# Lesson 11

Large Numbers on a Number Line

## Warm-up: Estimation Exploration: What Number Could This Be? (10 minutes)

### Narrative

The purpose of this Estimation Exploration is to practice the skill of making a reasonable estimate for a number based on its location on a number line. Students give a range of reasonable answers when given incomplete information. They have the opportunity to revise their thinking as additional information is provided. The synthesis should focus on discussing what other benchmarks (multiples of 10) would help make a better estimate. The actual number is revealed in the launch of Activity 1.

This estimation exploration encourages students to use what they know about place value to determine the value of the two tick marks the point lies between and then reason about where it is located between them (MP7).

### Launch

• Groups of 2
• Display the image.
• “What number is represented by the point?”
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Record responses in the table.

### Activity

• 1 minute: partner discussion
• Record responses.

### Student Facing

What number is represented by the point?

Record an estimate that is:

too low about right too high
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### Activity Synthesis

• “What information would help you make a more precise estimate?” (Additional tick marks or other numbers around the point)
• Consider providing new information. “Would you like to revise your estimates?”
• Record new or revised estimates.
• “What other information would you need to be more confident with your estimate?”

## Activity 1: Locate Large Numbers (20 minutes)

### Narrative

The purpose of this activity is for students to use their understanding of place value and the relative position of numbers within 1,000,000 to partition and place numbers on a number line. Students place four related numbers on a number line and consider relationships between digits to determine how to partition a number line.

The numbers have the same non-zero digits but with different place values, allowing students to observe the closely related values of the tick marks (MP7) and the identical location on the different number lines of the numbers they plot (MP8).

### Launch

• Groups of 2
• “What do you notice and wonder about the first four number lines?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion
• “Think about where you would place the first number on the number line.”
• “Explain to a partner how you decided where to place the number.”

### Activity

• 10 minutes: independent work time
• 3 minutes: partner discussion
• Monitor for students who:
• add tick marks to show the halfway mark, and the labeled number slightly less than half on each number line in the first problem
• label the seventh tick mark on each number line for the second problem

### Student Facing

1. Locate and label each number on the number line.

2. Locate and label each number on the number line.

3. What do you notice about the location of these numbers on the number lines? Make two observations and discuss them with your partner.

### Student Response

If students label the number line by ones, consider asking: “How might you use the relationship between 100 and 1,000 to help label the number line?”

### Activity Synthesis

• Ask 2–3 students to share their responses and their reasoning for each problem.
• “How did you partition the number line in the first problem?” (I know that 350 is halfway between 300 and 400, so I marked the halfway point, and then estimated where 3 down from that would be.)
• “How do the number lines help you to see the relationship between the numbers?” (The number lines have endpoints that are ten times as much as the number line before. Also, each number is ten times as much as the number before. The place values changed, but the numbers are located in the same relative position.)

## Activity 2: So Many Numbers, So Little Line (15 minutes)

### Narrative

In this activity, students place a set of numbers that are each ten times as much the one before it on the same number line. In doing so, they notice the impact of multiplying a number by ten on its magnitude. Unlike before, the number lines here have no or fewer intermediate tick marks, prompting students to think about how to partition the lines in order to facilitate plotting their assigned number.

MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain their approach to the problem. Invite groups to rehearse what they will say when they share with the whole class.
Representation: Access for Perception. Begin by demonstrating the relative magnitude of numbers in the hundreds, thousands, ten-thousands, and hundred-thousands using millimeters. Invite students to examine a meter stick and notice the size of one millimeter, ten millimeters, one hundred millimeters, and one thousand millimeters. Invite students to guess the length of ten-thousand and one hundred-thousand millimeters. If time and space allow, prepare a walk outside the classroom with stops at 10,000 millimeters from the door and 100,000 millimeters from the door.
Supports accessibility for: Conceptual Processing, Visual Spatial Processing, Attention

### Launch

• Groups of 4
• Assign each student in a group a letter A–D.

### Activity

• “Take a few quiet minutes to think about where your assigned number should go on the number line.”
• “Then, discuss your thinking with your group and work together to locate all four numbers on the number line.”
• 3–4 minutes: independent work time
• 7–8 minutes: group work time
• Monitor for students who:
• partition the number line into hundred-thousands or ten-thousands
• use benchmarks such as 50,000, 200,000, or 350,000

### Student Facing

Your teacher will assign a number for you to locate on the given number line.

1. 347
2. 3,470
3. 34,700
4. 347,000
1. Decide where your assigned number will fall on this number line. Explain your reasoning.

2. Work with your group to label the tick marks and agree on where each of the numbers should be placed.

### Student Response

If students run out of room and only place some numbers on the number line, consider asking: “What is the halfway mark?” Then, follow up with: “Which numbers would fall before the halfway mark? What about after the halfway mark?”

### Activity Synthesis

• Ask 2–3 small groups to share their number line.
• “How did you decide to partition the number line?” (I partitioned the number line by tens, hundreds, thousands, ten-thousands, hundred-thousands—not by ones.)
• “Which numbers were easier to locate? Why?” (34,700 and 347,000, were easier to locate because they were further away from zero.)
• “What would have made it easier to locate the other numbers?” (A longer number line would have made it easier to include more partitions)
• “Make some observations about where the numbers are positioned on the number line.” (Most of the numbers we located are much closer to zero than to 400,000)
• “You located the same four numbers here as you did in the first activity. How are the locations of the points different from those in Activity 1?” (Sample response: Ten times as much looks different when they are all on the same number line.)

## Lesson Synthesis

### Lesson Synthesis

“Today we located and analyzed sets of large numbers on a number line. In each set, each number was 10 times as much as the number before it. Let’s look at the number lines from the first activity.”

“How might we use multiplication equations to show the relationship between each point on the number line?”

• $$347 \times 10 = 3,\!470$$
• $$3,\!470 \times 10 = 34,\!700$$
• $$34,\!700 \times 10 = 347,\!000$$

“What is the relationship between the values of the labels on each number line?” (Each new number line has tick marks that are valued at 10 times as much as the labels on the previous number line.)

## Student Section Summary

### Student Facing

In this section, we worked with numbers to the hundred-thousands.

First, we used base-ten blocks, 10-by-10 grids, and base-ten diagrams to name, write, and represent multi-digit numbers within 1,000,000. We wrote the numbers in expanded form so that we can see the value of each digit. For instance:

$$725,\!400=700,\!000 + 20,\!000 + 5,\!000 + 400$$

Next, we learned that the value of a digit in a multi-digit number is ten times the value of the same digit in the place to its right. For example:

• Both 14,800 and 148,000 have 4 in them.
• The 4 in 14,800 is in the thousands place. Its value is 4,000.
• The 4 in 148,000 is in the ten-thousands place. Its value is 40,000.
• The value of the 4 in 148,000 is ten times the value of the 4 in 14,800.

We used both multiplication and division equations to represent this relationship.

$$10 \times 4,\!000 = 40,\!000$$

$$40,\!000 \div 10 = 4,\!000$$

Finally, we analyzed the “ten times” relationships by locating numbers on number lines.