# Lesson 9

The Difference Between Numbers

## Warm-up: Number Talk: Add To (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for different ways to find the value of a difference. Students often think of subtraction only as taking away. These equations were chosen to encourage students to show their understanding of subtraction as an unknown addend problem and express strategies based on counting on from the smaller number. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to interpret and represent subtraction strategies as counting on or counting back on the number line.

### Launch

• Display one problem.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Record answers and strategies using a number line.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$20 - 2$$

• $$20 - 17$$

• $$49 - 3$$

• $$67 - 64$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “I noticed that some students counted on for $$20 - 17$$ instead of subtracting or counting back.”
• “How do you decide when to count back or subtract and when to count on?” (If the numbers are close together, I count on. 17 is only 3 away from 20.)
• “We are going to continue to think about when it is helpful to add to find the difference.”

## Activity 1: Add or Subtract (15 minutes)

### Narrative

The purpose of this activity is for students to describe how representations of subtraction on the number line show the difference between two numbers in different ways. The number choices in the activity encourage students to use methods based on taking away or counting back that may be represented as a jump to the left from the larger number. Other number choices encourage students to consider methods that show their understanding of subtraction as an unknown addend problem. Students may start at the smaller addend and find the length of the jump to the right to reach the total. Students may also place both known numbers on the number line and find the length of the space between them by counting from the smaller to larger number or larger number to smaller number. Monitor for each strategy and the ways students represent them on the number line. All of these strategies show an important understanding of numbers and operations and how they can be represented on the number line (MP7).

The synthesis focuses on describing and comparing how the difference is represented on the number line. If a student uses a method that would be difficult to represent on the number line, pair the student with another student who uses a conceptually similar method. For example, if a student finds the value of $$57-24$$ as $$7-4=3$$, $$50-20=30$$, $$30+3=33$$, acknowledge the method would be hard to show on a single number line, and partner them with someone who shows jumps to the left based on place on the number line to compare methods.

### Required Materials

Materials to Gather

Materials to Copy

• Number Line to 100

### Required Preparation

• Place the number line recording sheets in sheet protectors. They will be used in the next activity and future lessons.

### Launch

• Groups of 2
• Give each student a copy of the number lines.

### Activity

• “You are going to find the number that makes each equation true in a way that makes sense to you.”
• “Then, use the number line to show your thinking.”
• 6 minutes: independent work time
• “Compare your methods, solutions, and number line representations with a partner.”
• 4 minutes: partner discussion
• For $$75-68={?}$$, monitor for students who:
• represent taking away or counting back with arrow(s) moving to the left from 75
• represent an unknown addend problem with arrow(s) pointing to the right from 68
• locate 75 and 68 on the number line and count the length of the space between by counting from the smaller to larger number or larger to smaller number

### Student Facing

1. What number makes this equation true?_____

$$38 - 4 = {?}$$

Represent your thinking on the number line.

2. What number makes this equation true?_____

$$75 - 68 = {?}$$

Represent your thinking on the number line.

3. What number makes this equation true?_____

$$57 - 24 = {?}$$

Represent your thinking on the number line.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite previously identified students to share their response and reasoning for $$75 - 68 = {?}$$ or draw the number lines and select the identified students to share their reasoning.
• “Both of these number lines show strategies for finding $$75 - 68 = {?}$$
• If it does not come up when students share, ask:
• “Which representation shows subtraction as taking away?”
• “Which representation shows subtraction as finding an unknown addend?”
• “How do both representations show the difference?” (In the first number line, the arrow is pointing to the difference. In the second one, the length of the jump shows the difference.)
• As needed, gesture and restate student responses to emphasize how the difference is represented as a length in each representation.
• “When we show subtraction as taking away on the number line, the difference is the number the last arrow points to. It’s represented as a length from 0 to where the arrow is pointing.”
• “When we show subtraction as an unknown addend problem on the number line, the difference is shown as the length of the space between the two numbers.”

## Activity 2: Different Ways to Find the Difference (20 minutes)

### Narrative

The purpose of this activity is for students to compare methods for solving subtraction problems. Students compare representations of methods that show subtraction as taking away and subtraction as an unknown addend problem. Students discuss how some representations may better show the actions in a problem and others may show a different way to find the unknown value (MP2). In the synthesis, students consider how to select a strategy based on the numbers in the problem.

Action and Expression: Develop Expression and Communication. Provide access to poster paper and different colored markers, or whiteboards and different colored markers. Invite students to represent jumping forward on the top of their number line in one color, and jumping backward on the bottom of the number line in another color. Write the equation that matches the work shown.
Supports accessibility for: Organization

### Required Materials

Materials to Copy

• Number Line to 100

### Launch

• Groups of 2
• Give each student a copy of the number lines.
• Display only the problem stem for Elena’s string problem without revealing the question.
• “We are going to read this problem 3 times.”
• 1st Read: “Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches.”
• “What is this story about?”
• 1 minute: partner discussion.
• Listen for and clarify any questions about the context.
• 2nd Read: “Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches.”
• “What can we count or measure in this situation?” (Elena’s string. The piece she cut off. The piece she has left.)
• 30 seconds: quiet think time
• 2 minutes: partner discussion
• Share and record all quantities.
• Reveal the question.
• “Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches. How long is the string now?”
• “Based on this story, what are some methods we can use to solve the problem?”
• 30 seconds: quiet think time
• 1–2 minutes: partner discussion

### Activity

• “Work with your partner to choose two number lines that show a method that could be used to find the length of Elena’s string.”
• “Then solve Han’s problem on your own. When you finish, think of how you could explain your method to others.”
• 3 minutes: partner work time
• 3 minutes: independent work time
• “Now, find someone who used a different method than you to solve Han’s problem. Take turns sharing and then try their method on your own.”
• 5 minutes: group work time
• Monitor for students who:
• represent taking away or counting back with arrow(s) moving to the left from 87 to 2.
• represent an unknown addend problem with arrow(s) pointing to the right from 85 to 87.
• locate 85 and 87 on the number line and count the length of the space between by counting from the smaller to larger number or larger to smaller number

### Student Facing

1. Elena had a length of string that was much too long for her project. The string was 65 inches long. Elena cut off 33 inches. How long is the string now?

Choose 2 number lines that show a way to find the length of Elena’s string.

2. Han had 87 inches of string. He cut off 85 inches of it. How much string does he have left?

1. Write an equation to represent the problem with a ? for the unknown.

2. Find the number that makes the equation true.

3. Represent your thinking on the number line.

3. Find someone who used a different method.

Show their method on the number line.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share responses and reasoning for Elena’s string.
• Invite previously identified students to share their thinking and number lines for Han’s string.
• If it doesn’t come up, show how you could locate the 85 and 87 on the number line and find the difference.
• “Which method would you prefer to use to solve Elena’s string problem? Why?” (Taking away 33 because it matches the story. Adding on from 33 because I know I could add tens and ones and not have to regroup.)
• “Which method would you prefer to use for Han’s string problem? Why?” (Finding the unknown addend because I know it was small because the numbers are so close together.)

## Lesson Synthesis

### Lesson Synthesis

“Today we saw different ways to solve subtraction problems represented on the number line. You can think about subtraction as taking away or as finding an unknown addend.”

“Describe to your partner the different ways you can represent and think about $$35 - 3 = ?$$ and discuss how you would find the number that makes the equation true.”

## Cool-down: What’s the Difference? (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.