# Lesson 9

Same Digit, Different Value

## Warm-up: True or False: Expanded Expressions (10 minutes)

### Narrative

The purpose of this True or False is for students to consider the value of the same digit in different places. This reasoning will also be helpful later in this lesson when students describe the relationship between different places in multi-digit numbers.

In this activity, students have an opportunity to look for and make use of structure (MP7) as they use commutative and associative properties of addition to compose numbers and determine equivalent sums.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$4,\!000 + 600 + 70,\!000 = 70,\!460$$
• $$900,\!000 + 20,\!000 + 3,\!000 = 920,\!000 + 3,\!000$$
• $$80,\!000 + 800 + 8,\!000 = 800,\!000 + 80 + 8$$

### Activity Synthesis

Focus question:

• “How can you explain your answer without finding the value of both sides?”
• “We can write numbers in different forms.”
• “What form is used to represent the numbers in this True or False?” (expanded form)

## Activity 1: Card Sort: Large Numbers (20 minutes)

### Narrative

In this activity, students sort a set of multi-digit numbers and describe the place-value relationships they notice in the sorted numbers. They analyze numbers that have the same digits and write the numbers in expanded form, highlighting the value of each digit. Students then describe relationships they see between the digits in each number.

For example, students may note that the value of the 2 in 46,200 is 200, in 462,000 it is 2,000, and that 2,000 is ten times as much as 200. In the synthesis, they learn that the observed relationship can be expressed with multiplication and division equations, such as $$2,\!000 = 200 \times 10$$, $$2,\!000 \div 200 = 10$$, or other equivalent equations.

When students sort the cards, they look for how the numbers are the same and different, including their overall value or the digits that make up the numbers (MP7).

Here are the numbers on the blackline master, for reference:

186,000

375,000

18,600

37,500

499,000

3,750

49,900

1,860

4,990

Representation: Access for Perception. Synthesis: Use base-ten blocks to demonstrate the relationship between the value of the same digit in different numbers. For example, represent 200 with two large square blocks and 2,000 with two large cubes.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Required Materials

Materials to Copy

• Card Sort: Large Numbers (4 to 6 digits)

### Required Preparation

• Create a set of cards from the blackline master for each group of 2 students.

### Launch

• Groups of 2
• “Read the directions for the first two problems and explain them to your partner in your own words.”
• Collect explanations and clarify any confusion about directions.

### Activity

• Give each group a set of cards from the blackline master.
• 5 minutes: partner and group work time on the first two problems
• As students work, listen for place-value language such as: value of the digit, ten times, thousands, ten-thousands, and hundred-thousands.
• Record any place-value language students use to describe how they sorted the numbers and display for all to see.
• “Now work independently to write the numbers in the next problem in expanded form. Then, talk with your partner about the value of the digits.”
• 3 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• accurately write the numbers in expanded form
• describe the relationship between the value of the digits in multiplicative terms (“ten times”)

### Student Facing

Your teacher will give you and your partner a set of cards with multi-digit numbers on them.

1. Sort the cards in a way that makes sense to you. Be prepared to explain your reasoning.
2. Join with another group and explain how you sorted your cards.
3. Write each number in expanded form.

1. 4,620
2. 46,200
3. 462,000
4. Write the value of the 4 in each number.
5. Compare the value of the 4 in two of the numbers. Write two statements to describe what you notice about the values.
6. How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?

### Student Response

Students may describe the relationship between digits only in terms of “more” or “less.” Consider asking: “How might we describe the relationship between the digits using multiplication?”

### Activity Synthesis

• Invite students to share their expressions in expanded form and what they noticed about the value of the 4.
• “What do you notice about the value of the 6 in each number? The value of the 2?” (The value of the 6 is different in each number. It is first 600, then 6,000, then 60,000.)
• Students may talk about the number of zeros in each number. Shift their focus to the place value of the 6— hundreds, thousands, ten-thousands.
• “How is the value of the 2 in 46,200 related to the value of the 2 in 462,000?” (The value of the 2 in 462,000 is 2,000 and the same digit in 46,200 has a value of 200. 2,000 is ten times the value 200.)
• “What multiplication equation could we write to represent the relationship between the 2 in 46,200 and 462,000?” ($$2,\!000 = 200 \times 10$$)
• “We can also write this equation using division: $$2,\!000 \div 200 =10$$.”

## Activity 2: Expand Large Numbers (15 minutes)

### Narrative

In this activity, students read, write, and analyze multi-digit numbers and use expanded form to describe the relationship between the digits. The numbers in the activity are designed to highlight common errors in reading and writing large numbers. Students encounter numbers with the digit zero in the ten-thousands place and think about how to represent this in expanded and word forms.

MLR2 Collect and Display. Synthesis: Direct attention to words collected and displayed from the previous activity. Invite students to borrow language from the display as needed, and update it throughout the lesson.

### Launch

• Groups of 2
• “Read the heading in each column and look in the table for examples of each form of number.”
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses from students. Clarify any misunderstanding about each number form. Record on chart for future reference if needed.

### Activity

• “Work independently on the first three problems. Then find 2 classmates to work on the last problem with.”
• 10 minutes: work time

### Student Facing

1. Express each number in standard form, expanded form, and word form.

number expanded form word form
784,003
$$50,\!000 + 9,\!000 +$$ $$300 + 60 + 1$$
eight hundred three thousand, ninety-nine
310,060
nine hundred thirty-four thousand, nine hundred
2. Choose two numbers from the table to make this statement true:

The 3 in _______________ is ten times the value of the 3 in _______________.

3. Explain why you chose those numbers.
4. Find two classmates who chose different numbers than you did. Record their numbers. Take turns sharing your completed statements and explaining your reasoning.

• The 3 in _______________ is ten times the value of the 3 in _______________.

• The 3 in _______________ is ten times the value of the 3 in _______________.

### Student Response

Students may find a number with a digit that is not ten times the value of the digit in another number. Consider asking students to record the values of the 3 in each number and arrange them side-by-side. For example, the 3s in 784,003, 59,361, 803,099, 934,900, and 310,060 have values of 3, 300, 3,000, 30,000, and 300,000, respectively. Ask:

• “Which value is ten times another value?”
• “How does this help you determine what numbers can you include in the statement, ‘The 3 in _______________ is ten times the value of the 3 in _______________.’?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we described the relationship between the same digit in different places in multi-digit numbers.”

“Share with a neighbor something you learned about the relationship between digits from today’s lesson.” (I learned that a digit in the ten-thousand place is ten times the value of the same digit in the thousands place.)

Record students’ ideas using words and ask, “What equation could we write to show how many groups of 80,000 there are in 800,000?” ($$800,\!000 \div 80,\!000 = 10$$ or $$10 \times 80,\!000 = 800,\!000$$)