Grade 8

End-of-Unit Assessment

Select all the points that are on the graph of the line $y=\text-2x+8$ .

$(0,8)$

$(\text-4,6)$

$(\text-2,8)$

$(10,\text-1)$

$(4,0)$

(3,2)

Answer: A, E, F

Teaching Notes

Students who select B or D may be reversing the $x$ - and $y$ -coordinates. Students who select C may be misled by the fact that the slope is -2 and the $y$ -intercept is 8. Students who do not select A, E, or F may not know that the graph of a line is the set of all solutions to the corresponding equation.

For one month, a music service tracked the number of downloads each day for 4 songs, represented by the lines $\ell$ , $j$ , $m$ , and $d$ . Which statement is true?

<p>A graph. Song downloads. Days. Line l. Line j. Line m. Line d.</p>

The number of song downloads represented by lines $j$ and $d$ both decreased over the month.

Initially, the number of song downloads represented by lines $\ell$ and $d$ was the same.

The number of song downloads represented by line $m$ remained constant throughout the month.

The number of song downloads represented by line $\ell$ steadily increased over the month.

Answer:

The number of song downloads represented by line $m$ remained constant throughout the month.

Teaching Notes

Students identify descriptions that could match a given graph. The descriptions indicate the rate of change or slope of the linear graph, and include positive, zero, and negative slope. They also compare two positive slopes.

Students who select A may not understand the interpretation of a positive slope as the number of song downloads increasing over time. Students who select B may not understand the connection between the vertical intercept and an initial amount. Students who do not select C may not understand that horizontal lines have a slope of 0, indicating that the number of song downloads is neither increasing nor decreasing over time. Students who select D may not understand the interpretation of a negative slope as the number of song downloads decreasing over time.

A pool holds 19,900 gallons of water. The pool can be filled using a large hose or a small hose. On an average day, it takes 5 hours to fill the pool with the large hose and 12 hours with the small hose. Which graph best represents this scenario?

<p>A graph. Water used. Gallons. Time. Hours. Large hose. Small hose.</p> — A

<p>A graph. Water used. Gallons. Time. Hours. Small hose. Large hose. </p> — B

<p>A graph. Time. Hours. Water used. Gallons. Large hose. Small hose.</p> — C

<p>A graph. Time. Hours. Water used. Gallons. Small hose. Large hose.</p> — D

Graph A

Graph B

Graph C

Graph D

Answer:

Graph A

Teaching Notes

Students interpret proportional relationships from given lines. They must identify the slope of the lines as the unit rate but also quantitatively compare these unit rates in the absence of a given scale on the axes.

Students who select C instead of A may be associating the slope with the gallons of water, but the axes do not reflect this. Students who select B may be misinterpreting the meaning of the slope of each line. Students who select D may not understand the horizontal line represents no change over time.

Write an equation for each line.

<p>A graph. Line k. Line m. Line p. Line r.</p>

Answer:

line $k$ : $y=x+4$ (or equivalent), line $m$ : $y=\text-2x+4$ (or equivalent), line $p$ : $y=\text-6$ (or equivalent), line $r$ : $x=1$

Teaching Notes

Students write equations for four lines given their graphs. One line is vertical, one is horizontal, one has positive slope, and one has negative slope.

Three different airplanes take off from an airport, and each maintains a constant speed until they near their destination.

The equation $d=9.5t$ represents the distance of the first airplane (in miles), $d$ , from the airport after $t$ minutes.

The second airplane's information is in the table:

time (minutes)	distance from airport (miles)
2	16
10	80
35	280
60	480

The graph shows the distance from the airport (in miles) of the third plane with respect to the time in minutes.

<p>A graph. time in minutes, distance in miles </p>

Which airplane is flying the slowest? Explain how you know.

Answer:

The second airplane is flying the slowest. Sample reasoning: The equation for the first airplane shows that it is flying at a speed of 9.5 miles per minute. Within the table, the unit rate for the second airplane is 8 miles per minute, because $16\div2=8$ . Using the points $(0,0)$ and $(30,300)$ from the graph for the third airplane, the slope is 10, showing that it is flying at a speed of 10 miles per minute.

Since the second airplane only travels 8 miles in 1 minute, it is flying the slowest.

Minimal Tier 1 response:

Work is complete and correct.
Sample: The first airplane is flying at a speed of 9.5 miles per minute. The second airplane is flying at a speed of 8 miles per minute. The third airplane is flying at a speed of 10 miles per minute. The second airplane is flying the slowest because it travels the shortest distance in one minute.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Work contains correct unit rates for all three airplanes but concludes that the first or third airplane is flying the slowest or does not name the slowest airplane; one unit rate is incorrect (possibly with an incorrect slowest airplane identified as a consequence); insufficient explanation of work.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Two or more incorrect unit rates; the correct airplane is identified but with no justification; response to the question is not based on unit rates or on similar methods.

Teaching Notes

Students compare the speed of three different airplanes where the proportional relationship between distance and time is represented in three different ways.

A sandwich store delivers sandwiches for a fee. The total cost to order from this store can be described by the equation $y=10+4.5x$ where $x$ is the number of sandwiches ordered and $y$ is the total cost including the delivery fee.

A second store also delivers sandwiches for a fee. A graph representing the number of cost to order sandwiches from this store has a vertical intercept at $(0,5)$ and is parallel to a graph describing the first sandwich store.

Jada wants to order 6 sandwiches and says that it would cost less to order from the second sandwich store. Do you agree or disagree with Jada? Explain your reasoning.

Answer:

I agree with Jada. Sample reasoning: The equation that represents the first store has a slope of 4.5 and a $y$ -intercept of 10, which means that sandwiches cost $4.50 each and there is a $10 delivery fee. The graph representing the second store has a $y$ -intercept of 5 and is parallel to a graph of the first store, which means that the second store also charges $4.50 per sandwich but there is only a $5 delivery fee. The cost per sandwich is the same but the second store has a lower delivery fee, making it less expensive for 6 sandwiches.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
An explanation that compares the slopes and vertical intercepts without providing numbers, or an explanation that includes a graph or equation is also acceptable.
Sample: See solution above.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Work does not explicitly state agreement or disagreement with Jada but can be inferred from the reasoning; work in the form of a table, graph, or calculations are done that show Jada is correct but the explanation is incomplete.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not include any explanation; work shows that Jada's statement is incorrect.

Teaching Notes

Students analyze the descriptions of two nonproportional linear relationships and use what they know about slopes and vertical intercepts to determine whether a given statement is true.

A truck is shipping jugs of drinking water and cases of paper towels. A jug of drinking water weighs 40 pounds and a case of paper towels weighs 16 pounds. The truck can carry 2,000 pounds of cargo altogether.

Complete the table showing three ways the truck could be packed with jugs of water and cases of paper towels so that it is carrying 2,000 pounds of cargo.

jugs of drinking water ( $w$ ) cases of paper towels ( $t$ )

10

50

5
Write an equation describing the number of jugs of water $w$ , and cases of paper towels $t$ , the truck can carry.
Draw a graph of the solutions to your equation.
A different truck can carry 3,000 pounds of cargo altogether. How would a graph representing this new truck be the same and different from the graph representing the truck that could carry 2,000 pounds?

jugs of drinking water ( $w$ )	cases of paper towels ( $t$ )
10
	50
	5

Answer:

jugs of drinking water ( $w$ ) cases of paper towels ( $t$ )

10 100

30 50

48 5
$40w+16t=2000$ (or equivalent)
Sample graph:
Sample responses: The two graphs would have the same slope but different vertical intercepts. The graph of the new line would be parallel to the graph of the original line.

jugs of drinking water ( $w$ )	cases of paper towels ( $t$ )
10	100
30	50
48	5

Minimal Tier 1 response:

Work is complete and correct.
The graph may be a continuous line, or it may consist only of points representing whole numbers of jugs of water and cases of paper towels.
The graph may have the axes and labels reversed, where the cases of paper towels is on the horizontal axis, and the jugs of drinking water is on the vertical axis.
Sample: See solution above.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Graph contains only the three points from the table; scale chosen does not allow for all relevant data to be displayed; reasonable work in part 1 but results in an incorrect equation.
Acceptable errors: Equation and graph are correct based on an incorrect proportional relationship in the table; Graph is correct based on an incorrect equation.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not factor in the 2,000 pound constraint; equation is not linear or does not make sense in this situation; incorrect answers in 2 or more representations.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Does not come up with an equation or graph; omission of or incorrect work in three or more problem parts.

Teaching Notes

A linear equation is described in terms of a constraint on the total weight of two different types of freight. The constraint gives an equation that students produce. Students then graph the solutions to the equation.