# Lesson 6

Ten Times as Many

## Warm-up: Choral Count: 12, 15 and 24 (10 minutes)

### Narrative

The purpose of this Choral Count is for students to practice counting by 12, 15, and 24, and notice patterns in the count after 10 multiples. This understanding will help students later in this lesson when they represent quantities that are 10 times as many using tape diagrams.

When they use the words multiple, value, and place students use language precisely (MP6).

### Launch

• “Count by 12 starting at 12.”
• Record as students count.
• Stop counting and recording after the tenth multiple has been said.
• Repeat these steps with 15 and 24.

### Activity

• Circle the tenth multiple in from each count.
• “What do you notice about the tenth multiple?” (It has a zero in the ones place.)
• “Why do you think this is?” (The value is now ten times as much as the first value.)
• “Can you predict the twentieth multiple?”(It is the double of the tenth multiple and it also ends in zero.)
• 1–2 minutes: quiet think time
• 1 minute: partner discussion
• Share and record responses.

### Activity Synthesis

• “What would the tenth multiple be if we were counting by 192? 1092?” (1,920, 10,920)

## Activity 1: Ten Times as Many (20 minutes)

### Narrative

In this activity, students are given a diagram that shows two quantities, one of which is 10 times as much as the other. They identify possible values and possible equations that the diagram could represent.

Students see that a single unmarked diagram could represent many possible pairs of values that have the same relationship (in this case, one is 10 times the other) and be expressed with many equations.

The activity also reinforces what students previously learned about the product of a number and 10—namely, that it ends in zero and each digit in the original number is shifted one place to the left because its value is ten times as much (MP7).

### Launch

• Groups of 2
• Read the first problem aloud.
• “Think of two possibilities for the value of A and B.”
• 2 minutes: quiet think time

### Activity

• 2 minutes: partner discussion
• “Work with a partner on the rest of the activity.”
• Monitor for students who:
• use the tape diagram to reason about the equations.
• rely on numerical patterns in multiples of 10 to reason about 10 times as many

### Student Facing

Here is a diagram that represents two quantities, A and B.

1. What are some possible values of A and B?

2. Select the equations that could be represented by the diagram.

1. $$15 \times 10 = 150$$
2. $$16 \times 100 = 1,\!600$$
3. $$30 \div 3 = 10$$
4. $$5,\!000 \div 5 = 1,\!000$$
5. $$80 \times 10 = 800$$
6. $$12,\!000 \div 10 = 1,\!200$$
3. For the equations that can't be represented by the diagram:

1. Explain why the diagram does not represent these equations.
2. How would you change the equations so the diagram could represent them?

### Student Response

Students may assume that it is impossible for the same diagram to represent multiple expressions. Consider asking:

• “How does the value of B change if A represents 6? How do you know?”
• “What expression could we write to represent this?”

Consider repeating with another number to reinforce the idea.

### Activity Synthesis

Display possible values for A and corresponding values for B in a table such as this:

value of A value of B
• “What do you notice about the values of each set? How do the values compare?” (The values for B are all multiples of 10. Each value for B is ten times the corresponding value for A.)
• Select students to share their responses and reasoning.
• For students who used the diagram to reason about the equations, consider asking, “How might you label the diagram to show that it represents the equations you selected?” (Sample response: For the equation $$30 \div 3 = 10$$ I would label A “3” and B “30.” I know that 30 is ten times as much as 3, so the diagram represents the equation.)
• For students who used their understanding of numerical patterns to support their reasoning, consider asking, “How did you know that the equation could be represented as a comparison involving ten times as many?” (Sample response: I know that when we multiply a number by 10, the product will be ten times the value. I also know that division is the inverse of multiplication, so I looked for equations that were multiplying or dividing by 10 or had ten as a quotient.)

## Activity 2: What Remains the Same? (15 minutes)

### Narrative

In this activity, students analyze situations in which one quantity is ten times as much as another quantity. Students may use different strategies to determine the missing quantity. For example, they may rely on counting as a strategy, or use place value understanding to explain regularity in the products of numbers with 10 (MP8). The reasoning in this lesson prepares students to consider quantities that are 100 times and 1,000 times as many in the next section.

MLR8 Discussion Supports. Synthesis. Display sentence frames to support whole-group discussion: “I noticed _____ so I . . .” and “First, I _____ because . . . .”
Representation: Access for Perception. Invite students to use a class set of base-ten cubes to explore how the value of each digit changes when a number is multiplied by ten.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

• Groups of 2

### Activity

• 5–6 minutes: quiet work time
• 3 minutes: partners discussion
• Monitor for students who:
• use multiplication expressions to represent the relationships between values of A and B, for example, $$14 \times 10$$ to represent the first pair of quantities
• use place value relationships to determine the value of each quantity

### Student Facing

1. Use the diagram to complete the table.

value of A value of B
14 $$\phantom{0000000000}$$
$$\phantom{0000000000}$$ 1,000
160
850
1,000
2,070
3,900
2. Select some values from your table to explain or show:

1. How you found the value of B when the value of A is known.

2. How you found the value of A when the value of B is known.

### Student Response

If students write a value other than 10,000 for B when A is 1,000, consider asking:

• “What do you notice about the pairs of smaller numbers you found that can help you find the value of B?”
• “What about the choral count from today’s warm-up can help you know what number to write?”

### Activity Synthesis

• “What were some things you noticed about the values of A and B?” (Sample responses:
• A was less and B was always greater.) “How many times as much as A was B?” (Ten times)
• The digits of A are all in B and are in the same order, but they are not in the same places in B. There's an extra 0 at the end of the value for B. “Why do you think this is?” (Because they are multiplying 10 by A.)
• “Could we represent both $$4 \times 10$$ and $$10 \times 10$$ using the same diagram of A and B? Why or why not?” (Yes, because A could represent 4 or 10, and B represents 10 times that value.)

## Lesson Synthesis

### Lesson Synthesis

“Today we used diagrams to represent values that are ten times as much as different values. We noticed some patterns when we analyzed the values.”

Display:

“What are some other statements we can make about this diagram that would always be true?” (The value of B is always the ten times the value of A. If you know the value of A you can always figure out the value of B using multiplication. If you know the value of B you can always figure out the value of A.)

Focus discussion on how the diagram shows that the value of A is ten times as much as the value of B no matter what the value of each rectangle is.

## Student Section Summary

### Student Facing

In this section, we learned to use multiplication and the phrase “_____ times as many” or “_____ times as much” for comparing two quantities.

At first, we used cubes and drawings to represent the quantities. For example: Andre has 3 cubes and Han has 12. We compared the number of cubes by:

• saying “Han has 4 times as many cubes as Andre”
• drawing diagrams that shows 3 pieces for Andre and 4 times as many pieces for Han
• writing an equation such as $$4 \times 3 = 12$$

As the numbers got larger, drawing every unit of each quantity became less convenient, so we used simpler diagrams with numbers to represent the size of the quantities.

If Andre has 30 cubes and Han has 4 times as many, we can represent the comparison with a diagram like this:

We ended by comparing quantities in which one quantity is ten times as much as another. We also recalled some patterns in the numbers when we multiplied a number by 10.