Multiple-Choice Items:

1. Which set of dimensions would give you a prism with the same volume as the rectangular prism shown below?

 

A	length: 3 in.     width: 4 in.      height: 3 in.
B	length: 4 in.     width: 2 in.      height: 9 in.
C	length: 3 in.     width: 2 in.      height: 6 in.
D	length: 10 in.   width: 7 in.      height: 9 in.

 

 

 

2. How many centimeter cubes would fit inside a cube like the one shown below?

 

 

A	10 cubes
B	15 cubes
C	25 cubes
D	125 cubes

 

 

 

3.    A candy store worker packed a total of 40 chocolate drops in a box. The box had 2 layers of chocolate drops. Each layer had the same number of chocolate drops. Which correctly describes the arrangement of the chocolate drops in one layer?

 

A	4 chocolate drops by 5 chocolate drops
B	8 chocolate drops by 5 chocolate drops
C	10 chocolate drops by 10 chocolate drops
D	20 chocolate drops by 20 chocolate drops

 

Use the figure below to answer question 4.

 

 

4. Maria is computing the volume of this compound figure. Which figure shows a way Maria can separate the figure into rectangular prisms?

 

 

Maria chose to separate the figure into two rectangular prisms as shown below. Use these figures to answer questions 5 and 6.

 

 

                              

 

5. What are the dimensions of prism A?

A
B
C
D

 

6. What is the volume of prism B?

 

A	15 cubic feet
B	48 cubic feet
C	84 cubic feet
D	120 cubic feet

 

7. How many milliliters are in a cubic centimeter?

 

A	1 mL
B	10 mL
C	100 mL
D	1,000 mL

 

8. The water level before you place an object in a graduated cylinder is 32 mL. The water level after you place the object in the graduated cylinder is 41 mL. What is the volume of the object?

A	9 cubic centimeters
B	12 cubic centimeters
C	19 cubic centimeters
D	41 cubic centimeters

 

 

9. Object A was placed in a graduated cylinder with a water level of 19 mL. The water level rose to 36 mL. What is the volume of object A?

 

A	7 cubic centimeters
B	17 cubic centimeters
C	23 cubic centimeters
D	27 cubic centimeters

 

 

Multiple-Choice Key:

1. B	2. D	3. A	4. C	5. C
6. D	7. A	8. A	9. B

 

 

 

Short-Answer Items:

 

10. Why is volume measured in cubic units, such as cubic centimeters? Why is volume not measured only in units, such as centimeters?

 

 

 

 

 

11. Why can the volume formula for a rectangular prism, , also be written as ?

 

 

 

 

 

12. When finding the volume of an object using displacement, why do you need to measure the water level before and after the object is put in the graduated cylinder?

 

 

 

 Short-Answer Key and Scoring Rubrics:

 

10. Why is volume measured in cubic units, such as cubic centimeters? Why is volume not measured only in units, such as centimeters?

Volume is a measure of how much fits inside an object or of how much space an object takes up. Centimeters or units only measure one dimension: distance. Cubic centimeters or cubic units measure three dimensions (length, width, and height) or how much fits inside an object.

 

Points	Description
2	The student provides an accurate explanation, describes the difference between units and cubic units, and makes the connection to volume instead of simply distance.
1	The student provides an accurate explanation but does not describe the difference between units and cubic units nor make the connection to volume instead of simply distance.
0	The student does not provide an accurate explanation.

 

11. Why can the volume formula , also be written as ?

 

A complete explanation would include a description of the product of length and width resulting in the area of the base and for this reason  can be replaced by B, or a description of how this product  determines how many cubic units are in one layer and h indicates how many layers.

 

Points	Description
2	The student explains that both formulas result in the same solution and provides clear and accurate evidence of the reasons why.
1	The student explains that both formulas result in the same solution and provides some evidence of understanding the reasons why.
0	The student provides no correct explanation of why both formulas result in the same solution.

 

 

12. When finding the volume of an object using displacement, why do you need to measure the water level before and after the object is put in the graduated cylinder?

 

The volume of the object is equivalent to the amount of water it displaces. For this reason, you must record the level of the water before and after the object is put into the cylinder to determine how much water the object displaced.

 

Points	Description
2	The student provides evidence of a clear understanding of the equivalence of the volume of the object to the amount of water displaced.
1	The student provides limited evidence of understanding of the equivalence of the volume of the object to the amount of water displaced.
0	The student provides no evidence of understanding of the equivalence of the volume of the object to the amount of water displaced.

 

 

 

 

 

 

 

Performance Assessment:

A cereal company has asked you to redesign a cereal box that would better fit the contents of the product inside. The cost of cardboard used to make the packaging has increased. The company does not want to increase nor decrease the amount of cereal it puts in each package. The company just wants to reduce the size of the box.

You have been given the following information by the cereal company:

• The original box has the following dimensions: 8 inches × 11 inches × 2 inches.
• The volume of the cereal inside the box is approximately 110 cubic inches. (Note that a cereal box always has some empty space. This is why the volume of the space inside the box and the volume of the cereal are different.)

Task:

• Determine the volume of a redesigned cereal box that would best fit the actual cereal.
• Show the dimensions (using whole numbers) for different boxes that would give you the volume necessary to package the cereal.
• Pick the box that would give the best dimensions in terms of volume, storage, and visual appeal.
• Make a net of your box design on chart graph paper located by the board.
• Explain why that box design would be the best.

 

Questions to answer:

 

1.   What is the volume of the original cereal box?

 

_____________________________________________________________________

 

 

 

2.   What do you notice about the volume of the original cereal box compared to the volume of the cereal?

 

     _____________________________________________________________________

 

 

 

3. Why do you think the cereal company would design the original cereal box in this way?

 

 

_____________________________________________________________________

 

_____________________________________________________________________

 

4. Determine dimensions for at least 4 different boxes that would give the volume necessary to package the cereal.

 

        

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

 

 

 

5. What strategies did you use to determine these different dimensions?

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

 

 

 

6. Which dimensions for the cereal box do you think would be the best in terms of volume, storage, and visual appeal?

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

 

 

 

 

7. Explain why you think this box design would be the best. Use specific detail and mathematical data to support your answer.

 

      _____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

 

 

8. Which box design do you think would be an ineffective box and why?

_____________________________________________________________________

 

_____________________________________________________________________

 

_____________________________________________________________________

 

 

 

9.   Make a net of the box design you consider best on grid paper. Be sure to label your net with the dimensions and units used.

 

Performance Assessment Key and Scoring Rubric:

Many of the answers will vary based on student thinking.

Look for mathematically sound, logical, and reasonable thought processes in student answers.

 

Question # 1: 176 cubic inches

 

Question #2: The volume of the cereal is smaller than the volume of the original cereal box. Some extra space is needed, however, so cereal has some room to move around and does not get crushed in packaging.

 

Question #3: Points to consider: Companies may want customers to think they are getting more than they actually are. Companies may want to protect the product inside from being crushed during packaging by leaving extra room inside the box.

 

Question #4: Any rectangular prism that has a volume between 110 and 176 cubic inches would be considered correct.

 

Question #5: Strategies that students might use will vary. Some strategies students use may include the volume formula, multiplication, building with cubes, or using drawings.

 

Questions #6 and #7: Points to consider: Students should choose a box that has a little more room in the box than the actual cereal volume so the cereal does not get damaged in shipping. The box size chosen should be easily stackable. The front face should be large enough to design a logo. The box should not be too tall or too short.

 

Question #8: Points to consider: Students should mention that a box whose base is too narrow might not be stackable. If the front face of a box is not large enough there may not be enough room to design a logo. This may affect the visual appeal of the cereal. A box that is too tall may be clumsy to get cereal out of, especially for smaller children. A box that is too short may not be effective because the cereal might spill while being poured.

 

Question #9: Answers will vary. Verify that the net is appropriate for the dimensions of the new cereal box the student chose.

Performance Assessment Scoring Rubric:

Points	Description
4	Understanding of the concept of volume is clearly evident. The student uses effective strategies to get accurate and reasonable answers. Many varied accurate dimensions are shown. The student’s response clearly shows logical thinking and the steps used to arrive at conclusions. Explanations are clear and supported with specific details and mathematical data. Accurate answers are shown for dimensions and the net.
3	Understanding of the concept of volume is evident. The student uses appropriate strategies to arrive at reasonable answers. Some varied and accurate dimensions are shown. The student’s response shows thinking and some steps used to arrive at conclusions. Explanations have some details and mathematical data. Minor errors in computation may result in dimensions that are close.
2	Limited understanding of the concept of volume is evident. The student uses strategies that may be ineffective or inaccurate. Steps show some evidence of understanding how to calculate volume. A few dimensions are shown. Explanations may not be developed with detail or may be difficult to follow. The student may arrive at an answer using flawed reasoning.
1	Lack of understanding of the concept of volume is evident. Attempts are made to solve the problem, but the student is unable to work through steps to arrive at reasonable answers. Information from the problem is not used correctly. Explanation is limited and may contain incorrect reasoning.
0	Complete lack of understanding of the concept of volume is evident. No attempts are made to solve the problem. The student shows no understanding of the questions. No explanation is given.