Mid-Unit Assessment

1.

Which estimate is closest to the actual value of $(2.99548) \boldcdot (1.8342)$ ?

A.

4.8

B.

5.5

C.

6.2

D.

8.3

Answer:

5.5

Teaching Notes

Students estimate a product of two decimals. The numbers are sufficiently complicated such that finding the product will be time consuming. Students can reason that the product is less than $3 \boldcdot 2$ , which eliminates choices C and D. The product is close to $3 \boldcdot (1.8)$ , or 5.4, which indicates that choice B is correct.

2.

For part of a science experiment, Andre adds 0.25 milliliters of food coloring to 12.3 milliliters of water. How many milliliters does the mixture contain?

A.

12.05

B.

12.325

C.

12.55

D.

14.8

Answer:

12.55

Teaching Notes

This problem requires students to add decimals while paying careful attention to place value. Students who select choice A used subtraction instead of addition. Students who select choices B or D have not lined up the decimal places correctly.

3.

Select all the expressions that have a value greater than 3.

A.

$1.67 + 1.4$

B.

$2.97 + 0.004$

C.

$3.017 - 0.05$

D.

$4.5 - 1.47$

E.

$5.503 - 2.52$

Answer: A, D

Teaching Notes

Students estimate or perform addition and subtraction of decimals. Students who select choice B may be thinking of 2.97+0.04, which is 3.01. Likewise, students who select choices C and E may have made small mistakes in lining up the decimal places or accidentally added or omitted the digit 0. Students who do not select choice A may have calculated 2.07, or may have lined up the decimal places incorrectly. Students who do not select choice D may have a misunderstanding about the values after a decimal point, thinking that 0.47 is greater than 0.5.

4.

Four runners are training for long races. Noah ran 5.123 miles, Andre ran 6.34 miles, Jada ran 7.1 miles, and Diego ran 8 miles.

What is the total running distance of the four runners?
Compared to Noah, how much farther did Jada run?

Answer:

26.563 miles
1.977 miles

Teaching Notes

Students use a context to recognize when to add and subtract decimals.

5.

One way to compute a 15% tip for a bill is to multiply it by 0.15.

A restaurant bill was $42.40. Calculate the tip. Explain or show your reasoning.

Answer:

$6.36. Sample reasoning: I multiplied 42.40 by 0.1 to get 4.24 and then found half of that to get $(0.05) \boldcdot (42.40) = 2.12$ . Then I added the two answers to get $6.36.

Minimal Tier 1 response:

Work is complete and correct.
Sample: 6.36, because $(42.4) \boldcdot (0.15) = 6.36$ using the algorithm for multiplication.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Minor incorrect arithmetic on correct setup; incorrect decimal point of product.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Choice of operations invalid; incorrect decimal point of multiplicand (such as $(42.4) \boldcdot (1.5)$ ); multiple arithmetic errors.

Teaching Notes

Students use the suggested method of multiplying an amount by a decimal to compute the tip on a bill and then show their reasoning process.

6.

Find the product: $(0.061) \boldcdot (0.43)$
Find the difference:

Answer:

0.02623
0.5692

Teaching Notes

This problem asks students to compute a product and a difference. While students are not required to use an efficient algorithm for subtraction, it is likely they will do so, given the format provided.

7.

Two students have different ways of calculating $(0.25) \boldcdot (0.044)$ .

Elena says, “I find $25 \boldcdot 44$ and then place the decimal point.”

Show how Elena might calculate $(0.25) \boldcdot (0.044)$ .
Diego says, “0.25 is the same as $\frac{1}{4}$ , so I found $\frac{1}{4}$ of 0.044.”

Show how Diego might calculate $(0.25) \boldcdot (0.044)$ .

Answer:

Sample response: $25 \boldcdot 44 = 1,100$ . Since 0.25 was to the hundredths place and 0.044 was to the thousandths place, the last digit of the number 1,100 should be in the hundred-thousandths place, meaning that the digits need to move 5 places to the right. The answer is 0.011.
Sample response: $0.044 = \frac{44}{1000}$ , so $\frac{1}{4}$ of $\frac{44}{1000}$ is $\frac{11}{1000}$ . This is 11 thousandths, which is $0.011$ .

Minimal Tier 1 response:

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

$25 \boldcdot 44 = 1,100$ and the digits of 1,100 need to move 5 decimal places to the right.
$\frac 1 4$ of 44 is 11, so $\frac 1 4$ of $0.044$ is $0.011$ .

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Incomplete explanation of how $25 \boldcdot 44$ relates; calculation of 5 decimal places off by 1; incomplete description of how to calculate $\frac 1 4$ of 0.044.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Two or more error types from Tier 2 response; one omitted or nonsensical response; relationship to $25 \boldcdot 44$ off by 2 or more decimal places or otherwise disconnected; incorrect operation used. If either part is completely correct, the response earns at least Tier 3.

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

Students explain two different methods for multiplying decimals. In the first method, Elena multiplies the digits and then places the decimal point based on the decimal places in the original values. In the second method, Diego converts one of the decimals to a fraction and multiplies the numbers.