# Lesson 8

Beyond 100,000

## Warm-up: How Many Do You See? (10 minutes)

### Narrative

The purpose of this How Many Do You See is to allow students to use place value language to describe the value of the base-ten blocks they see. Students may provide answers that indicate the number of blocks they see, while others may indicate the value of the blocks.

If students do not bring it up, ask about the value of the blocks.

### Launch

• Groups of 2
• “How many do you see? How do you see them?”
• Display image.
• 1 minute: quiet think time

### Activity

• Display image.
• 1 minute: partner discussion
• Record responses.

### Student Facing

How many do you see? How do you see them?

### Activity Synthesis

• “What amount do the blocks represent?”
• “What relationships do you notice between the blocks?”

## Activity 1: Lin’s Representation (15 minutes)

### Narrative

In this activity, students use base-ten blocks or base-ten diagrams to represent large numbers in the ten-thousands and hundred-thousands. They learn that when they assign a new value, 10, to the small cube, larger numbers are more accessible and can be represented with fewer blocks. The limitation of blocks in the classroom will create a need to represent large numbers in a different way. Blocks should be made available and students should be invited to use them if needed. Students should also be encouraged to represent base-ten blocks in diagrams in ways that make sense to them.

When students interpret and use Lin's strategy, they state the meaning of each base-ten block or part of their diagram in a strategic way allowing them to represent large numbers (MP6).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Representation: Access for Perception. Provide access to labels for the base-ten blocks. Invite students to act out Lin’s strategy by shifting the labels so that the small cube is labeled 10 (ten), the long rectangle block is labeled 100 (hundred), and so on. Extra labels can also support students who need a more concrete representation of the numbers, but who may have run out of base-ten blocks. Invite students to make connections between the blocks, the labels, and the diagrams they draw on paper.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Memory

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give each group a set of base-ten blocks.
• “How would you use base-ten blocks or a base-ten diagram to represent 15,710?”
• 2 minutes: independent work time
• “Although you may not be finished, please share your plan for representing 15,710 with a partner.”
• 2 minutes: partner discussion

### Activity

• “Lin was working on this same task when she came up with a strategy to build large numbers without using or drawing so many blocks.”
• “Work with a partner to complete the rest of the activity.”
• “As you complete the second problem, include details such as notes or labels to help others understand your thinking.”
• 5–10 minutes: independent work time
• As students are working, monitor for students who:
• attempt to draw or represent each unit of the number
• create a key or communicate groups of each unit (for example, represent 15 thousands using 150 large squares, or draw a new image to represent 1 ten-thousand and use 5 large squares to represent 5 thousands)

### Student Facing

1. Use base-ten blocks or draw a base-ten diagram to represent 15,710.
2. Lin is using blocks like these to represent 15,710. She decided to change the value of the small cube to represent 10.

What is the value of each block if the value of the small cube is 10?

1. Small cube: 10

2. Long rectangular block: __________

3. Large square block: __________

4. Large cube: __________

3. Use Lin’s strategy to represent 15,710.
4. Use Lin’s strategy to represent each number.

1. 23,000
2. 58,100
3. 69,470
5. Using her strategy, which base-ten blocks would be used to represent 100,000?

### Student Response

Students may try to build 15,710 and run out of blocks to build or represent. Consider asking: “What blocks are missing?” and “What can you draw to represent these missing blocks?”

### Activity Synthesis

MLR7 Compare and Connect
• 2–3 minutes: gallery walk
• “What connections do you see between the representations you made using Lin's strategy?” (I see 15 of the large squares for 15,000 and in another drawing. I see 1 large cube and 5 large squares.)
• Collect student responses and revoice connections between groups of ten that are equivalent to other units.
• “Which blocks would we use to represent 100,000?” (100 large squares or ten large cubes. We could also create a new block to represent 100,000.)

## Activity 2: What Number is Represented? (10 minutes)

### Narrative

In this activity, students interpret a collection of blocks in which a small cube represents different values. They notice a pattern in the value of the digits when the small cube represents 1 and then represents 10. Although students are not required to articulate this relationship until the next lesson, the reasoning here elicits observations about the relationship between the digits in the multi-digit number and the number of each type of block (MP7, MP8).

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• 3 minutes: independent work time

### Activity

• 5 minutes: partner work
• Monitor for students who can describe the relationship between the numbers in the first two problems.

### Student Facing

1. A small cube represents 1. What value do the blocks in the picture represent?

2. A small cube is now worth 10. What is the new value that the blocks in the picture represent?

3. Write two statements comparing the numbers in the previous problems.

### Student Response

Students may say that the value of the collection of blocks remains the same even when the small cube has changed in value. Consider asking: “If each small cube is now worth 10, what is the value of the long rectangle?”

### Activity Synthesis

• Invite students to share their answers to the last problem.
• Create a chart of statements students make when comparing the two numbers for reference in a future lesson.

## Activity 3: Build Hundred-thousands (10 minutes)

### Narrative

The purpose of this activity is to remind students the meaning of expanded form so they can write numbers to the ten-thousands in expanded form.

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• “Complete the first problem on your own and then we'll talk about it as a class.”
• 3 minutes: independent work time
• Select 1–2 students to share their responses to the first problem.
• “How would we write this number using expanded form?” $$(40,\!000 + 9,\!000 + 800 +30)$$
• “Remember, when we write a number as a sum of hundreds, tens, and ones, we are using expanded form.“

### Student Facing

1. To represent large numbers, Lin changed the value of the small cube to 10. She used the following blocks to represent her first number.
1. What number did Lin represent? Show or explain your reasoning.
2. Write an equation to represent the value of the blocks.
2. She used more blocks to represent another number.
1. What number did Lin represent? Show or explain your reasoning.
2. Write an equation to represent the value of the blocks.

### Student Response

If students lose track of the value of the blocks, consider asking: “What is the value of each long block when the small cube has a value of 10?” and “How might this help you find the value of 10 large cubes?”

### Activity Synthesis

• Select 1–2 students to share equations for the second problem.
• “120,450: Let’s practice saying this number together as a class.”
• “What digit is in the thousands place in this number?” (zero)
• “How did Lin end up with a 0 in the thousands place, when she had 20 blocks with a value of 1,000?” (Each group of 10 thousands makes 1 unit of ten-thousand. Since there are 2 groups of 10 thousands, there are 2 ten-thousands.)
• “How can we explain the number represented by 10 blocks with the value of 10,000 each?” (Ten groups of 10,000 is 100,000. We can also reason by counting by 10,000. Nine blocks with a value of 10,000 is 90,000, so 10 blocks would be 10,000 more than that, or 100,000.)
• Record the reasoning about the value of the blocks using equations:
$$10 \times 10 = 100\\ 10 \times100 = 1,\!000\\ 10 \times 1,\!000 = 10,\!000\\ 10 \times 10,\!000 = 100,\!000$$

## Lesson Synthesis

### Lesson Synthesis

Consider using whiteboards during the synthesis to poll the class informally.

“Today we wrote multi-digit numbers using expanded form. Explain expanded form to a partner.”

Write 115,000 for students to see.

“How many hundred-thousands are in this number?” (1)

“How many groups of 10,000 make 100,000?” (10)

“What equation could we write to show 10 groups of 10,000 are equivalent to 100,000?” ($$10 \times 10,\!000 = 100,\!000$$ or $$100,\!000 \div 10 = 1,\!000$$)

“How would we write 115,000 using expanded form?” $$(100,\!000 + 10,\!000 + 5,\!000)$$