Multiplying and Dividing

Multiplying and Dividing

Use Reasoning

Opening

Use Reasoning

Watch the video that shows Jan and Carlos using reasoning about multiplication.

  • What does Jan say about one thing she knows about multiplication?
  • Does Carlos agree with what she says?
  • What reasoning does Carlos use to support his opinion?
  • What conclusion does Carlos first come up with?
  • How do Jan and Carlos refine that conclusion?
  • Do you agree with their conclusion?
  • Do you think you would get a different conclusion if you started with a number less than 1?
  • Test this idea.

VIDEO: Mathematical Practice 2

Math Mission

Opening

Investigate and generalize about the effects of using different types of numbers when dividing and multiplying.

Reason About Numbers

Work Time

Reason About Numbers

Carlos says, “When you divide by a number, the result is always smaller than the number you started with. For example, when I divide 10 by 2, I get 5, which is smaller than 10.”

  • Without doing any calculations, say whether you think Carlos’s statement is true.
  • Now investigate Carlos’s statement. Try several different starting numbers (including whole numbers, fractions, and decimals) and divide by numbers of different types and different sizes. Find a way to organize your work so you can keep track of the results.

Pick a whole number to start with. Divide it by another whole number and then by fractions and decimals greater than and less than 1. Then start with a decimal or a fraction and repeat this process.

Carlos’s Statement

Work Time

Carlos’s Statement

Carlos says, “When you divide by a number, the result is always smaller than the number you started with. For example, when I divide 10 by 2, I get 5, which is smaller than 10.”

  • Is Carlos’s statement true? When you divide, do you always get a smaller number than you started with?
  • If not, when do you get smaller numbers and when do you get larger numbers? Use examples and reasoning to support your conclusions.

To help you think about how to organize your thoughts, think of how Jan and Carlos organized their thoughts in the video.

Prepare a Presentation

Work Time

Prepare a Presentation

Explain your thinking about dividing numbers. Use your work to support your explanation.

Challenge Problem

  • Describe a real-world situation that shows that multiplying does not always give a larger result.
  • Describe a real-world situation that shows that dividing does not always give a smaller result.

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes about your classmates' conclusions and reasoning about multiplication and division.

As your classmates present, ask questions such as:

  • Why did you organize your work that way?
  • How did organizing your work that way help you make a conclusion?
  • Where do you see the patterns?
  • Why did you make those conclusions?
  • Do you think you tried enough different examples?

Multiplication and Division

Formative Assessment

Summary of the Math: Multiplication and Division

Write a summary about the conclusions you made about multiplication and division.

Check your summary.

  • Do you explain in which situations multiplying results in a number greater than the original number, and in which situations it results in a number less than the original number?
  • Do you explain in which situations dividing results in a number greater than the original number, and in which situations it results in a number less than the original number?

Beanbag Elephants

Formative Assessment

Self Check: Beanbag Elephants

Jan makes and sells beanbag animals. Her most popular animal is her beanbag elephant.

  • For each elephant, she uses 13 yard of gray fabric, 134 pounds of dried beans, and a pair of plastic eyes.
  • Fabric costs $3 per yard, beans cost $1 per pound, and plastic eyes cost $0.89 per pair.
  • She sells each elephant for $17.20.

Show all of your work.

  1. Jan has 423 yards of gray fabric. How many elephants can she make?
  2. Jan has 28 pounds of dried beans. Does she have enough beans to stuff all of the elephants that she made with the fabric?
  3. Jan spends $22.25 on plastic eyes. How many pairs did she buy? Is this enough for all of the elephants she stuffed?
  4. After subtracting the cost of the materials that she used to make the elephants, how much money will Jan earn if she sells all of the elephants?

Reflect On Your Work

Work Time

Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

The most important thing I learned in this lesson is …