End-of-Unit Assessment

1.

Select all the true statements.

<p>Three rows of shapes. First row contains four smiley faces. Second row contains two squares. Third row contains six triangles. </p>

A.

The ratio of triangles to squares is 2 to 6.

B.

The ratio of squares to smiley faces is $2:4$ .

C.

The ratio of smiley faces to triangles is 4 to 2.

D.

There are three triangles for every square.

E.

There are two smiley faces for every square.

F.

There are two triangles for every smiley face.

Answer: B, D, E

Teaching Notes

Students who select Statement A have the ratio backwards ( $2:6$ vs. $6:2$ ). Students who fail to select Statements B, D, or E may be confused about ratio language, either reversing the order of comparison or not understanding the general concept. Students who select Statement C might be mistakenly looking at the ratio of smiley faces to squares. Students who select Statement F probably did not correctly count smiley faces to triangles. They see there are more triangles than smiley faces but do not accurately see that 4 to 6 cannot be equivalent to 1 to 2.

2.

Select all the ratios that are equivalent to $9:6$ .

A.

$6:9$

B.

$3:2$

C.

$13:10$

D.

$5:2$

E.

$18:12$

Answer:

B, E

Teaching Notes

Students who select Statement A have the order of the ratio reversed, possibly believing the reverse is equivalent. Students who select Statement C and D may be mistaking pairs of numbers with a common difference (in this case, a difference of 4) for equivalent ratios. Students who fail to select Statement B may think an equivalent ratio must use multiples of the numbers in the ratio. Students who fail to select Statement E may be unclear about the concept or may have made an arithmetic error.

3.

A mixture of orange paint contains 8 teaspoons of red paint and 12 teaspoons of yellow paint. To make the same shade of orange paint using 18 teaspoons of red paint, how much yellow paint would you need? Use the double number line diagram to help if needed.

<p>A double number line diagram. Red paint. Teaspoons. Yellow paint. Teaspoons.</p>

A.

27 teaspoons

B.

24 teaspoons

C.

22 teaspoons

D.

12 teaspoons

Answer:

27 teaspoons

Teaching Notes

Students who select Statement D may have reversed the quantities and found the number of teaspoons of red paint given 18 teaspoons of yellow paint. Students who select Statement C may be using a constant difference between the red paint and yellow paint. Students who select Statement B have doubled the amount of yellow paint, possibly because the amount of red paint has approximately doubled.

4.

Elena rode her bike 2 miles in 10 minutes. She rode at a constant speed. Complete the table to show the time it took her to travel different distances at this speed.

distance traveled (miles)	elapsed time (minutes)
2	10
1
	1

Answer:

distance traveled (miles)	elapsed time (minutes)
2	10
1	5
$\frac15$ or equivalent	1

Teaching Notes

This problem asks students to find two unit rates using a table—minutes per mile and miles per minute.

5.

Three concert tickets cost $45. At this rate, what is the cost per ticket?
Three milkshakes cost $9.75. At this rate, how much do 2 milkshakes cost?
Three oranges cost $1.35. At this rate, how much do 5 oranges cost?

Answer:

$15
$6.50
$2.25

Teaching Notes

The numbers in this problem are messy enough that most students will need to calculate the unit price first, then multiply the unit price by the number of items. Look out for responses that involve calculating the unit price but do not go further.

6.

A bag contains 150 marbles. Some are blue, and the rest are white. There are 21 blue marbles for every 4 white marbles. How many blue marbles are in the bag? Explain your reasoning.

Answer:

126 blue marbles. Sample reasoning: Make a table in which the first row has 21 blue marbles and 4 white marbles. To find the row where the total number of marbles is 150, multiply the first row by 6, giving 126 blue marbles and 24 white marbles, which add to the correct total of 150.

Minimal Tier 1 response:

Work is complete and correct.
Sample: $21\boldcdot6=126$ , so there are 126 blue marbles and 24 white marbles.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Blue and white marbles are in the correct ratio but do not add up to 150. Correct answer is given but with no work or explanation shown. Work involves some correct reasoning about equivalent ratios with some errors.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Work does not show evidence of an understanding of equivalent ratios. Answer is incorrect and no work is shown.

Teaching Notes

Students will need to use some guesswork to find a pair of numbers that are in the correct ratio and add up to 150.

7.

To make fruit punch, Priya mixes 3 scoops of powder with 5 cups of water. Mai mixes 4 scoops of powder with 6 cups of water.

Create a double number line or a table that shows different amounts of powder and water that taste the same as Priya’s mixture.
Create a double number line or a table that shows different amounts of powder and water that taste the same as Mai’s mixture.
How do their two mixtures compare in taste? Explain your reasoning.

Answer:

Sample response:

Priya’s recipe

scoops of powder	cups of water
3	5
6	10
9	15
12	20

Mai’s recipe

scoops of powder	cups of water
4	6
8	12
12	18

Mai’s recipe tastes stronger. Sample reasoning: As shown in the tables, Mai’s recipe uses less water for the same amount of powder at 12 scoops.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Acceptable errors: Some mixing up of the terms “cup” and “scoop.”

Sample:
- See table for Priya’s recipe.
- See table for Mai’s recipe.
- Mai’s tastes stronger because more powder is used per cup, 1 cup of water needs $\frac23$ of a scoop of powder, but for Priya, 1 cup of water needs only $\frac35$ of a scoop of powder and $\frac23$ is greater than $\frac35$ .

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: A good explanation to Part C is based on incorrect powder-water combinations found in Parts A and B.
Sample errors: Work shows arithmetic errors in otherwise reasonable tables or double number lines. Explanation for Part C is on track but leaves the reader to connect too many dots, for instance, “Mai’s tastes stronger because $\frac23 >\frac35$ .”

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: A good explanation to Part C is based on incorrect powder-water combinations found in parts a and b.
Sample errors: Representations in Parts A and B do not show evidence of an understanding of equivalent ratios. Correct representations are used in Parts A and B but explanation in Part C is missing or does not involve a comparison of unit rates.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: None of the work contains evidence of an understanding of equivalent ratios.

Teaching Notes

Make sure students include at least two different powder-water combinations in their representations for each type of drink.