End-of-Unit Assessment

1.

A path of a race is divided into 5 equal sections. The race is 42.195 km long. How long is each section?

A.

210.975 km

B.

21.0975 km

C.

8.439 km

D.

0.8439 km

Answer:

8.439 km

Teaching Notes

Students divide a decimal by a whole number in the context of a situation. The method is not specified. Students who select choice A multiplied instead of divided. Students who select choice B multiplied instead of divided and placed the decimal point incorrectly. Students who select choice D placed the decimal point incorrectly in the quotient.

2.

Select all of the expressions that are equal to 2.

A.

$1,200 \div 60$

B.

$1.2 \div 0.6$

C.

$0.012 \div 0.06$

D.

$0.5 \div 0.25$

E.

$0.05 \div 0.025$

Answer:

B, D, E

Teaching Notes

Students examine division expressions involving decimals and identify those that have the same value. The key understanding sought is that multiplying both numbers of the quotient by the same power of 10 does not change the quotient.

Students who select A may be working too fast, neglecting to use number sense to check their answers. Students who select choice C or don't select choice B, D, or E may have made errors in placing the decimal point. Students who don't select D or E may have rejected it out of hand because they reversed the dividend and divisor and thought about $25 \div 5$ , which is 5.

3.

Which value is closest to the quotient of $5,278 \div 0.05$ ?

A.

1,000,000

B.

100,000

C.

10,000

D.

1,000

Answer:

B

Teaching Notes

Students estimate a quotient of decimals. While they could find the exact answer, the numbers are not friendly and the expectation is that students will estimate and use their knowledge of place value. The quotient $5,278 \div 0.05$ has the same value as $527,800 \div 5$ , which they can estimate more readily.

Students who select choice A or C may have miscounted the number of decimal places. Students who select choice D may just be dividing by 5, without paying attention to place value. Students who select choice C may have looked at the one zero after the decimal point in 0.05 and thought that the digits should move one place to the left instead of two.

4.

It takes 4.12 gallons of paint to cover a fence. How much paint is needed for $\frac15$ of the fence?

Answer:

0.824 gallons (or equivalent)

Teaching Notes

Students multiply a decimal by a fraction. Some approaches students may take:

Convert 4.12 to a fraction and multiply fractions (which they may or may not convert back to a decimal).
Convert $\frac15$ to a decimal and multiply two decimals.
Calculate $4.12 \div 5$ .

5.

Find each quotient using long division.

$3,328 \div 8$
$868 \div 14$

Answer:

416
62

Minimal Tier 1 response:

Work is complete and correct.
Sample: See calculations.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Acceptable errors: A small mistake early on results in different calculations (with no other errors).
Sample errors: One or two arithmetic mistakes, such as subtracting incorrectly; one or two mistakes in the algorithm such as bringing down the wrong number, provided there is evidence elsewhere in the work that the student knows how to do this; use of the method of partial quotients rather than long division.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Correct answers found by a different method, such as guess-and-check multiplication; consistent mistakes with the algorithm; work is only partially completed.

Teaching Notes

This problem assesses students’ knowledge of the long-division algorithm. Because a method is specified, full credit should be given if the method is used.

6.

A sign in an art store gives these options:

12 markers for $25
24 markers for $46
50 markers for $94

Find each unit price to the nearest cent, and show your reasoning.
Which option gives the lowest unit price?

Answer:

The unit prices are $2.08, $1.92, and $1.88. Reasoning varies, but long division is a reasonable approach.
50 markers for $94 has the lowest unit price.

Minimal Tier 1 response:

Work is complete and correct.
Sample: (Accompanied by work showing long division or other calculation methods.) The unit prices are $2.08, $1.92, and $1.88. 50 markers for $94 is the best deal.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Acceptable errors: Incorrect selection of lowest unit rate comes from errors in calculation of unit rates.
Sample errors: One or two errors in long division; incorrect selection of the best deal despite having calculated all three unit rates correctly.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Calculation of unit rates involves a conceptual error, such as dividing the number of markers by the price; three or more errors in long division; correct selection of the best deal without calculation of each unit rate.

Teaching Notes

Students calculate quotients of whole numbers resulting in decimal quotients, then compare the resulting decimals to look for the best deal. In addition to using long division, a student might strategically notice that equivalent fractions are very effective for the third calculation: $\frac{94}{50}=\frac{188}{100}$ , which gives the unit price of $1.88 for 50 markers.

7.

A stack of 200 file folders is 5.125 inches tall.

Elena guesses that each file folder is 0.035 inch thick. Explain how you know that Elena’s answer is not correct.
Compute the thickness of each file folder. Show your reasoning.

Answer:

If each folder were 0.035 inch thick, the stack would be 7 inches high: $200 \boldcdot (0.035)=7$ .
0.025625 inch, because $5.125 \div 200=0.025625$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:
- This can’t be right, because $200 \boldcdot (0.035)$ is 7.
- $5.125 \div 200$ is $(5.125 \boldcdot 5) \div 1,000$ , which is $25.625 \div 1,000$ . The digits of 25.625 need to move 3 places to the right, so each folder is 0.025625 inches thick.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Arithmetic error in multiplication or division; result off by one decimal place but otherwise correct.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Invalid or omitted work on one of the two problem parts; major error in division; division result off by more than one decimal place; incorrect choice of operation.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Two or more error types from Tier 3 response.

Teaching Notes

In the second part of this problem, students will need to divide a decimal by a whole number. The numbers are chosen so that either partial quotients or long division is the most likely effective choice. One challenge with this problem will be placing the decimal correctly in the quotient.