Grade 8

Mid-Unit Assessment

Select all the functions whose graphs include the point $(16,4)$ .

$y = 2x$

$y = x^2$

$y = x + 12$

$y = x - 12$

$y = \frac 1 4 x$

Answer: D, E

Teaching Notes

Students who select choice A misunderstand the overall concept relating a function to its graph. Students who select choice B or C may be mixing variables or working backwards from the point $(16,4)$ , looking for relationships.

This graph shows the temperature in Diego’s house between noon and midnight one day.

Select all the true statements.

Time is a function of temperature.

The lowest temperature occurred between 4:00 and 5:00.

The temperature was increasing between 9:00 and 10:00.

The temperature was 74 degrees Fahrenheit twice during the 12-hour period.

There was a four-hour period during which the temperature did not change.

Answer: D, E

Teaching Notes

It is easy to read choice A too quickly—time is not a function of temperature, but temperature is a function of time. Students who select choice A may either be making this error or may be having genuine trouble with the definition of a function. Students who select choice B have identified the time period in which the temperature decreases most quickly, not the period that contains the lowest temperature. Students who select choice C are having trouble distinguishing between increasing and decreasing intervals on a graph, or perhaps they are looking at the wrong time interval. Students who do not select choice D may be having trouble thinking about where to look for multiple occurrences of 74 degrees Fahrenheit (visualizing a horizontal line drawn at the mark for 74 degrees Fahrenheit is helpful). Students who do not select choice E may be unsure how to look for intervals of no change.

This table shows a linear relationship between the amount of water in a tank and time.

time (minutes)	water (gallons)
0	30
5	20
10	10

Which of these statements is true?

The water in the tank is increasing at a rate of 2 gallons per minute.

The water in the tank is increasing at a rate of 10 gallons per minute.

The water in the tank is decreasing at a rate of 2 gallons per minute.

The water in the tank is decreasing at a rate of 10 gallons per minute.

Answer:

The water in the tank is decreasing at a rate of 2 gallons per minute.

Teaching Notes

Students who select choice A have calculated the correct rate but are interpreting it incorrectly as an increase instead of a decrease. Students who select choice D are correctly interpreting the change as a decrease, but have only calculated the change in gallons, not the rate of change. Students who select choice B have made both these errors.

Elena goes for a long walk. This graph shows her time and distance traveled throughout the walk.

Coordinate plane, x, time in hours, 0 to 5 by ones, y, distance in miles, 0 to 10 by ones. Line segments connect the origin to 1 comma 3, to 1 point 5 comma 3, to 2 comma 5, to 4 comma 9.

What was her fastest speed, in miles per hour?

Answer:

Teaching Notes

Students who answer “2” may be using the longest section of the graph, where Elena walks 4 miles in 2 hours. Students who answer “3” may think the first section is the steepest, but it is not. Students who answer something other than “2,” “3,” or “4” may have a deeper misunderstanding about rates of change.

Lin counts 5 bacteria under a microscope. She counts them again each day for four days and finds that the number of bacteria doubles each day—from 5 to 10, then from 10 to 20, and so on.

Is the population of bacteria a function of the number of days? If so, is it linear? Explain your reasoning.

Answer:

It is a function, but it is not a linear function. It is a function because there is a single output (the number of bacteria) for each input (the number of days). It is not a linear function because the rate of change does not stay the same throughout. Alternatively, the points on the graph of this function clearly do not make a straight line.

Minimal Tier 1 response:

Work is complete and correct.
Sample: Yes, because there is one output for every input (or yes, because each day has only one number of bacteria). No, because the number of bacteria doesn’t go up by the same amount each day.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Explanation appeals to the fact that the day 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

If necessary, clarify for students that it does not matter that the population of bacteria may vary throughout each day—we are only using the population at each time of day that Lin checks.

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

When the football play started, Andre ran forward 20 yards, then turned around and ran 5 yards back. He stood in that spot for 3 seconds, then walked back to where he began.

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" and "time” or “yards" and "seconds.”

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: First, the slope of the run forward and run back should be similar, with the stopping point for the run back about $\frac 1 4$ of the way back from the maximum. Second, there should be a clear horizontal line corresponding to standing in place. Third, there should be a less steep line back to the horizontal axis to indicate the walk back to the start.

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

A-Plus Plumbing charges according to this table:

time (hours)	cost (dollars)
1	140
4	320
6	440

Quality Plumbing charges according to this graph:

Coordinate plane, x, time in hours, 0 to 6 by ones, y, cost in dollars, 0 to 250 by fifties. Line begins at 0 comma 150 and extends through 1 comma 200, 2 comma 250, and ends at 3 comma 300.

How much does A-Plus Plumbing cost for each hour of work, and what is the one-time fee? Explain or show your reasoning.
How much does Quality Plumbing charge for each hour of work, and what is the one-time fee? Explain or show your reasoning.
Can A-Plus Plumbing and Quality Plumbing ever charge the same total for the same amount of time? Explain or show your reasoning.

Answer:

The cost for each hour of work is $60, and the one-time fee is $80. Sample reasoning: Determine the cost per hour by finding the rate of change: $\frac{320-140}{3} = 60$ . Then determine the one-time fee by subtracting $60 from $140.
The cost for each hour of work is $50, and the one-time fee is $150. Sample reasoning: Determine the cost per hour by finding the slope of the line, which is 50. Determine the one-time fee by using the $y$ -intercept of the graph.
Yes. Sample reasoning: A-Plus Plumbing has a lower one-time fee but costs more per hour, so it will eventually catch up to Quality Plumbing.

Minimal Tier 1 response:

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

The cost is $60 because $140 + 3 \boldcdot 60 = 320.$ The fee is $80 because $140 - 60 = 80$ .
The cost is $50 because the graph goes up by 50 every hour. The fee is $150 because for 0 hours, they charge $150.
A-Plus Plumbing starts with a lower price but costs more each hour. This means the graphs must intersect.

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 the goal is to find a constant number to add each hour that will result in the numbers in the table, 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

Look for students’ work in part c in particular; it is not necessary to calculate the amount of time. Errors in part a or part b will reveal whether students have difficulty interpreting functions defined by ordered pairs or by graphs. In part c, it is sufficient justification to add a reasonably accurate graph of A Plus Plumbing’s pricing scheme to the existing graph, showing an unlabeled intersection point.