Section A Practice Problems

Problem 1

Han says that the value of the 7 in 735,208 is 10 times the value of the 7 in 137,342. Do you agree with Han? Explain or show your reasoning.

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Solution

No. Sample response: The 7 in 735,208 represents 700,000 and that is 100 times the value of 7,000, which is the value of 7 in 137,342.

Problem 2

Find the value of each product. Explain or show your reasoning.

$27 \times 53$
$518 \times 6$

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Solution

1,431. Sample response:

The value of the product is 1,431 since $1,000+350+60+21=1,431$ .
3,108. Sample response:

The value of the product is 3,108 since $3,000 + 60 + 48 = 3,108$ .

Problem 3

Find the value of

7,518 \div 6

. Explain or show your reasoning.

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Solution

1,253. Sample response:

1,000 \times 6 = 6,000

200 \times 6 = 1,200

50 \times 6 = 300

3 \times 6 = 18

1,000 +200 + 50 + 3=1,253

Problem 4

What is the volume of this rectangular prism? Explain or show your reasoning.

Rectangular prism. 40 by sixty by eighty centimeters.

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Solution

192,000 cubic centimeters. Sample response:

40 \times 60 \times 80 = 192,000

Problem 5

2 diagrams of equal length. 5 equal parts, 1 part shaded. Total length, 1.

How does the drawing show $2 \div 5$ ? Explain or show your reasoning.
How does the drawing show $\frac{2}{5}$ ? Explain or show your reasoning.

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Solution

Sample responses:

There are 2 wholes total and 1 out of 5 equal parts is shaded in each, so that's $2 \div 5$ .
Each shaded part is 1 of 5 equal parts in a whole or $\frac{1}{5}$ so that's $\frac{2}{5}$ .

Problem 6

Find the value of each product. Explain or show your reasoning.

$100 \times 50$
$120 \times 50$
$127 \times 50$

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Solution

5,000. Sample response: I just put 2 zeros on 50.
6,000. Sample response: I added 20 more 50s to $100 \times 50$ , and that was 1,000 more.
6,350. Sample response: I added 7 more 50s to $120 \times 50$ , and that was 350 more.

Problem 7

Complete the diagrams. Use each diagram to find the value of $253 \times 31$ .

Diagram, rectangle partitioned vertically and horizontally into 6 rectangles. — A

Diagram, rectangle partitioned horizontally into 2 rectangles. Top rectangle, vertical side, 30, horizontal side, two hundred fifty three. Bottom rectangle, vertical side, 1. — B

How are the strategies alike? How are they different?

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Solution

7,843. Sample responses:

I added up all of the numbers to get 7,843.

I added up the numbers and got 7,843.

With the first strategy, I had more products to find and they were simpler, but then there were more products to add at the end. With the second strategy, there were fewer products to find but they were harder.

Problem 8

Find the value of

315 \times 43

, using partial products.

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Solution

13,545. Sample response:

Problem 9

Find the value of

16,452 \times 6

. Use the standard algorithm.

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Solution

98,712. Sample response:

Problem 10

Find the value of $322 \times 41$ . Use the standard algorithm.

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Solution

13,202. Sample response:

Problem 11

Find the value of $562 \times 34$ . Use the standard algorithm.

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Solution

19,108. Sample response:

Problem 12

Andre is playing Greatest Product. He says the greatest product to make in the game is

987 \times 65

. Do you agree? Explain or show your reasoning.

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Solution

Sample responses:

No. I calculated the product $987 \times 65$ and got 64,155. I think if I put the biggest digit in the greatest place value of the bigger number, and then put the second-biggest digit in the greater place value of the other number, I will get a greater product.
No. I can use the same numbers to get a bigger number. For example $976 \times 85$ is more than 80,000 while $987 \times 65$ is less than 70,000.

Problem 13

Use the digits 1, 2, 3, 4, and 5 to make a product with a value close to 8,000.

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Solution

Sample response:

Problem 14

The recommended side lengths for a birdhouse for a yellow-bellied sapsucker are 13 centimeters by 13 centimeters for the floor and a height of from 31 centimeters to 38 centimeters. What are the least and greatest volumes for this birdhouse? Explain or show your reasoning.

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Solution

5,239 cubic centimeters and 6,422 cubic centimeters. Sample responses:

The area of the floor is $13 \times 13$ , which is 169 square centimeters. The smallest birdhouse is $169 \times 31$ cubic centimeters and the biggest birdhouse is $169 \times 38$ cubic centimeters.

Problem 15

Jada remembers that the partial-products algorithm can go from left to right or from right to left. She wonders if the standard algorithm also can go in either direction.

Calculate $418 \times 53$ , using partial products right to left and left to right.
Calculate $418 \times 53$ , with the standard algorithm. What happens if you try to make the calculation from left to right?

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Solution

22,154. Sample response:
Sample response:

I can multiply 50 by 400 and get 20,000 and record the 2 and the 0. But when I get to $50 \times 10$ , if I record the 5 in the hundreds place, then I have to change it to 9 when I get to $50 \times 8$ , which is 4 hundred more. I have the same problem when I multiply 418 by 3 since $3 \times 10 = 30$ , but there are 2 more tens coming from $3 \times 8$ .

Problem 16

Clare has a strategy for multiplying a number by 99. To find $648 \times 99$ , she calculates $648 \times 100$ and then subtracts $648$ .

Use Clare's strategy to calculate $648 \times 99$ .
Use the standard algorithm to calculate $648 \times 99$ .
Which strategy do you prefer? Explain your reasoning.

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Solution

64,152. Sample response: $648 \times 100 = 64,800$ and $64,800 - 648 = 64,152$
64,152. Sample response:
Sample response: I liked Clare's method because I could do most of the math in my head. There was a lot of composing, with the standard multiplication algorithm. A nice thing about the standard algorithm was that I could use my answer, 5,832, for $9 \times 648$ to find $90 \times 648$ , so I did not have to recalculate this.