End-of-Unit Assessment

Answer:

26 cm$^3$

The graph shows the relationship between the radius and volume for many cones whose height is 6 inches.

A curve in an x y plane. Origin 0. Vertical axis, volume, cubic inches, from 0 to 45 by 5s. Horizontal axis, radius, inches, from 0 to 4 by 0 point 25’s. 

Select all the true statements about such cones.

Answer: B, D, E

A sphere has radius 2.7 centimeters.

What is its volume, to the nearest cubic centimeter?

Answer:

82

For cones with radius 6 units, let $h$ represent the cone's height, in units and $v$ represent the cone's volume in cubic units.

1. Sketch the graph of this relationship on the axes.
2. Is there a linear relationship between height and volume? Explain how you know.

Answer:

1. See graph.
2. Yes. Sample explanations: There is a linear relationship because the equation relating height and volume is in the same form as $y = mx + b$ with $m=12\pi$ and $b=0$; There is a linear relationship because there is a proportional relationship, and all proportional relationships are linear.

Minimal Tier 1 response:

• Work is complete and correct.
• Sample:

1. See graph.
2. Yes, because the volume is $12 \pi$ times the height.

Tier 2 response:

• Work shows general conceptual understanding and mastery, with some errors.
• Sample errors: Some points in part a are incorrectly plotted, but the explanation for part b is correct based on independent justification; answer for part b is something like “the graph looks like a line” without further justification that the graph is linear.

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; an incorrect graph in part a, with an incorrect answer to part b (including answers based on the nonlinearity of the graph).

Answer:

11.8 meters. If the height in meters is $h$, then the equation $\pi \boldcdot (1.6)^2 \boldcdot h = 95$ is true. Using 3.14 as an approximation for $\pi$ gives the equation $8.04h = 95$, and the solution to this equation is $h \approx 11.8$.

Answer:

Sample explanations:

• Cone B has radius 3 units and height 16 units. When the radius is halved, the height must be multiplied by 4 to keep the volume the same.
• Cone B could have any radius $r$ and height $h$ as long as $\frac 1 3 r^2h = 48$, or $r^2h = 144$. One possible solution is radius 2 units and height 36 units, but there are others.

Minimal Tier 1 response:

• Work is complete and correct.
• Acceptable errors: Omission of units; work demonstrates that the volume of Cone B is $48\pi$ but does not directly compare to Cone A.
• Sample: Radius 4 and height 9, which has volume $\frac13 \cdot\pi \cdot 4^2 \cdot 9$. This is $48\pi$ cubic units, the same as Cone A.

Tier 2 response:

• Work shows general conceptual understanding and mastery, with some errors.
• Sample errors: Arithmetic errors involving scaling; carefully written work reveals arithmetic errors involving the volume formula of a cone; scaling arguments involving a similar but incorrect volume formula, such as the formula for the volume of a cylinder.

Tier 3 response:

• Significant errors in work demonstrate lack of conceptual understanding or mastery.
• Sample errors: Conceptual errors involving scaling; calculations are done using an incorrect volume formula; incorrect answer with no or little work shown; correct answer with no work shown and no comparison to Cone A.

Answer:

1. $r$	$V$
   1	$9 \pi$ or 28
   2	$36 \pi$ or 113
   3	$81 \pi$ or 254

2. No. Sample reasonings:
   1. The three points in the table 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 85 cubic inches per inch, but the slope between $r=2$ and $r=3$ is about 141 cubic inches per inch.
   2. Since the point $(0,0)$ is also a solution to the equation, if this relationship is linear, then doubling the value of $r$ from 1 to 2 means doubling the value of $V$ from $9\pi$ to $18\pi$. Since the value of $V$ at $r=2$ is not $18\pi$, this relationship isn't linear.
3. 4 small cups. Sample reasoning: When the radius is scaled by a value, it is scaled in two dimensions. When it is scaled in two dimensions, the scale factor gets squared: $2^2=4$.

Minimal Tier 1 response:

• Work is complete and correct, with complete explanation or justification.
• Acceptable errors: Minor rounding mistakes, omission of units.
• Sample:

1. See table.
2. No. Increasing from a 1-inch to a 2-inch radius, the volume goes up by 85 cubic inches, but increasing from a 2-inch to a 3-inch radius, the volume goes up by 141.
3. 4 cups, because if the radius of a cylinder is doubled, the volume gets quadrupled.

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; 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.
• Acceptable errors: A good argument about linearity in part b is based on incorrect table entries in part a.

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 given for part c; answer to part c is “2 cups,” 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; two or more error types under Tier 3 response.