# Lesson 3

Unlabeled Tick Marks

## Warm-up: Notice and Wonder: From 0 to 30 (10 minutes)

### Narrative

The purpose of this warm-up is to elicit ideas about what a number line can be used to represent. The sequence of diagrams emphasizes the position of multiples of 5 and 10 on the number line, which will be useful when students represent numbers on number lines that only include labeled tick marks at these positions. While students may notice and wonder many things about these number line diagrams, ideas about how the diagrams may represent counting are the important discussion points.

### Launch

• Groups of 2
• Display the number lines.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “These number lines help us see what it looks like when we count by different numbers.”
• “Count to 30 by 5 starting with 0.”
• “Which number line represents our count? Explain.” (B because the arrows show moving from 0 to 5 to 10 to 15...)
• “Count to 30 by 10 starting with 0.”
• “Which number line represents our count? Explain.” (C because the arrows show moving from 0 to 10 to 20 to 30).

## Activity 1: Locate the Numbers (15 minutes)

### Narrative

The purpose of this activity is for students to work with number lines that only have multiples of 5 or 10 labeled and do not start with 0. Students reason about how a number line can be accurate without all the numbers labeled. Students use the numbers that are labeled to locate specific numbers on number lines. Students rely on the regular structure of the number line and the counting sequence in order to accurately place numbers (MP7).

MLR2 Collect and Display. Collect the language students use as they work with the number lines and discuss the number patterns. Display words, phrases, and representations such as: number line, distance from zero, in order, interval, spaces, tick mark, point, and pattern. During the synthesis, invite students to suggest ways to update the display: “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.

### Launch

• Groups of 2
• Display the first number line.
• “What do you notice about this number line?” (Not all the tick marks are labeled. Only the fives are labeled.)
• 30 seconds: quiet think time
• Share responses.
• “Are the labeled tick marks in the right spots on the number line? Explain.” (Yes because 5 is 5 length units from 0. 10 is 10 units from 0. There are 5 length units between each labeled mark.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses.
• “Today we are going to make sense of and use number lines that do not label every tick mark.”

### Activity

• “Locate each number on the number line and mark it with a point. Be ready to show your partner how you know you located the numbers and how you know they are in the right spots.”
• 4 minutes: independent work time
• 4 minutes: partner work time
• “How did you locate this number?”
• “How did you use the labeled tick marks?”
• “How do you know your number is at the right length from 0?”
• Monitor for students who show their number is at the right position by:
• counting on from 0 or referencing the length from 0
• counting on or back from a labeled tick mark
• describing the length between each labeled tick mark

### Student Facing

1. Locate 24 on the number line. Mark it with a point.

2. Locate 37 on the number line. Mark it with a point.

3. Locate 48 on the number line. Mark it with a point.

4. Locate 83 on the number line. Mark it with a point.

5. Explain how you know the numbers you located are at the right spot on each number line.

### Student Response

If students count every tick mark to locate numbers, consider asking:
• “How could you use the labeled tick marks to help you find a number?”

### Activity Synthesis

• Display the number line labeled 20, 30, 40, 50, 60.
• Invite 12 previously identified students to share how they located 48.
• “How do you know your point is the right distance from 0?” (It’s like when we measured from different spots on a ruler. The tick marks are spaced the same space apart and count by 1 length unit. We could draw the line back and find where 0 is.)
• “A number line representation cannot represent all numbers with tick marks, points, or labels. We can use what we know based on the numbers that are labeled to locate other numbers.”

## Activity 2: Are You Missing Something? (20 minutes)

### Narrative

The purpose of this activity is for students to locate numbers up to 100 on a number line. They complete number lines that are labeled with multiples of 5 or 10 by using what they know about length on the number line and counting by 5 and 10. They use the labeled tick marks to locate and represent numbers within 100. When students explain to one another how the located different numbers on the number lines they construct viable arguments  and may critique each other's reasoning (MP3).

Action and Expression: Develop Expression and Communication. Give students access to two colors of connecting cubes. Build a number line that changes color back and forth at intervals of 5. For example, 5 red cubes, 5 yellow cubes, 5 red cubes, 5 yellow cubes, and so on. The concrete visual of color representing intervals of 5 can be seen clearly.
Supports accessibility for: Conceptual Processing, Organization

• Groups of 2

### Activity

• “On your own, complete each number line by filling in the missing labels with the number the tick mark represents. Then, locate each number, mark it with a point, and label the point with the number it represents.”
• “When you finish, think of how you can explain to your partner how you know your labels and points are at the right spots on the number lines.”
• 5 minutes: independent work time
• 5 minutes: partner discussion
• Monitor for students who:
• explain their labeled tick marks based on counting by 5 or 10
• explain their labeled tick marks based on the equal lengths between each labeled tick mark
• use labeled tick marks to explain how they locate numbers

### Student Facing

Complete each number line by filling in the labels with the number the tick mark represents. Then, locate each number, mark it with a point, and label it with the number it represents.

1. Locate and label 17 on the number line.

2. Locate and label 59 on the number line.

3. Locate and label 43 on the number line.

4. Locate and label 35 on the number line.

### Student Response

If students' labels are numbers other than the specified numbers, consider asking:
• “I see you labeled this point as ___. How can you use the number line to prove that is the location of that number?”

### Activity Synthesis

• Display the image of an incomplete number line labeled with 10 and 50.
• “How could you find 35 without filling in the missing labels?” (We could count back from 50. We could count on from 10.)
• Invite previously identified students to explain how they found the labels and used them to locate 35.
• Complete the number line as students explain.

## Lesson Synthesis

### Lesson Synthesis

“Today we made sense of number lines that do not have each number labeled and number lines that do not start with 0.”

Display the number line from the second activity synthesis with multiples of 10 labeled.

“Why might someone want to label only the tick marks that show the numbers you say when you skip-count by 10? Why not always label every tick mark?” (It might take too long to label every tick mark when there are a lot of numbers. It might be hard to read if all the numbers are there. You might not have enough space to label every tick mark.)

“What other tick marks could we label on this number line to make it easier to find other numbers?” (The numbers we say when we count by 5.)