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# Multiplying and Dividing

## Overview

Students explore whether multiplying by a number always results in a greater number. Students explore whether dividing by a number always results in a smaller number.

# Key Concepts

In early grades, students learn that multiplication represents the total when several equal groups are combined. For this reason, some students think that multiplying always “makes things bigger.” In this lesson, students will investigate the case where a number is multiplied by a factor less than 1.

Students are introduced to division in early grades in the context of dividing a group into smaller, equal groups. In whole number situations like these, the quotient is smaller than the starting number. For this reason, some students think that dividing always “makes things smaller.” In this lesson, students will investigate the case where a number is divided by a divisor less than 1.

# Goals and Learning Objectives

- Determine when multiplying a number by a factor gives a result greater than the number and when it gives a result less than the number.
- Determine when dividing a number by a divisor gives a result greater than the number and when it gives a result less than the number.

# Use Reasoning

# Mathematical Practices in Action

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Have students watch the video that shows Jan and Carlos engaged in Mathematical Practice 2: Reason abstractly and quantitatively. In the video, Jan says that one thing she knows about multiplication is that you always get a bigger number than what you started with. Carlos doubts that this is true and gives an example (2 × $\frac{1}{2}$ = 1) where this does not happen. Carlos first comes up with the conclusion that whenever you multiply by a fraction it’s less than the number you started with, and whenever you multiply by a whole number it’s greater than the number you started with. Jan and Carlos refine this conclusion to: If you multiply by a number less than 1, the product will be less than the number you started with, and if you multiply by a number greater than 1, the product will be greater than the the number you started with. Jan and Carlos use reasoning as they think through this issue.

Give partners a few moments to talk about the discussion questions. Then lead a whole class discussion.

ELL: When showing the video, be sure that ELLs are following the explanations. Pause the video at key points to allow ELLs time to process the information. Ask students if they need to watch it a second time. Ask questions to check for understanding. Allow students to discuss the video prior to writing a summary.

## 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

# Lesson Guide

Discuss the Math Mission. Students will investigate and generalize about the effects of using different types of numbers when dividing and multiplying.

## Opening

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

# Reason About Numbers

# Lesson Guide

Have students work in groups of two or three on the Work Time problems. Allow students to use a calculator for their calculations.

Encourage students to find a way to organize their work so they can look for patterns in their results. If students struggle to find an organization scheme, suggest they use a table like the one shown here.

# Interventions

**Student does not work systematically.**

- Can you organize your work to make it easy to look for patterns?
- Can you organize your results in a table?

**Student does not test enough numbers to see a pattern.**

- Did you try starting with a whole number?
- How many numbers did you divide it by?
- Did you try dividing by at least one fraction and one decimal?
- Did you try dividing by a very small number and a very large number?

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

While investigating Carlos’s statement, students can break their work into cases. They must recognize counterexamples to the statement as well as examples that support it. Based on the results of their explorations, students summarize their findings, citing examples and counterexamples and using reasoning to support their conclusions.

**Mathematical Practice 7: Look for and make use of structure.**

Students look for patterns in their work to help them draw conclusions. For example, they look at all the cases for which dividing produces a smaller result and try to determine what is similar about those cases.

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

As students test various numbers, they may begin to see patterns. For example, they may see that dividing by a number less than 1 but greater than 0 always gives a larger result, while dividing by a number greater than 1 or less than 0 gives a smaller result. They may use this pattern to work more efficiently—choosing numbers specifically to test this pattern rather than choosing numbers at random. In the final part of each problem, students express the patterns they discovered in a general summary.

# Answers

- Answers will vary. Sample investigation:

## 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

# Interventions

**Student has difficulty summarizing a pattern.**

- Look back and find all the examples that gave you a result greater than the original number.
- Do these examples have something in common? What do you notice about the number you divided by?
- Look at all the examples that gave you a result less than the original number.
- Do these examples have something in common? What do you notice about the number you divided by?

# Answers

- Carlos’s statement is not true. Dividing by a number greater than 1 gives a result less than the original number, but dividing by a number less than 1 and greater than 0 gives a result greater than the original number. Examples and reasoning will vary.

## 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

# Preparing for Ways of Thinking

Look for the following to be presented during Ways of Thinking:

- Useful or interesting ways students organize results in the first Work Time task
- Ways students use their organized results to help test Carlos's statement
- Correct and incorrect conclusions and reasoning about when and why a quotient is greater or less than the original divisor

Choose some students to present their real-world situations from the Challenge Problem as well.

SWD: Students with disabilities may require support in recognizing patterns for multiplication and division and in making conclusions about their observations. Have students who easily understand the task demonstrate the patterns they observed to struggling students. This promotes cooperative learning and is beneficial to both the stronger student and the student who is struggling to learn the concept.

# Challenge Problem

## Possible Answers

- Answers will vary. Sample situation: Suppose Dan is 15 years old, and Sally’s age is $\frac{2}{3}$ Dan’s age. Then Sally’s age is $\frac{2}{3}$ x 15 years = 10 years, which is less than 15.
- Answers will vary. Sample situation: Suppose you cut 0.1 meter strips from a 1-meter ribbon. Then you get 1 ÷ 0.1 = 10 strips, which is greater than 1.

## 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

# Lesson Guide

To start, have students share some of the ways they organized their work and discuss how this helped them to test Carlos’s statement. Choose other students to present their conclusion, supporting with reasoning and examples.

Have students who solved the Challenge Problem share their situations.

ELL: Present some of the Intervention questions in writing to support ELLs. Provide sentence frames such as the following (in the order the questions were given):

- “When I multiply a number by….”
- “I found when I divide a number….”

# Mathematics

Ask students questions to help them think about real-life situations in which multiplying makes a quantity smaller and dividing makes a quantity bigger.

Situation 1: Suppose a recipe calls for $1\frac{1}{2}$ cups of flour, but I am making only half the recipe.

- What multiplication shows how much flour I need? (Answer: $\frac{1}{2}$ × $1\frac{1}{2}$)
- Will the answer be greater or less than $1\frac{1}{2}$? (Answer: less)

Situation 2: Suppose I want to measure the $1\frac{1}{2}$ cups of flour using a $\frac{1}{4}$-cup measure.

- What division shows how many times I have to fill the quarter cup? (Answer $1\frac{1}{2}$ ÷ $\frac{1}{4}$)
- Will the answer be greater or less than $1\frac{1}{2}$? (Answer: greater)

Situation 3: Suppose a store donates 2 cents to charity for every dollar a customer spends.

- What multiplication shows how much the store will donate if a customer spends $5? (Answer: 0.02 × 5)
- Will the answer be more or less than 5? (Answer: less)

Situation 4: Suppose I want to buy something that costs $3 and all I have are nickels.

- What division shows how many nickels are in $3? (Answer: 3 ÷ 0.05)
- Will the answer be greater than or less than 3? (Answer: greater)

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Ask students when they were reasoning quantitatively and when they were reasoning abstractly, and what that means. Here is one way of reasoning abstractly:

Dividing a number *n* by 1 gives the quotient *n*. This means 1 divides into *n* exactly *n* times. For example, 3.6 ÷ 1 = 3.6. If we divide *n* by a number bigger than 1, it should go in fewer than *n* times. If we divide by a number smaller than *n*, it should go in more than *n* times.

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Students can critique the presentations of others and offer suggestions for improvement to their classmates.

**Mathematical Practice 6: Attend to precision.**

Encourage students to present their conclusions clearly and precisely. Work as a class to correct errors.

## 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

# A Possible Summary

When you multiply a positive number by a factor less than 1, the result will be less than the starting number.

When you multiply a positive number by a factor greater than 1, the result will be greater than the starting number.

When you divide a positive number by a divisor less than 1 but greater than 0, the result will be greater than the starting number.

When you divide a positive number by a divisor greater than 1, the result will be less than the starting number.

ELL: Ask questions to all students about the summary, but especially to ELLs, to check for understanding before moving on. In addition, make sure that you provide the summary in written form or as an anchor chart.

## 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

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery problems and which students would benefit from review before the assessment. Have students work on the Self Check individually.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

**Student multiplies by $\frac{1}{3}$ instead of dividing by $\frac{1}{3}$.**

- Think about a simpler problem with whole numbers: What if Jan had 6 yards of fabric and used 2 yards for each elephant? What operation would you use to solve this problem?
- Finding a fraction times a number is the same as finding that fraction of the number. Is that what you want to do?

**Student tries to divide without changing 1$\frac{3}{4}$ to a fraction.**

- What method can you use to divide a whole number by a fraction?
- Can you use that method for this problem?
- How can you write a mixed number as a fraction?

**Student has trouble getting started.**

- What do you know?
- What are you trying to find?
- List all the materials Jan spends money on.
- How can you find the cost of these materials for each elephant?

**Student’s work is hard to follow.**

- How can you organize your work so someone else can understand what you did?
- How can you label your work so it is clear what the numbers and calculations represent?

**Student gets an unreasonable answer.**

- Go back and read the problem. Is your answer reasonable?
- How can you use estimation to check your answer?

# Answers

- Jan has enough gray fabric for 4$\frac{2}{3}$ ÷ $\frac{1}{3}$ = 14 elephants.
- Jan has more than enough beans to stuff 14 elephants. She actually has enough for 28 ÷ 1$\frac{3}{4}$ = 16 elephants.
- Jan bought $22.25 ÷ $0.89 per pair = 25 pairs of eyes. She has more than enough eyes for the 14 elephants she has sewn and stuffed.
- One elephant costs ($\frac{1}{3}$ × 3) × (1$\frac{3}{4}$ × 1) + (1 × 0.89) = 1 + 1.75 + 0.89 = $3.64 to make. If Jan sells all of her elephants, then she will make (14 × 17.20) − (14 × 3.64) = 240.80 − 50.96 = $189.84.

## 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 $\frac{1}{3}$ yard of gray fabric, 1$\frac{3}{4}$ 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.

- Jan has 4$\frac{2}{3}$ yards of gray fabric. How many elephants can she make?
- Jan has 28 pounds of dried beans. Does she have enough beans to stuff all of the elephants that she made with the fabric?
- Jan spends $22.25 on plastic eyes. How many pairs did she buy? Is this enough for all of the elephants she stuffed?
- 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

# Lesson Guide

Have students write a brief reflection before the end of the class. Review the reflections to find out what important points they gleaned from the lesson.

## 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 …**