End-of-Unit Assessment

1.

A woodworker wants to cut a board that is 8.225 feet long into 5 equal-length pieces. How long will each piece be?

A.

0.1645 foot

B.

1.645 feet

C.

4.1125 feet

D.

41.125 feet

Answer:

1.645 feet

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 placed the decimal point incorrectly in the quotient. Students who select choice C or D multiplied instead of divided.

2.

Select all of the expressions that have the same value as $892 \div 8$ .

A.

$8920 \div 80$

B.

$894 \div 10$

C.

$89.2 \div 0.08$

D.

$8.92 \div 0.8$

E.

$0.892 \div 0.008$

Answer: A, 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 choice C or D have used incorrect powers of 10. Students who select choice B may have a deeper misunderstanding of division because this choice adds the same amount (2) to both numbers. Students who do not select choice A or E may have made an error in counting digits or (in the case of E) may have placed the decimal point correctly.

3.

Which value is closest to the quotient of $4,367 \div 0.004$ ?

A.

1,000

B.

10,000

C.

100,000

D.

1,000,000

Answer:

1,000,000

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 $4,367 \div 0.004$ has the same value has $4{,}367{,}000 \div 4$ , which they can estimate more readily.

Students who select choice A may just be dividing by 4, without paying attention to place value. Students who select choice B or C may have miscounted the number of decimal places. Students who select choice C may have looked at the two zeros after the decimal point in 0.004 and thought that the digits should move two places to the left instead of three.

4.

One way to convert a measurement from inches to centimeters is to multiply the number of inches by 2.54. How many centimeters are there in $\frac{1}{4}$ inch?

Answer:

0.635 centimeters (or equivalent)

Teaching Notes

Students multiply a decimal by a fraction in a unit-conversion context. Some approaches students may take:

Convert 2.54 to a fraction and multiply the fractions (which they may or may not convert back to a decimal).
Convert $\frac{1}{4}$ to a decimal and multiply two decimals.
Calculate $2.54 \div 4$ .

5.

Find each quotient using long division.

$2,247 \div 7$
$676 \div 13$

Answer:

321
52

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 poster boards for $29
24 poster boards for $56
50 poster boards for $129

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

Answer:

The unit prices are $2.42, $2.33, and $2.58. Reasoning varies, but most students will use long division.
24 poster boards for $56 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 $2.42, $2.33, and $2.58. 24 poster boards for $56 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 poster boards 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: $\frac{129}{50} = \frac{258}{100}$ , which gives the unit price of $2.58 for 50 poster boards.

7.

A stack of 500 pieces of paper is 1.875 inches tall.

Diego guesses that each piece of paper is 0.015 inch thick. Explain how you know that Diego’s answer is not correct.
Compute the thickness of each piece of paper. Show your reasoning.

Answer:

If each piece of paper were 0.015 inch thick, the stack would be 7.5 inches high: $500 \boldcdot (0.015) = 7.5$ .
0.00375 inch, because $1.875 \div 500 = 0.00375$ .

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

This can’t be right, because $500 \boldcdot (0.015)$ is 7.5.
$1.875 \div 500$ is $(1.875) \boldcdot 2 \div 1000$ , which is $3.75 \div 1000$ . The digits of 3.75 need to move 3 places to the right, so each piece is 0.00375 inch 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.