Section B Practice Problems

Problem 1

Mai has a sheet of stickers with 23 rows and 8 stickers in each row.

Does Mai have more or less than 100 stickers? Explain your reasoning.
How many stickers does Mai have? Explain or show your reasoning.

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Solution

Sample responses:

She has more than 100 because $5 \times 20 = 100$ , so $8 \times 20$ and $8 \times 23$ must be more than 100.
Mai has 184 stickers: $8 \times 20$ is 160, $8 \times 3$ is 24 and $160 + 24 = 184$ .

Problem 2

Find the value of $7 \times 64$ . Use a diagram if it is helpful.

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Solution

Sample response: 448. I partitioned the longer side of the diagram into 60 and 4 and then found $7 \times 60$ and $7 \times 4$ . Then I added 420 and 28, which is 448.

Problem 3

Use the diagram to find the value of $8 \times 573$ .

Diagram, rectangle partitioned vertically into 3 rectangles. Left rectangle, vertical side, 8, horizontal side, 5 hundred. Middle rectangle, horizontal side, seventy. Right rectangle, horizontal side, 3.
Find the value of $4 \times 3,516$ .

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Solution

4,584. Sample response: $4,000 + 560 + 24=4,584$
14,064. Sample response: $12,000+2,000+40+24=14,064$

Problem 4

Use the diagram to find the value of $47 \times 62$ .

Diagram, rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 40, horizontal side, sixty. Top right rectangle, horizontal side, 5. Bottom 2 rectangles, vertical side, 7.
Is this diagram helpful to finding the value of $47 \times 62$ ? Explain your reasoning.

Diagram, rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 34, horizontal side, 48. Top right rectangle, horizontal side, 14. Bottom 2 rectangles, vertical side, 13.

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Solution

2,914, because $2,400 + 420 + 80 + 14 = 2,914$ .
Sample response: No, because the partial products I need to find here are difficult.

Problem 5

The diagram and calculations show two ways for finding the value of $2,518 \times 6$ .

Diagram, rectangle partitioned vertically into 4 rectangles.

multiply. 2 thousand 5 hundred 18 times 6.

How does each part of the vertical calculation relate to the diagram?
Find the value of $3,172 \times 5$ using a method of your choice.

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Solution

Sample response: The 12,000 is $6 \times 2,000$ , the 3,000 is $6 \times 500$ , the 60 is $6 \times 10$ and the 48 is $6 \times 8$ .
15,860. Sample response:

Problem 6

Here is an incomplete calculation that uses partial products of $65 \times 43$ .

Write multiplication expressions that the numbers 15, 180, 200, and 2,400 each represent. Then find the value of $65 \times 43$ .

multiply. sixty 5 times 43. 6 rows. First row: sixty 5. Second row: multiplication symbol, 43. Horizontal line. Third row: 15. Fourth row: one hundred 80. Fifth row: two hundred. Sixth row: plus two thousand 4 hundred. Horizontal line.
Find the value of the product $45 \times 38$ .

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Solution

Sample responses:

Problem 7

Here is how Elena calculates the value of $723 \times 3$ .

multiply. 7 hundred 23 times 3, equals 2 thousand 1 hundred sixty 9.

Where does the 9 in Elena's calculation come from? What about the 6?
Where do the 2 and the 1 in calculation come from?
Use Elena's method to find the value of $534 \times 2$ .

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Solution

Sample response: The 9 is from $3 \times 3$ and the 6 represents 60, which is $3 \times 20$ .
The 2 and 1 represent 2,100 and this is $3 \times 700$ .
1,068. Sample response:

Problem 8

There are 4,218 students in school district A. School district B has 3 times as many students as school district A. How many students are in school district B? Explain or show your reasoning.

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Solution

12,654. Sample response:

Problem 9

Clare checks her answers for some products. Without doing the computation again, she knows that these answers are incorrect. How might Clare have known?

$5 \times 5,783 = 27,914$
$7 \times 8,419= 54,253$
$9 \times 9,999= 99,999$

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Solution

Sample responses:

The value of the product is a multiple of 5, so the last digit will always be a 0 or a 5, never a 4.
$7 \times 8,000 = 56,000$ so the answer is too small.
The answer is too large because $9 \times 10,000$ is only 90,000.

Problem 10

Here is Mai's strategy to find the value of $9 \times 8,235$ .

Explain why Mai's method works.
Use Mai's method to find the value of $9 \times 6,789$ .
Find the value of $9 \times 6,789$ using a strategy you learned. How is Mai's method like yours? How is it different?

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Solution

Sample responses:

$9 = 10 - 1$ , so Mai found $9 \times 8,235$ by finding $10 \times 8,235$ and then subtracting $1 \times 8,235$ .
Alike: Both methods use multiplication and then another operation.

Different: Mai's method involves using multiples of 6,789 that are easy to find and subtracting one from the other. My method involves multiplying the value of each digit in 6,789 by 9 and then adding the partial products.