Grade 6

End-of-Unit Assessment

Lin’s family has completed 60% of a trip. They have traveled 30 miles. How long is the whole trip?

50 miles

48 miles

30 miles

18 miles

Answer:

Teaching Notes

Students are not expected to take an algorithmic approach to find $B$ in $A$ % of $B$ is $C$ . They may use a double number line diagram or table to keep track of values. Students who select choice D are computing 60% of 30, rather than using the information that 30 miles is 60% of the trip. Students who select choice C are subtracting (60−30). Students who select choice B are calculating 60% of 30, then added 30 more miles (because that distance is already traveled).

It took 4 hours to drive 240 miles. At this rate, how long does it take to drive 90 miles?

60 hours

6 hours

$2 \frac23$ hours

$1 \frac12$ hours

Answer:

$1 \frac12$ hours

Teaching Notes

Students who incorrectly select choice A are dividing distance (240) by time (4), finding the speed of travel in miles per hour. Students who select choice B may be dividing 240 by 4, but realizing that 60 hours is implausible, are modifying their answer by dividing by 10. Students who select choice C are dividing 240 by 90, perhaps by performing operations on numbers in the problem until they arrive at a plausible answer.

There are 9 children in a class who take the bus to school, and there are 15 total children in the class. What percentage of the children take the bus to school?

90%

60%

0.6%

Answer:

60%

Teaching Notes

This problem is possible to solve without any calculations because 9 is a bit more than half of 15, and choice B is the only answer choice that is reasonably close. Students who select choice A or C are using the given information of “9 students” without calculating the percentage of the whole. Students who select choice D are dividing 9 by 15, but are not expressing the rate as a percentage.

It takes Andre 4 minutes to swim 5 laps.

How many laps per minute is that?
How many minutes per lap is that?
At that rate, how long does it take Andre to swim 22 laps?

Answer:

$\frac{5}{4}$ or 1.25 laps per minute
$\frac45$ minutes, or 48 seconds per lap
17.6 minutes ( $22 \boldcdot \frac{4}{5} = 17.6$ )

Teaching Notes

This problem has students calculate both unit rates that describe a situation. In the third part, students need to decide which of the two unit rates makes sense to use.

A train is traveling at a constant speed. The table shows how far it travels in some amounts of time.

What is the speed of the train?

time (hours)	distance (miles)
3	135
4	180

Answer:

45 miles per hour

Teaching Notes

To find the speed, students need to look for the distance traveled in 1 hour. Students may create a double number line diagram or add an additional row to the table to find the number of miles corresponding to 1 hour. They may also divide a given distance by the corresponding time. Some students may observe the difference in time and in distance shown in the two rows—the distance traveled increases 45 miles with 1 additional hour of travel—and conclude that this is the speed.

Which weighs more: a pumpkin that weighs 3.2 kilograms, or a cat that weighs 9 pounds? Explain your reasoning.

Note: 1 pound is about 0.45 kilograms.

Answer:

The cat weighs more. Possible strategies:

9 pounds is about 4.05 kg.
Without computing, a student might reason that $9\boldcdot (0.45)$ is a little less than 4.5, so the cat must weigh more than the pumpkin.

Minimal Tier 1 response:

Work is complete and correct.
Sample: A 9-pound cat weighs a little less than 4.5 kilograms, so the cat weighs more.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Correct unit conversion with incorrect interpretation; unit conversion multiplies kilograms by 0.45 or otherwise “goes the wrong way”; arithmetic errors in unit conversion.

Tier 3 response:

Significant errors in work demonstrate a lack of conceptual understanding or mastery.
Sample errors: Work does not involve unit conversion, either through estimation (as in the minimal Tier 1 response) or explicitly; unit conversion is attempted but with an incorrect conversion factor.

Teaching Notes

This problem can be solved quickly using estimation, though students may do computations to convert pounds to kilograms or kilograms to pounds.

Noah and Andre plan to exercise for a certain amount of time this week. So far, Noah has exercised for 35% of his goal of 200 minutes. Andre has exercised 40% of this goal of 180 minutes.

Who exercised for a longer amount of time so far? Explain or show your reasoning.

Answer:

Andre exercised for a couple of minutes longer. Noah exercised for 70 minutes. Andre exercised for 72 minutes. Sample reasoning:

Calculate $\frac{P}{100}$ times the duration for each person.
Create tables, double number line diagrams, or tape diagrams to find the duration for an intermediate percentage (1% or 5% for Noah, and 5%, 10%, or 20% for Andre), and then scale those numbers to 35% and 40%, respectively.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: $\frac{35}{100} \boldcdot 200=70$ , so Noah exercised for 70 minutes. $\frac{40}{100} \boldcdot 180 = 72$ , so Andre exercised for 72 minutes.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Work uses a valid representation or sequence of calculations but contains one error that propagates. A written-only response is reasoned correctly but contains arithmetic errors.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: A representation is used to keep track of the information but contains flawed reasoning. Calculations show incorrect values being associated with 100% or misinterpretation of the values to be found and compared.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: No visual representation or written explanation. Explanation does not involve reasoning about ratios and rates.

Teaching Notes

This problem prompts students to compare percentages of two different values. Students can find each percentage in a variety of ways, with or without using representations such as tables, double number line diagrams, or tape diagrams. Because 100% corresponds to different values for each person in the problem, students will need to create two representations (either of the same kind or different kinds) if they choose to use them.