Grade 8

Mid-Unit Assessment

Which point is on the graph of the function $y = 3x + 2$ ?

$(1,5)$

$(2,3)$

$(3,2)$

$(5,1)$

Answer:

Teaching Notes

Students who select choice B or C are probably unsure of how to approach the problem and take the numbers directly from the equation. Students who select choice D have switched the input-output pair.

This table shows a linear relationship between the the age of a plant in weeks and its height in centimeters for the first 12 weeks.

age (weeks)	height (centimeters)
1	6
3	8
10	15
12	17

Select all the true statements.

The height of the plant increases at a constant rate of 2 cm every week.

The height of the plant increases at a constant rate of 1 cm every week.

The height of the plant increases at a constant rate of $\frac12$ cm every week.

Age is a function of height.

Height is a function of age.

Answer:

B, D, E

Teaching Notes

Students who select choice A have looked at the first two rows of the height column and have not realized that the increase was over two weeks. Students who select choice C may not understand how to find the constant rate of change. Students who select only one of choices D or E, but not both, may not realize that both can be true at once, or they may think only choice D is true because age is the independent variable.

The graph shows the amount of water in a bathtub starting at 10:00. Select all the true statements.

<p>A graph. Amount of water, in gallons. Time, in minutes after 10 o clock p m.</p>

The tub was filling faster at 10:14 p.m. than at 10:01 p.m.

At 10:06 p.m., the tub was neither filling nor draining.

The maximum amount of water in the tub was about 35 gallons.

The amount of water in the tub stayed the same from 10:16 p.m. to 10:20 p.m.

It took 4 minutes for the tub to drain.

Answer:

B, C, E

Teaching Notes

Students who select choice A may think that because the point with $x$ -coordinate of 14 is higher on the graph, it corresponds to a faster rate of change. Students who select choice D may be confusing a constant rate of change with a constant amount of water in the tub, or they may have a deeper misunderstanding about interpreting graphs.

Mai hikes up a trail for 40 minutes. The graph shows the elevation in feet that she reaches throughout her hike. Name the time period where Mai gains elevation at the fastest rate.

<p>A graph. Elevation, in feet. Time, in minutes.</p>

Answer:

22-26 minutes

Teaching Notes

Check to see if students understand what “time period” means in this context. Some students may answer “10–22 minutes,” looking at the longest section of the graph. Students who answer “32–40 minutes” may be choosing the section of the graph corresponding to the greatest height.

A vine planted today has a height of 5 feet and grows 1 foot each month up the side of a brick building.

Is the height of the vine a function of the number of months? If so, is it a linear or a nonlinear function? Explain your reasoning.

Answer:

It is a linear function. It is a function because there is a single output (the height of the vine on the building) for each input (the number of months). It is linear because the growth rate of 1 foot per month is a constant rate of change. Alternately, we know this is a linear function because it has equation $ℎ = M + 5$ .

Minimal Tier 1 response:

Work is complete and correct.
Sample: Yes, because there is one output for every input (or yes, because each month has only one recorded height). Yes, because the vine has a constant rate of growth each month.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Explanation appeals to the fact that the month is the independent variable but does not get at the "one output for each input" definition of function; one well-explained correct answer along with another answer that is not well explained, but correct, or it is along with an incorrect answer that shows some understanding.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: An incorrect answer to one or both questions that does not show significant understanding; both responses are flawed in some way.

Teaching Notes

Students may want to include a table or a graph to get a visual of how tall the vines get each month.

Draw a graph of Lin’s distance as a function of time for this situation:

Lin walked a half mile to school at a constant rate. Five minutes after arriving at school, Lin realized that she had left her permission slip at home. Lin began sprinting home, but when she was halfway there, she got tired and walked the rest of the way.

Label the axes appropriately. You do not have to include numbers on the axes or the coordinates of points on your graph.

Answer:

Sample response:

Minimal Tier 1 response:

Work is complete and correct.
Sample: See diagram as well as notes in narrative.
Acceptable errors: Axes are labeled only as “distance/time” .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Axes unlabeled or labeled incorrectly; graph does not meet one of the criteria mentioned in the narrative.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Graph does not meet two or more criteria mentioned in the narrative; graph does not meet one of the criteria mentioned in the narrative, and the axes are unlabeled/incorrect.

Teaching Notes

When judging the quality of a student’s graph, look for a few details: The slopes of the walking segments should be similar (with opposite signs). A horizontal line should represent the 5-minute wait time. The slope of the sprint home should be steeper than the walking slopes. The change from sprinting from walking should occur at “halfway home”—that is, at half of the maximum distance.

Two cleaning services charge money at an hourly rate, plus an initial one-time fee that is the same no matter how long the job takes.

Sparkle Team Cleaners charges according to this table:

time (hours)	cost (dollars)
1	56
2	72
4	104
5	120
8	168

So Fresh & So Clean charges according to this graph:

<p>A graph. Cost, in dollars. Time, in hours.</p>

A customer would like a company to visit their home and only give a quote for their services, without doing further work. Which company is the most cost effective for this customer? Explain or show your reasoning.
Another customer would like the company to perform 7 hours of work. Which company should this customer choose?
What is the smallest number of hours of work for which Sparkle Team Cleaners is cheaper than So Fresh & So Clean?

Answer:

So Fresh & So Clean. Sample reasoning: To find the one-time fee for Sparkle Team Cleaners, first notice that they charge $16 per hour. Subtract $16 from $56, the cost for 1 hour of cleaning, to find the one-time fee of $40. The one-time fee for So Fresh and So Clean is the $y$ -intercept, $30.
Sparkle Team Cleaners. Sample reasoning: Since Sparkle Team Cleaners charges $16 per hour, subtract $16 from $168, the cost for 8 hours of cleaning, to find that 7 hours of cleaning costs $152. From the graph, we can see that So Fresh & So Clean charges $20 per hour. The point $(6,150)$ is on the graph, so 7 hours of cleaning must cost $170.
3 hours. Sparkle Team Cleaners charges $90. (about 2.5 hours is also an acceptable solution: It’s the point when both companies charge the same amount and the answer students will find if they solve a system of equations.)

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: Omitting units ($).
Sample:

It’s $30 vs. $40, so So Fresh & So Clean costs less.
Sparkle Team Cleaners is $152, and So Fresh & So Clean is $170, so Sparkle Team Cleaners costs less.
3 hours, because Sparkle Team Cleaners just barely passes So Fresh & So Clean ($88 vs. $90)

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Correct work and explanation for part c based on mistakes in parts a and b; approach to part c involves a correct system of equations with arithmetic mistakes in the solution method; work calculating slope or rate of change involves arithmetic mistakes.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Approach to parts a and b shows some understanding that finding a starting point and a constant rate of change is needed, but the work is not systematic and involves errors; misinterpret the table or graph, such as reversing the columns or coordinates, but has reasonable work following that; approach to part c involves a correct system of equations but no reasonable approach to solving the system; work for parts a and b is correct, but work for part c is conceptually flawed.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: No evidence of understanding the connection between charge per hour and the information in the table and graph; errors on parts a, b, and c from Tier 3 response.

Teaching Notes

Students should be able to use a graph, table, and equation interchangeably. Look for students who write equations for the two companies’ pricing schemes and use the equations throughout the problem. Other methods involve finding the hourly rate for each company and using this to find data points not visible in the table or graph, and plotting the points from the table directly onto the graph for comparison.