Interpret multiplication as scaling (resizing)--5th Grade

In 5th grade, students are introduced to the concept of multiplication as scaling (resizing). This means that when you multiply a number by another number, you are changing the size of the original number.¹ The size of the change depends on the number you are multiplying by.

Here are some examples:

• Multiplying by a whole number greater than 1: When you multiply a number by a whole number greater than 1, the product is larger than the original number.² For example, 3 x 4 = 12.³ The product, 12, is larger than the original number, 3.
• Multiplying by a fraction less than 1: When you multiply a number by a fraction less than 1, the product is smaller than the original number. For example, 1/2 x 4 = 2. The product, 2, is smaller than the original number, 4.
• Multiplying by 1: When you multiply a number by 1, the product is the same as the original number.⁴ For example, 1 x 4 = 4. The product, 4, is the same as the original number, 4.⁵

Students can use visual models to help them understand multiplication as scaling.⁶ For example, they can use a number line to represent the original number and the product. They can also use area models to represent the original number and the product.

Understanding multiplication as scaling is an important concept for students to learn in 5th grade. It will help them to understand more complex mathematical concepts in the future.

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Imagine you have a recipe for cookies that calls for 2 cups of flour. If you want to make twice as many cookies, you need to double the recipe, so you multiply the amount of flour by 2. This means you need 2 * 2 = 4 cups of flour.

Multiplication can also make things smaller. Let's say you want to make half as many cookies. You would multiply the amount of flour by 1/2. This means you need 2 * 1/2 = 1 cup of flour.

So, multiplication is like scaling, which means changing the size of something. When you multiply a number by a number greater than 1, you make it bigger. When you multiply a number by a number less than 1, you make it smaller.

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Here's an illustration to help explain multiplication as scaling:

Imagine a rubber band:

• Original Size: This represents your starting number.

  • Let's say the original rubber band is 5 inches long.

• Multiplying by 2: This is like stretching the rubber band twice as long.

  • 5 inches * 2 = 10 inches
  • The rubber band is now bigger!

• Multiplying by 1/2: This is like shrinking the rubber band to half its size.

  • 5 inches * 1/2 = 2.5 inches
  • The rubber band is now smaller!

Visual Representation:

• Draw a short line to represent the original rubber band (5 inches).
• Draw a longer line twice the length of the first to show multiplying by 2.
• Draw a shorter line half the length of the first to show multiplying by 1/2.

Key Points for the Illustration:

• Use different colors for the original rubber band and the scaled versions.
• Label each line with its length.
• Add arrows to show the direction of the scaling (stretching or shrinking).

This visual will help 5th graders understand that multiplication doesn't always just make numbers bigger. It changes the size of a number, like scaling an object.

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