Section C Practice Problems

Problem 1

Diagram, square. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.

Shade the first diagram to represent $5 \times 0.07$ .
What is the value of $5 \times 0.07$ ? Explain or show your reasoning.
What is the value of $5 \times 0.2$ ? Use the second diagram if it is helpful.

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Solution

Sample responses:

0.35. Sample response: There are 35 or $5 \times 7$ small squares shaded and each one is 0.01.
1. Sample response: It's $5 \times 2$ or 10 tenths, which is 1.

Problem 2

Mai says that $7 \times 0.4$ and $7 \times 0.04$ have the same value of 28. Do you agree? Explain or show your reasoning.
Explain why $8 \times 0.03 = (8 \times 3) \times 0.01$ .

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Solution

No. Sample response: She is right that there are 28 in both cases, but for $7 \times 0.4$ it's 28 tenths and for $7 \times 0.04$ it's 28 hundredths. So, $7 \times 0.4 = 2.8$ and $7 \times 0.04 = 0.28$ .
Sample response: 0.03 is 3 hundredths, $8 \times 0.03$ is $8 \times 3$ hundredths, and a hundredth is 0.01 so $8 \times 0.03 = (8 \times 3) \times 0.01$ .

Problem 3

Explain why each expression has the same value as $9 \times 0.45$ .

$(9 \times 0.4) + (9 \times 0.05)$

$(9 \times 45) \div 100$

$(10 \times 0.45) - (1 \times 0.45)$
Find the value of $9 \times 0.45$ using one of the expressions or your own strategy.

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Solution

Sample response: This is true using the distributive property because $0.45$ is 4 tenths and 5 hundredths, $0.4 + 0.05$ .
Dividing by 100 is the same as multiplying by $\frac{1}{100}$ and $45 \times \frac{1}{100} = 0.45$ .
This is true using the distributive property again because $10 - 1 = 9$ .
4.05. Sample response: $10 \times 0.45 = 4.5$ and $4.5 - 0.45 = 4.05$ so $9 \times 0.45 = 4.05$ .

Problem 4

Shade the diagram to represent $0.7 \times 0.4$ .

What is the value of $0.7 \times 0.4$ ?

Diagram, square. Length and width, 1. Partitioned into 10 rows of 10 of the same size squares. No squares shaded.

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Solution

0.28. Sample response: There are 28 shaded pieces and each is 0.01. The diagram shows 0.7 of 0.4 shaded so that is $0.7 \times 0.4$ .

Problem 5

Explain or show why $5.6 \times 3.4 = (56 \times 34) \times 0.01$ .
Use this strategy to calculate $5.6 \times 3.4$ .

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Solution

Sample response: $5.6 = 56 \times 0.1$ and $3.4 = 34 \times 0.1$ so $5.6 \times 3.4 = (56 \times 34) \times 0.01$ .
19.04. Sample response: I found $56 \times 34 = 1,904$ so $5.6 \times 3.4 = 19.04$ .

Problem 6

Diego finds the value of $17.5 \times 3.3$ . He says, "I know $\frac{175}{10} \times \frac{33}{10} = \frac{175~ \times ~33}{100}$ , so I just find $175 \times 33$ and then divide by 100."

Explain or show why Diego's method works.
Use his method to find the value of $17.5 \times 3.3$ .

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Solution

Sample response: 17.5 is equal to $\frac{175}{10}$ and 3.3 is equal to $\frac{33}{10}$ so that means $17.5 \times 3.3 = \frac{175}{10} \times \frac{33}{10}$ . The numerator of the product is the product of the numerators and the denominator of the product is the product of the denominators so $\frac{175}{10} \times \frac{33}{10} = \frac{175~ \times ~33}{100}$ and this is $(175 \times 33) \div 100$ .
57.75. Sample response: $175 \times 33 = 5,775$ and then dividing by 100 gives 57.75.

Problem 7

Han says the diagram shows $4 \times 0.5 = 2$ . Label the diagram to show Han's reasoning.
Mai says it shows $10 \times 0.2 = 2$ . Label the diagram to show Mai's reasoning.
What other products can the diagram represent? Explain or show your reasoning.

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Solution

Sample responses:

Each large square represents 1 whole.

If I outlined every small square, it shows $200 \times 0.01 = 2$ . If I outlined groups of 2 small squares, it shows $100 \times 0.02 = 2$ . If I outline groups of 5 small squares, it shows $40 \times 0.05 = 2$ .