End-of-Unit Assessment

1.

Select all the points that are on the graph of the line $2x + 4y = 20$ .

A.

$(0,5)$

B.

$(0,10)$

C.

$(1,2)$

D.

$(4,2)$

E.

$(5,0)$

F.

$(10,0)$

Answer: A, F

Teaching Notes

Students who select B, D, or E may be reversing the $x$ - and $y$ -coordinates. Students who select C may be misled by the fact that the coefficients of $x$ and $y$ are in a 1:2 ratio. Students who do not select A or F may not know that the graph of a line is the set of all solutions to the corresponding equation.

2.

For two weeks, the highest temperature each day was recorded in four different cities, represented by the lines $\ell$ , $m$ , $n$ , and $p$ . Which statement is true?

graph. horizontal axis, time passed in days. vertical axis, temperature in degrees. 4 lines labeled l, m, n, p.

A.

The high temperature in the city represented by line $\ell$ increased as time passed.

B.

The high temperature in the city represented by line $m$ decreased steadily.

C.

Initially, the high temperature was warmer in the city represented by line $p$ than in the city represented by line $m$ .

D.

The high temperature in the city represented by line $p$ increased faster than the high temperature in the city represented by line $n$ .

Answer:

The high temperature in the city represented by line $m$ decreased steadily.

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 that horizontal lines have a slope of 0, indicating that the temperature is neither increasing nor decreasing over time. Students who do not select B may not understand the interpretation of a negative slope as the temperature decreasing over time. Students who select C may not understand the connection between the vertical intercept and an initial amount. Students who select D may not understand that a greater slope means a faster rise in temperature.

3.

Jada earns twice as much money per hour as Diego. Which graph best represents this scenario?

graph with no grid. Horizontal axis money earned in dollars. Vertical axis time worked in hours. 2 lines starting at origin. — A

graph with no grid. Horizontal axis time worked in hours. Vertical axis money earned in dollars. 2 lines starting at origin. — B

A.

Graph A

B.

Graph B

C.

Graph C

D.

Graph D

Answer: Graph B

Teaching Notes

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

Students who select A instead of B may be associating the slope with the amount of money earned per hour, but the axes do not reflect this. Students who select C have misinterpreted the meaning of the slope of each line. Students who select D may not understand what earning twice as much per hour means.

4.

Write an equation for each line.

Answer:

line $\ell$ : $y = 4$ (or equivalent), line $m$ : $y = 4 - 2x$ (or equivalent), line $n$ : $y = x - 1$ (or equivalent), line $p$ : $x = \text -4$

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.

5.

Three runners are training for a race. One day, they all run a lap around a track, each at their own constant speed.

The graph shows the distance in meters that Runner #1 runs with respect to the time in seconds.

graph. horizontal axis, time in seconds, scale = 0 to 70 by 10's. vertical axis, distance in meters, scale 0 to 400, by 50's. line passing through origin and 50 comma 200.

Runner #3’s information is in the table:

time (seconds)	distance (meters)
9	45
11	55
25	125
45	225
60	300

The equation that relates Runner #2’s distance (in meters) with time (in seconds) is $d=6.5t$ .

Which of the three runners runs the fastest? Explain your reasoning.

Answer:

Runner #2 runs the fastest. Sample reasoning: Using the points $(0,0)$ and $(50,200)$ from Runner #1’s graph, the slope is 4, showing that they run 4 meters every second. Runner #2’s equation shows that they run 6.5 meters every second. Within the table, the unit rate for Runner #3 is 5 meters per second because $45\div9=5$ .

Since Runner #2 travels farther every second, Runner #2 is the fastest.

Minimal Tier 1 response:

Work is complete and correct.
Sample: Runner #1 goes at 4 meters per second, Runner #2 goes at 6.5 meters per second, and Runner #3 goes at 5 meters per second. Runner #2 is the fastest because they travel the greatest distance in 1 second.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Work contains correct unit rates for all three runners but concludes that runner #1 or #3 is the fastest or does not name a fastest runner; one unit rate is incorrect (possibly with an incorrect fastest runner 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 runner is identified but with no justification; response to the question is not based on unit rates or on similar methods, such as calculating which runner has gone the farthest after 10 miles.

Teaching Notes

Students compare the pace of three different runners. The proportional relationship between time and distance is represented in three different ways. There is more than one way to do this problem correctly. For example, students could determine how long it takes each runner to run 5 miles to determine the fastest runner, or could determine each runner's speed in miles per minute or miles per hour.

6.

A store is selling notebooks for $1.00 and pencils for $0.25. Jada has $10.00 to spend on school supplies.

Complete the table showing three ways Jada can spend all $10.00 on notebooks and pencils.

number of notebooks ( $n$ ) number of pencils ( $p$ )

6

4

2
Write an equation describing the number of notebooks $n$ , and pencils $p$ that Jada can buy for $10.00.
Draw a graph of the solutions to your equation.
Notebooks still cost $1.00 and pencils still cost $0.25, but now Jada has $15 to spend on supplies. How would a graph representing this new situation be the same and different from the graph representing when Jada had $10 to spend?

number of notebooks ( $n$ )	number of pencils ( $p$ )
6
	4
2

Answer:

number of notebooks ( $n$ ) number of pencils ( $p$ )

6 16

9 4

2 32
$n+0.25p=10$ (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.

number of notebooks ( $n$ )	number of pencils ( $p$ )
6	16
9	4
2	32

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 notebooks and pencils.
The graph may have the axes and labels reversed, where the number of pencils is on the horizontal axis, and the number of notebooks is on the vertical axis.
Sample: See solution on graph.

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 $10 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 cost to purchase a combination of notebooks and pencils with a given budget. The constraint gives an equation that students produce. Students then graph the solutions to the equation.

7.

Han has a music playlist and each day he adds more songs to his list. The equation $y=4x+20$ describes Han’s playlist, where $x$ is the number of days, and $y$ is the total number of songs.

Tyler also has a music playlist. A graph representing the number of songs on Tyler’s playlist each day has a vertical intercept at $(0,12)$ and is parallel to the graph describing Han’s music playlist.

Tyler says that in 3 days he will have more songs on his playlist than Han will have on his. Do you agree or disagree with Tyler? Explain your reasoning.

Answer:

I disagree with Tyler. Sample reasoning: The equation that represents Han’s music playlist has a slope of 4 and a $y$ -intercept of 20, which means that Han starts out with 20 songs on his playlist and adds 4 songs each day. The graph representing Tyler’s playlist has a $y$ -intercept of 12 and is parallel to a graph of Han’s playlist, which means that Tyler starts out with 12 songs on his playlist and also adds 4 songs each day. Both Han and Tyler add the same amount of songs each day, so the rate of change for both situations is the same. However, since Han started out with more songs, Tyler will never catch up or have more songs on his playlist than Han.

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 Han but can be inferred from the reasoning; work in the form of a table, graph, or completed calculations that shows Han is incorrect, 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 Han’s statement is correct.

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.