In the previous sections, you used models and algorithms to multiply fractions and mixed numbers. In this section, you will investigate how the first factor changes when it is multiplied by the second factor. Sometimes the number will increase, and the product will be greater than the first factor; sometimes, the number will decrease, and the product will be less than the first factor.

In the interactive below, sort the multiplication problems into three categories: one where the product is less than the first factor, one where the product is equal to the first factor, and one where the product is greater than the first factor. Drag the multiplication problems into the correct category.

Use the interactive to answer the following questions.

• What do you notice about the second factor for multiplication problems that generate a product that is less than the first factor?

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  Check Your Answer

  The second factor is less than 1.

• What do you notice about the second factor for multiplication problems that generate a product that is equal to the first factor?

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  Check Your Answer

  The second factor is equal to 1.

• What do you notice about the second factor for multiplication problems that generate a product that is greater than the first factor?

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  Check Your Answer

  The second factor is greater than 1.

Pause and Reflect

In general, how does the second factor determine whether or not the product will be less than, equal to, or greater than the first factor?

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Check Your Answer

If the second factor is less than 1, then the product will be less than the first factor. If the second factor is equal to 1, then the product will be equal to the first factor. If the second factor is greater than 1, then the product will be greater than the first factor.

How can you determine, without performing the multiplication, when the multiplication results in an increase or a decrease from the first number?

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Check Your Answer

Look at the number being multiplied by the original number. If that number is less than 1, then the multiplication will result in a decrease from the first number. If that number is greater than 1, then the multiplication will result in an increase from the first number.

Practice

Predict whether the product will be less than, equal to, or greater than the starting number.

1. On Monday, the recycling center collected 25 1 over 2 1 2 pounds of aluminum cans. On Tuesday, the recycling center collected 3 over 5 3 5 as many cans as they did on Monday. How many pounds of cans did they collect on Tuesday?

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   Need a hint?

   Is the multiplier less than, equal to, or greater than 1?
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   Check Your Answer

   The product (the number of pounds of cans collected on Tuesday) will be less than 251 over 2 1 2 .
2. Karen is training for a marathon. Last week, she ran 121 over 2 1 2 miles each day. This week, she will run 5 over 4 5 4 longer each day than she did last week. How many miles will Karen run each day this week?

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   Need a hint?

   Is the multiplier less than, equal to, or greater than 1?
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   Check Your Answer

   The product (the number of miles Karen will run each day this week) will be greater than 121 over 2 1 2 .
3. The normal elevation of Lake Lewisville when it is completely full is 520 feet. The current elevation of Lake Lewisville is 19 over 20 19 20 of the normal elevation. What is the current elevation of Lake Lewisville?

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   Need a hint?

   Is the multiplier less than, equal to, or greater than 1?
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   Check Your Answer

   The product (the current elevation of Lake Lewisville) will be less than 520.