# Lesson 13

Number Line Distances

These materials, when encountered before Algebra 1, Unit 4, Lesson 13 support success in that lesson.

## 13.1: Math Talk: How Far? (10 minutes)

### Warm-up

The purpose of this Math Talk is to elicit strategies and understandings students have for computing the distance between two points on the number line, in particular thinking about distance as the absolute value of a difference of two numbers. These understandings will be helpful later in this lesson when students learn about the absolute value function.

### Launch

Display one problem at a time. Give students quiet think time for each problem, and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

Evaluate mentally: How far away is each house from the school?

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

## 13.2: $a$ and $b$ (10 minutes)

### Activity

In this activity students subtract 2 values and recall that the distance between the values on the number line is given by the lesser value subtracted from the greater value.

### Student Facing

1. For each pair of values, find $$b - a$$. Be prepared to explain your reasoning.
1. $$a = 28, b = 57$$
2. $$a = \frac{4}{5}, b = \frac{1}{2}$$
3. $$a = 27, b = \text{-}17$$
4. $$a = \text{-}35, b = \text{-}19$$
5. $$a = 19, b = 35$$
6. $$a = \text{-}106, b = 43$$
2. For which pairs of values does the subtraction give the distance between the numbers on the number line?
1. What do you notice about these pairs of numbers?
3. Given 2 numbers, how can you find the distance between them on the number line?

### Activity Synthesis

The purpose of the discussion is to find an algorithm for finding the distance between two numbers on the number line. Select students to share their solutions and methods. Ask students,

• “When you have two values, $$a$$ and $$b$$, what is the difference between the solution to $$a - b$$ and $$b-a$$?” (The value is the same, but one of them will be negative.)
• “Is there a way to write a formula to find the distance between $$a$$ and $$b$$ on the number line if you don’t know which is greater?” (There are no ways that I remember right now.)

## 13.3: It’s That Far Away (20 minutes)

### Activity

In this activity, students find 2 numbers that are a given distance from a number. In the associated Algebra 1 lesson students find the positive difference between 2 values by figuring out how far away a guess is from an actual value.

### Student Facing

1. Find 2 numbers that are $$d$$ away from $$a$$ on the number line.
1. $$a = 14, d = 6$$
2. $$a = \text{-}7, d = 16$$
3. $$a = 103, d = 56$$
4. $$a = 4, d = 138$$
2. Use $$d$$ and $$a$$ to write 2 expressions that find the values that are $$d$$ away from $$a$$.
3. Kiran is looking at some old work where he did problems like this and found an answer that was marked correct. The answer is -18 and 46. Could Kiran figure out the values of $$a$$ and $$d$$ from the problem based on these values? If so, what are the values? If not, what additional information would help? Explain or show your reasoning.
4. In a planned neighborhood along Stepford Street, all of the houses are identical and equally distant from one another. The house at 102 Stepford Street is 2,250 feet from the house at 84 Stepford Street. Is there enough information to find the address of another house that is that same distance away from 84 Stepford Street? Explain your reasoning.

### Activity Synthesis

The purpose of the discussion is for students to think more deeply about distances on the number line. Select students to share their responses. Ask students,

• “How far apart are 85 and 31 on the number line? What is another number that is the same distance away from 31?” (They are 54 apart. -23 is also 54 away from 31.)
• “In most places in the United States, houses on the same side of the street are either all even numbers or odd numbers. Does this affect the answer to the question about the planned neighborhood?” (No. It would affect how I might draw the picture (for example, there may only be 9 houses between 84 and 102 Stepford Street), but the same numbering would happen on the other side of the house at 84 Stepford.)