End-of-Unit Assessment

1.

Which is closest to the difference in the volume of the two cylinders?

A.

15,795 cm³

B.

2,534 cm³

C.

806 cm³

D.

512 cm³

Answer:

806 cm³

Teaching Notes

Students who select choice A may have squared both the radius and the height before determining the difference. Students who select choice B may have squared $\pi$ as well as the radius. Students who select choice D may have calculated the volume of Cylinder B as with having a radius of 5 instead of 2.5.

2.

A sphere has a volume of $972\pi$ cm³. What is its radius in centimeters?

A.

36

B.

27

C.

10

D.

9

Answer:

9

Teaching Notes

Students who select choice B used the formula $V=\frac43 \pi r^2$ instead of the correct formula with $r^3$ . Students who select choice C may have multiplied by $\frac{4}{3}$ instead of the reciprocal of $\frac43$ while solving for $r$ . Students who select choice A may have made both errors mentioned above. Students should arrive at the same correct answer for any reasonable choice of the approximation of $\pi$ .

3.

The graph shows the relationship between a sphere’s radius and volume.

<p>A graph. Volume, in cubic inches. Radius, in inches.</p>

Select all the statements that are true about this relationship.

A.

If the radius doubles, the volume doubles.

B.

If the radius doubles the volume becomes about 4 times bigger.

C.

If the radius doubles, the volume becomes about 8 times bigger.

D.

The relationship between radius and volume is linear.

E.

The relationship between radius and volume is not linear.

Answer: C, E

Teaching Notes

Students who select choice A instead of choice C may have thought proportional reasoning applies, but it does not here. Students who select choice B instead of choice C may be remembering examples involving cones and cylinders of a given height, in which the volume does become 4 times bigger when doubling the radius. The difference here is that a sphere’s volume is a cubic function of its radius. Students who select choice D instead of choice E have a misunderstanding about linear functions.

4.

For cylinders with radius 2 units, let $h$ represent the cylinder's height, in units, and the $V$ represent the cylinder's volume in cubic units.

Complete the table relating the height and volume of cylinders that each have a radius of 2 inches. Write each volume as a multiple of $\pi$ , or round to the nearest cubic unit.
Is there a linear relationship between height and volume? Explain how you know.

$h$	$V$
1
2
3

Answer:

$h$ $V$

1 $4\pi$

2 $8\pi$

3 $12\pi$
Yes. Sample explanations: There is a linear relationship because the points would lie on the same line (the rate of change is the same); There is a linear relationship because there is a proportional relationship, and all proportional relationships are linear.

$h$	$V$
1	$4\pi$
2	$8\pi$
3	$12\pi$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

See table.
Yes, because the points would lie on the same line. The rate of change is the same.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: One volume in part a is incorrect, but the explanation for part b is correct based on independent justification; answer for part b is something like “because 8 is two times 4" but does not extend the proportionality argument to address the entire table.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Any explanation for part b that does not appeal to slope, proportionality, the form of the equation for a line, or other concepts related to linearity; the table completed incorrectly for part a, with an incorrect answer to part b (including answers based on an incorrect table).

Teaching Notes

While most students will argue that the relationship is linear because of the constant differences in the table, others may find a constant of proportionality or appeal to the equation relating height and volume, $V=4 \pi h$ .

5.

A cone and cylinder have the same height, and their bases are congruent circles. If the volume of the cylinder is 120 in³, what is the volume of the cone?

Answer:

40 in³

Teaching Notes

For this question, students can use the fact that the volume of a cone is $\frac{1}{3}$ of the volume of a cylinder with the same height and same base. Students may also use the volume of the cylinder to solve for the radius and in turn use the radius to determine the volume of the cone.

6.

Three students each calculated the volume of a sphere with a radius of 6 centimeters.

Diego found the volume to be $288\pi$ cubic centimeters.
Andre approximated 904 cubic centimeters.
Noah calculated 226 cubic centimeters.

Do you agree with any of them? Explain your reasoning.

Answer:

Diego and Andre are both correct. Sample explanation:

Diego: $V=\frac{4}{3}\pi r^3$ , $V=\frac{4}{3}\pi\boldcdot6^3$ , $V=\frac{4}{3}\boldcdot216\pi$ , $V=288\pi$ .

Andre also got $288\pi$ but multiplied 288 by 3.14 and rounded his answer to 904.

Diego expressed his answer in terms of $\pi$ , whereas Andre’s answer is the volume to the nearest cubic centimeter.

Minimal Tier 1 response:

Work is complete and correct.
Acceptable errors: Saying that Andre is incorrect with the rationale that using a multidigit approximation of $\pi$ yields an answer closer to 905 cm.
Sample: Diego and Andre are both correct. $V=\frac{4}{3}\pi\boldcdot6^3$ , so Diego got $288\pi$ and Andre multiplied 288 by 3.14 instead of $\pi$ to get 904.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Identifying only Andre or only Diego as being correct, not explaining that Andre’s answer involves an approximation for $\pi$ ; correctly identifying Andre and Diego as correct but with no or very little explanation.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Calculations are done using an incorrect volume formula; incorrect answer with no or little work shown; correct answer with no work shown.

Teaching Notes

There are two correct answers for this question. Both are approximations of the sphere’s volume, with one expressed in terms of $\pi$ . Students who select Noah have applied the wrong formula for volume of a sphere.

7.

There are many cones with a height of 12 inches. Let $r$ represent the radius and $V$ represent the volume of these cones. The relationship between $r$ and $V$ is given by $V=4\pi r^2$ .

Plot points that show the volume when $r=1$ , $r=2$ , $r=3$ , and $r=4$ .
Is there a linear relationship between the radius and the volume of these cones? Explain how you know.
A vendor at a street fair sells popcorn in cones, all of height 12 inches. The sharing-size cone has 3 times the radius of the single-size cone. About how many times more popcorn does the sharing-size cone hold than the single-size cone? Explain how you know.

Answer:

No. Sample reasonings:
1. The four points are not on the same line, because the slope between the pairs of points is not the same. The slope between $r=1$ and $r=2$ is about 38 cubic inches per inch, but the slope between $r=2$ and $r=3$ is about 63 cubic inches per inch.
2. Since the point $(0,0)$ is also on the graph, if this relationship is linear, then doubling the value of $r$ from 1 to 2 means doubling the value of $V$ from $4\pi$ to $8\pi$ . Since the value of $V$ at $r=2$ is $16\pi$ , this relationship is not linear.
9 times more popcorn. Sample reasoning: Since the sharing size cone is 3 times wider in two different dimensions, square the scale factor: $3^2=9$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: The equation in part a uses $\pi$ instead of 3.14.
Sample:

See graph.
No. Increasing from a 1-inch to a 2-inch radius, the volume goes up by 38, but increasing from a 2-inch to a 3-inch radius, the volume goes up by 63.
9 times more, because $3^2=9$ .

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Good work on parts b and c with errors in part a; one or two points on the graph is plotted incorrectly; work in part b states that points do not lie along the same line or do not fit a consistent slope but does not provide numerical evidence; correct answer to part c without justification

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Argument in part b does not involve comparing rates in some way; argument in part b confuses “linear” with “proportional”; little progress made on parts a and b but a good explanation for part c; points in part a are consistently off, perhaps because of an error applying the equation, such as squaring the coefficient of $r$ ; answer to part c is “3 times bigger,” regardless of justification.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Errors in using the formula for the volume of a cone prevent meaningful work on any of the problem parts; multiple Tier 3 error types.

Teaching Notes

Students will have the easiest time plotting the points if they approximate in their equation. To determine whether there is a linear relationship, students may plot points or calculate the slope (or rate of change) between two pairs of points. At this stage in their learning, it is not expected that students will use the form of an equation to reason that a relationship is linear or quadratic.