit by 1 unit by 18 units
  • 2 units by 1 unit by 9 units
  • 3 units by 1 unit by 6 units
  • 3 units by 2 units by 3 units

2. 6. Sample response:

  • 1 unit by 1 unit by 24 units
  • 2 units by 1 unit by 12 units
  • 3 units by 1 unit by 8 units
  • 4 units by 1 unit by 6 units
  • 2 units by 2 units by 6 units
  • 2 units by 3 units by 4 units

3. Sample response: The product of the three numbers is always the same. This means when one side length doubles, another side length is cut in half, and when one side length triples, another side length is divided by 3. This is also true for 36 cubes. Possible side lengths are:

  • 1 unit by 1 unit by 36 units 
  • 2 units by 1 unit by 18 units (doubling the first side length and halving the third) 
  • 4 units by 1 unit by 9 units (again doubling the first side length and halving the third) 
  • 3 units by 1 unit by 12 units 
  • 6 units by 1 unit by 6 units 
  • 2 units by 2 units by 9 units 
  • 2 units by 3 units by 6 units 
  • 4 units by 3 units by 3 units