Unit 5 Functions And Volume — Unit Plan

Title

Assessment

Lesson 1

Inputs and Outputs

What's the Rule?

Fill in the table for this input-output rule:

An input-output rule diagram. Input, 4, right arrow, rule is, divide by 2 and add 1, right arrow, output, 3.

input	output
0
2
-8
100

Show Solution

In each row, the output should be one more than half of the input.

input	output
0	1
2	2
-8	-3
100	51

Lesson 2

Introduction to Functions

Wait Time

You are in line to watch the volleyball championship. You are told that you will have to wait for 50 minutes in line before they open the doors to the gym and you can find a seat. Determine whether:

You know the number of seconds you have to wait.
You know the number of people in line.

For each statement, if you answer yes, draw an input-output diagram, and write a statement that describes the way one quantity depends on another.

If you answer no, give an example of 2 outputs that are possible for the same input.

Show Solution

Yes. Sample response: The number of seconds to wait depends on the number of minutes to wait.
No, if I know how many minutes I have to wait in line, I do not necessarily know how many people are in line. Sample response: The number of people who have to wait cannot be determined by the amount of time someone has to wait. For example, there could be 50 people waiting, or there could be 100 people waiting.

Section A Check

Section A Checkpoint

Problem 1

Here is a table of inputs and outputs for a relationship, but one of the numbers is missing.

What number could the missing input be if this relationship is a function?
What number could the missing input be if this relationship is not a function?

input	output
1	5
2	8
3	10
7	6
	15
20	14

Show Solution

any value not already listed as an input
any value already listed as an input

Lesson 3

Equations for Functions

The Value of Some Quarters

The value $v$ of your quarters (in cents) is a function of $n$ , the number of quarters you have.

Draw an input-output diagram to represent this function.
Write an equation that represents this function.
Find the output when the input is 10.
Identify the independent and dependent variables.

Show Solution

See diagram:
$v = 25n$ . This reflects the statement that the value (in cents) of my collection of quarters is always 25 times the number of quarters I have.
When the input is 10, the output is 250 (since $250=25\boldcdot 10$ ).
$n$ is the independent variable, and $v$ is the dependent variable.

Lesson 4

Tables, Equations, and Graphs of Functions

Subway Fare Card

Here is the graph of a function showing the amount of money remaining on a subway fare card as a function of the number of rides taken.

Coordinate plane, horizontal, number of rides, 0 to 20 by ones, vertical dollars on card, 0 to 50 by fives. Line begins at 0 comma 45, through labeled point P = 7 comma 27 point 5, ends at 18 comma 0.

What is the output of the function when the input is 10? On the graph, plot this point and label its coordinates.
What is the input to the function when the output is 5? On the graph, plot this point and label its coordinates.
What does point $P$ tell you about the situation?

Show Solution

20. See graph in part 2.
16
After taking 7 rides, there will be $27.50 remaining on the card.

Lesson 5

More Graphs of Functions

Diego’s 10K Race

Diego runs a 10-kilometer race and keeps track of his speed.

Coordinate plane, horizontal, distance in kilometers, 0 to 10 by twos, vertical, speed in kilometers per hour, 9 to 13 by ones.

What was Diego’s speed at the 5-kilometer mark in the race?
According to the graph, where was Diego when he was going the slowest during the race?
Describe what happened to Diego’s speed in the second half of the race (from 5 kilometers to 10 kilometers).

Show Solution

10 kilometers per hour
3 kilometers into the race
Sample response: From 5 kilometers to 6 kilometers, Diego went faster, but he slowed down from 6 kilometers to 8 kilometers. He sped up again from 8 kilometers to 9 kilometers and finished the last kilometer at the same speed.

Lesson 7

Connecting Representations of Functions

Comparing Different Areas

The table shows the area of a square for specific side lengths.

side length (inches)	0.5	1	2	3
area (square inches)	0.25	1	4	9

The area $A$ of a circle with radius $r$ is given by the equation $A = \pi \boldcdot r^2$ .

Is the area of a square with side length 2 inches greater than or less than the area of a circle with radius 1.2 inches?

Show Solution

Less than. From the table, we see that the area of a square of side length 2 inches is 4 square inches, whereas from the equation, we find that the area of a circle with radius 1.2 inches is about 4.52 square inches.

Section B Check

Section B Checkpoint

Problem 1

The relationship between the circumference of a circle and its radius is represented by this graph:

Graph of the relationship between the circumference of a circle and its radius. Horizontal, radius in centimeters, vertical circumference in centimeters.

The perimeter in centimeters, $P$ , of a rectangle whose length is twice the size of its width in centimeters, $w$ , is given by the equation $P=6w$ .

Compare the outputs of the two functions when the inputs for each function is 4.
For the relationship between $P$ and $w$ , name the independent and dependent variables.
Han says that the circumference is increasing faster than the perimeter. What do you think he means by that?

Show Solution

Sample response: When the radius of the circle is 4 cm, the circumference is $8\pi$ cm. When the width of the rectangle is 4 cm, the perimeter is 24 cm.
Sample response: For the equation $P=6w$ , we can input $w$ to calculate the value of $P$ , the output. So $w$ is the independent variable, and $P$ is the dependent variable.
Sample response: The rate of change for the circumference is $2\pi$ cm (or about 6.28 cm) per 1 cm increase in the input, while the rate of change for the perimeter of the rectangle is 6 cm per 1 cm increase in input. Since the output changes by more for each increase of the input by 1 cm, the circumference is increasing faster.

Problem 2

The graph shows Tyler’s distance from school as a function of time since school ended.

Coordinate plane, horizontal, time in hours, 0 to 5. Vertical, distance from school in miles, 0 to 2.

Clare walks home right after school. She stays home for an hour, then walks back to school to go to the volleyball game. After the game, she returns home.

Sketch a graph of Clare’s story.
Which quantity is a function of which? Explain your reasoning.
Based on your graph, is Clare’s house closer to school than Tyler’s house? Explain how you know.

Show Solution

Sample response:

Coordinate plane, horizontal, time in hours, 0 to 5. Vertical, distance from school in miles, 0 to 2. Tyler's distance from school is horizontal from 0 comma 0 to 1 comma 0, then linearly upward to 1.5 comma 0.5, then horizontal to 5 comma 0.5. Clare's distance from school is linearly upward from 0 comma 0 to 0.5 comma 0.75, horizontal to 1.5 comma 0.75, then linearly downward to 2 comma 0, then horizontal to 3.5 comma 0, then linearly upward to 4 comma 0.75, then horizontal to 5 comma 0.75.
The distance from the school is a function of time since school ended. Sample reasoning: Clare is at the school at different times, so time cannot depend on distance from the school. For each time, there is one and only one value of distance, so distance must depend on time.
Sample response: No, Clare’s house is farther from school than Tyler’s because the graphs show that Tyler’s house is 0.5 miles from the school and Clare’s house is 0.75 miles from the school.

Lesson 8

Linear Functions

Beginning to See Daylight

In a certain city in France, they gain 2 minutes of daylight each day after the spring equinox (usually in March), but after the autumnal equinox (usually in September), they lose 2 minutes of daylight each day.

Graph A, horizontal, days past the equinox, vertical, minutes of sunlight. Horizontal line above the x-axis. <br>
Graph B, horizontal, days past the equinox, vertical, minutes of sunlight. Begins above the origin and slopes down. <br>
Graph C, horizontal, days past the equinox, vertical, minutes of sunlight. Begins at the origin and slopes up. <br>
Graph D, horizontal, days past the equinox, vertical, minutes of sunlight. Begins above the origin and slopes up. — A

Which of the graphs is most likely to represent the graph of daylight for the month after the spring equinox?
Which of the graphs is most likely to represent the graph of daylight for the month after the autumnal equinox?
Why are the other graphs not likely to represent either month?

Show Solution

D
B
Graph A does not make sense because there is a constant amount of daylight. Graph C does not make sense because it goes through the origin, meaning it started with 0 minutes of daylight.

Lesson 9

Linear Models

Board Game Sales

A small company is selling a new board game, and they need to know how many to produce in the future.

After 12 months, they sold 4 thousand games. After 18 months, they sold 7 thousand games. And after 36 months, they sold 15 thousand games.

Could this information be reasonably estimated using a single linear model? If so, use the model to estimate the number of games sold after 48 months. If not, explain your reasoning.

Show Solution

Predictions between 20 and 22 thousand sales, depending on the data points used for the model, are reasonable.

Sample response: Yes. After 48 months, they sold about 20.5 thousand games. From Month 12 to Month 36, the rate of games sold was about $\frac{11}{24}$ thousand games per month. This means the amount sold during the 12 months from Month 36 to Month 48 was 5.5 thousand, since $\frac{11} {24} \boldcdot 12=5.5$ , and 5.5 thousand added to 15 thousand is 20.5 thousand.

Section C Check

Section C Checkpoint

Problem 1

Here is a piecewise linear model for the water height in feet of a reservoir that supplies water to a nearby town over a 12-month period.

A scatterplot, horizontal, months, 0 to 12, vertical, height of water in feet, 6660 to 6740.

Select all the true statements.

h = \text-4x + 6,676

is a reasonable model for the water level from Months 0 to 3.

B.The slope from Months 3 to 5 is 57 and means that during that time the height of the water increased 57 feet each month.

C.The time is a function of the height of the water.

D.The height of the water is a function of the time.

E.The water height decreased at about the same rate from Months 0 to 3 and from Months 7 to 12.

Show Solution

A, D, E

Problem 2

A large university campus has two bike sharing programs students can pay for each month.

Program A advertises that they charge 34 dollars to join and 1 dollar per mile traveled.
Program B advertises that their fee to join is less than 50 dollars and that they charge less per mile than Program A.

A graph of line l. Horizontal axis, distance traveled in miles. Vertical axis, cost in dollars.

Which program is represented by line $\ell$ ?
If $c$ is the cost in dollars and $m$ is the distance traveled in miles for a month, write a possible equation to represent Program B.

Show Solution

Program A is represented by line $\ell$ .
Any equation with an initial value less than 50 and a rate of change less than 1 dollar per mile. Sample response: $c=45+0.25m$ .

Lesson 12

How Much Will Fit?

Rectangle to Round

Here is a box of pasta and a cylindrical container.

A photo of two objects. The object on the left is a box of pasta that is in the shape of a rectangular prism. The object on the right is an empty, cylindrical container.

The two objects are the same height, and the cylinder is just wide enough for the box to fit inside with all 4 vertical edges of the box touching the inside of the cylinder.

If the box of pasta fits 8 cups of rice, estimate how many cups of rice will fit inside the cylinder. Explain or show your reasoning.

Show Solution

Sample response: About 11 cups of rice since it should be a little more than the box.

Lesson 13

The Volume of a Cylinder

Liquid Volume

The cylinder shown here has a height of 7 centimeters and a radius of 4 centimeters.

A drawing of a cylinder. A dashed line on the bottom base indicating the radius is drawn.

What is the area of the base of the cylinder? Express your answer in terms of $\pi$ .
How many cubic centimeters of fluid can fill this cylinder? Express your answer in terms of $\pi$ .
Give a decimal approximation of your answer to the previous question using 3.14 to approximate $\pi$ .

Show Solution

$16\pi$ cm². The square of the radius of the base is $4^2=16$ , which is multiplied by $\pi$ , giving $\pi\boldcdot 4^2=16\pi$ .
$112\pi$ cm³. The height of the cylinder is 7, which is multiplied by the area of the base, giving $16\pi\boldcdot 7=112\pi$ .
351.68 cm³, because $112\boldcdot 3.14 \approx 351.68$

Lesson 14

Finding Cylinder Dimensions

Find the Height

This cylinder has a volume of $12\pi$ cubic inches and a diameter of 4 inches.

Find the cylinder's radius and height.

An image of a right circular cylinder whit height labeled h and radius labeled r.

Show Solution

The radius is 2 inches, and the height is 3 inches. Since the diameter is 4 inches, the radius is half of 4 inches. The volume is $12\pi=2^2\pi h$ , which means $12\pi=4\pi h$ and $h=3$ .

Lesson 15

The Volume of a Cone

Calculate Volumes of Two Figures

There is a cone with the same base as the given cylinder but with a height that is 3 times taller.

What is the volume of each figure? Express your answers in terms of $\pi$ .

A right circular cylinder with a height of 4 and radius of 3.

Show Solution

Cylinder: $36\pi$ cubic units, because $\pi \boldcdot 3^2 \boldcdot 4 =36\pi$

Cone: $36\pi$ cubic units, because $\frac13 \pi \boldcdot 3^2 \boldcdot 12 = 36\pi$

Lesson 16

Finding Cone Dimensions

A Square Radius

Noah and Lin are making paper cones to hold popcorn to hand out at a family math night.
They want the cones to hold $9\pi$ cubic inches of popcorn.

What are two different possible values for height $h$ and radius $r$ for the cones?

Show Solution

Sample responses:

Height and radius both 3 inches since $\frac{1}{3} \pi \boldcdot 3^2 \boldcdot 3 = 9\pi$ .
Radius 2 inches and height 6.75 inches since $\frac{1}{3} \pi \boldcdot 2^2 \boldcdot 6.75 = 9\pi$ .
Radius 1 inch and height 27 inches since $\frac{1}{3} \pi \boldcdot 1^2 \boldcdot 27 = 9\pi$ .
Radius 9 inches and height $\frac{1}{3}$ inches since $\frac{1}{3} \pi \boldcdot 9^2 \boldcdot \frac{1}{3} = 9\pi$ . (This cone may look more like a plate, but it solves the problem.)

Section D Check

Section D Checkpoint

Problem 1

Two candles are shaped like cylinders.

Candle A has a diameter of 8 cm and a height of 12 cm. Candle B has a radius of 5 cm and a height of 8 cm.

Which candle takes more wax to make? Explain or show your reasoning.

Show Solution

Candle B. Sample reasoning: Candle A has a volume of about 602.88 cm³, since $V=\pi \boldcdot 4^2 \boldcdot 12 \approx 603$ , while Candle B has a volume of about 628 cm³, since $V=\pi \boldcdot 5^2 \boldcdot 8 \approx 628$ .

Problem 2

A cone has a height of 6 cm and a volume of $8 \pi$ cm³.

Sketch the cone.
Find its radius in centimeters. Explain or show your reasoning.
Label your sketch with the cone’s height and radius.

Show Solution

See image.
2 cm. Sample reasoning: The volume of a cone is $V=\frac13 \pi r^2 h$ , so $8\pi=\frac13 \pi r^2 \boldcdot 6$ . This means $8=2r^2$ , so $4=r^2$ , and $r=2$ .

Lesson 17

Scaling One Dimension

A Missing Radius

Here is a graph of the relationship between the height and volume of some cylinders that all have the same radius, 1 ft. An equation that represents this relationship is $V= \pi h$ .

Identify and plot another point on the line, and interpret its meaning.
How can you tell if this relationship is a function?

Coordinate plane, horizontal, height, feet, 0 to 19, vertical, volume, feet cubed, 0 to 65 by 5. Straight line from origin through points labeled 9 comma 28 point 26, 18 comma 56 point 52.

Show Solution

Sample response: The point $(10, 31.4)$ is on the line. A cylinder with radius 1ft and height 10 ft will have a volume of 31.4 ft³.
Sample response: I know this relationship is a function because the equation is in the form $y = mx+b$ and all linear relationships are functions.

Lesson 18

Scaling Two Dimensions

Halving Dimensions

There are many cylinders for which the height and radius are the same value.
Let $c$ represent the height and radius of a cylinder and $V$ represent the volume of the cylinder.

Write an equation that expresses the relationship between the volume, height, and radius of this cylinder using $c$ and $V$ .
If the value of $c$ is halved, what must happen to the value of the volume $V$ ?

Show Solution

$V = \pi c^3$
If the value of $c$ is halved, then the value of the volume would be $\frac18$ of the original volume since $\pi \left(\frac12 c\right)^3=\pi c^3 \left(\frac12\right)^3=\frac18 \pi c^3$ .

Lesson 19

Estimating a Hemisphere

A Reasonable Estimate

A hemisphere fits exactly inside a rectangular prism box with a square base that has edge length 10 inches.

What is a reasonable estimate for the volume of the hemisphere?

Show Solution

Sample responses:

Less than 500 cubic inches. The volume of the box that the hemisphere fits in is $10^2\boldcdot 5$ , and the hemisphere does not take up all the space in the box.
Less than $125\pi$ cubic inches. The volume of the cylinder that the hemisphere fits in is $\pi (5)^2 \boldcdot 5$ , and the hemisphere does not take up all the space in the cylinder.
More than $\frac{125}3\pi$ cubic inches. The volume of the cone that fits in the hemisphere is $\frac13 \pi (5)^2 \boldcdot 5$ , and the hemisphere is larger than the cone.

Lesson 20

The Volume of a Sphere

Volumes of Spheres

Recall that the volume of a sphere is given by the formula $V=\frac 43 \pi r^3$ .

A sphere. A dashed line is drawn from the center of the sphere to the edge of the sphere and is labeled "4."

Here is a sphere with radius 4 feet. What is the volume of the sphere? Express your answer in terms of $\pi$ .
A spherical balloon has a diameter of 4 feet. Approximate how many cubic feet of air this balloon holds. Use 3.14 as an approximation for $\pi$ , and give a numerical answer.

Show Solution

$\frac{256}{3} \pi$ (or $85.33\pi$ ) cubic feet, because $V=\frac43 \pi (4)^3$
33.49 cubic feet, because $V=\frac43 \pi (2)^3$

Lesson 21

Cylinders, Cones, and Spheres

New Four Spheres

Some information is given about each sphere. Order them from least volume to greatest volume. You may sketch a sphere to help you visualize if you prefer.

Sphere A has a radius of 4.

Sphere B has as a diameter of 6.

Sphere C has a volume of 64 $\pi$ .

Sphere D has a radius double that of sphere B.

Show Solution

B, C, A, D

Sphere A has a radius of 4, so its volume is $\frac{256}3\pi$ .

Sphere B has a diameter of 6, so its radius is 3, and its volume is $36\pi$ .

Sphere C has a volume of 64 $\pi$ .

Sphere D has a radius twice as large as sphere B, so its radius is 6, and its volume is $288\pi$ .

Section E Check

Section E Checkpoint

Problem 1

A sphere has a height of 24 inches. Calculate its volume to the nearest inch.

Show Solution

$2304\pi$ in³, or about 7238 in³

Problem 2

Put these figures in order by volume from least to greatest.

A sphere with radius $r$
A cone with radius $r$ and height $r$
A cylinder with radius $r$ and height $r$
A cube with side length $r$

Show Solution

Cube, cone, cylinder, sphere

Unit 5 Assessment

End-of-Unit Assessment

Problem 1

A cylinder has volume 78 cm $^3$ . What is the volume of a cone with the same radius and height?

26 cm $^3$

39 cm $^3$

156 cm $^3$

234 cm $^3$

Show Solution

26 cm $^3$

Problem 2

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. Vertical axis, volume, cubic inches. Horizontal axis, radius, inches.

Select all the true statements about such cones.

The relationship between radius and volume is linear.

The relationship between radius and volume is not linear.

If the radius of the cone doubles, the volume of the cone doubles.

If the radius of the cone doubles, the volume of the cone is multiplied by 4.

If the radius of the cone is 2 inches, the volume of the cone is about 25 cubic inches.

Show Solution

B, D, E

Problem 3

A sphere has radius 2.7 centimeters.

What is its volume, to the nearest cubic centimeter?

Show Solution

Problem 4

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.

Blank coordinate plane, horizontal, height of cone, 0 to 10, vertical, volume of cone, 0 to 500 by 100.

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

Show Solution

<p>A line graphed on a coordinate plane.</p>

See graph.
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:

See graph.
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).

Problem 5

A cylinder has a radius of 1.6 meters. Its volume is 95 cubic meters. Find its height to the nearest tenth of a meter.

Show Solution

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 $h \approx 11.8$ .

Problem 6

Cones A and B both have volume $48\pi$ cubic units but have different dimensions. Cone A has radius 6 units and height 4 units. Find one possible radius and height for Cone B. Explain how you know Cone B has the same volume as Cone A.

Show Solution

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.

Problem 7

There are many cylinders with a height of 9 inches. Let $r$ represent the radius in inches and $V$ represent the volume in cubic inches. The relationship between $r$ and $V$ is given by $V=9\pi r^2$ .

$r$	$V$
1
2
3

Complete the table relating the radius and volume of cylinders with height 9 inches. Write each volume as a multiple of $\pi$ or round to the nearest cubic inch.
Is there a linear relationship between the radius and the volume of these cylinders? Explain how you know.
A water station has two types, small and large, of cylinder cups to choose from. Both cups have a height of 9 inches, but one has twice the radius of the other. How many small cups would it take to fill the large cup? Explain how you know.

Show Solution

$r$ $V$

1 $9 \pi$ or 28

2 $36 \pi$ or 113

3 $81 \pi$ or 254
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.
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:

See table.
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.
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.