Mid-Unit Assessment

1.

Which estimate is closest to the actual value of $(3.99437) \boldcdot (2.6851)$ ?

A.

14.2

B.

12.6

C.

10.8

D.

8.3

Answer:

10.8

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 $4 \boldcdot 3$ , which eliminates choices A and B. The product is close to $4 \boldcdot (2.7)$ , or 10.8, which indicates that choice C is correct.

2.

Select all the expressions with a value greater than 5.

A.

$4.88 + 0.009$

B.

$3.76 + 1.3$

C.

$5.012 - 0.04$

D.

$7.702 - 2.71$

E.

$6.5 - 1.49$

Answer:

B, E

Teaching Notes

Students estimate or perform addition and subtraction of decimals. Students who select choice A may be thinking of $4.88+0.9$ , which is 5.78. Likewise, students who select choices C and D 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 B may have calculated 4.06, or may have lined up the decimal places incorrectly. Students who do not select choice E may have a misunderstanding about the values after a decimal point, thinking that 0.49 is greater than 0.5.

3.

Mai’s backpack weighs 9 pounds. Han’s backpack weighs 8.3 pounds. Priya’s backpack weighs 7.75 pounds. Kiran’s backpack weighs 6.125 pounds.

Select all true statements about the weights of the backpacks.

A.

Han’s backpack is 2.175 pounds heavier than Kiran’s.

B.

Mai’s backpack is 2.25 pounds heavier than Priya’s.

C.

Han’s backpack is 16.05 pounds heavier than Priya’s.

D.

Kiran’s backpack and Han’s backpack together weigh 14.3125 pounds.

E.

The backpacks weigh 31.175 pounds together.

Answer:

A,E

Teaching Notes

Students use a context to recognize when to add and subtract decimals. Students who select choice B may have been misled by the fact that $9−7=2$ . Students who select choice C have added rather than subtracted. Students who select choice D have concatenated the decimal parts of the two numbers to be added.

4.

To make a cleaning liquid, Jada adds 0.35 liter of vinegar to 16.4 liters of water. How many liters does the mixture contain?

Answer:

16.75 liters

Teaching Notes

This problem requires students to add decimals while paying careful attention to place value.

5.

One way to compute a 20% discount is to multiply the price by 0.20.

The price of a pair of sneakers is $84.50. Calculate the discount. Explain or show your reasoning.

Answer:

$16.90. Sample reasoning: I multiplied $84.50 by 0.1 to get $8.45. Since the discount was double that amount, I multiplied $8.45 by 2 to get $16.90.

Minimal Tier 1 response:

Work is complete and correct.
Sample: $16.90, because $(84.50) \boldcdot (0.20) = 16.90$ 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 $(84.5) \boldcdot (0.02)$ ); multiple arithmetic errors.

Teaching Notes

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

6.

Find the difference:
Find the product: $(0.072) \boldcdot (0.31)$

Answer:

0.4096
0.02232

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 $(2.4) \boldcdot (3.7)$ .

Andre says, “I find $24 \boldcdot 37$ and then place the decimal point.”

Show how Andre might calculate $(2.4) \boldcdot (3.7)$ .
Lin says, “I drew this diagram and then found the area of each part.”

Show how Lin might calculate $(2.4) \boldcdot (3.7)$ .

Answer:

Sample response: $24 \boldcdot 37 = 888$ . Since 2.4 was to the tenths place and 3.7 was also to the tenths place, the last digit of the number 888 should be in the hundredths place, meaning that the digits need to move 2 places to the right. The answer is 8.88.
Sample response: The areas of the rectangles in the diagram are 6, 1.4, 1.2, and 0.28. Their sum is 8.88.

Minimal Tier 1 response:

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

$24 \boldcdot 37=888$ , and the digits of 888 need to move 2 places to the right.
$6+1.4+1.2+0.28=8.88$

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 $24\boldcdot 37$ relates; calculation of 2 decimal places off by 1; one or two errors in calculating the areas of the pieces of the rectangle.

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 $24\boldcdot 37$ 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, Andre multiplies the digits and then places the decimal point based on the decimal places in the original values. In the second method, Lin draws an area diagram with the factors as the side lengths, which she decomposes by place value, and then finds the partial areas.