Warm-up: Number Talk: Related Numbers (10 minutes)
- Display one expression.
- “Give me a signal when you have an answer and can explain how you got it.”
- 1 minute: quiet think time
- Record answers and strategy.
- Keep expressions and work displayed.
- Repeat with each expression.
Find the value of each expression mentally.
- \(650 + 75\)
- \(5,\!650 + 75\)
- \(50,\!650 + 75\)
- \(500,\!650 + 75\)
- “Which parts of the number change when we add 75?” (Just the hundreds, tens and ones.)
- “How do we know without adding if digits in a number are going to change?” (Because we are not adding more than 10 tens to any of the numbers and there's only 6 hundreds, we know the thousands are not going to change.)
Activity 1: Alike but Not the Same (15 minutes)
In this activity, students make sense of the relationships between the values of the same digit in different numbers, and write multiplication and division equations to represent these relationships.
As they complete and analyze the table, students recognize that the value of the digit in one row is ten times as much as the value of the digit in the row below. Students work to articulate these relationships precisely, using words and equations, and receive feedback from their peers on the equations they are writing. During the synthesis, students discuss why a multiplication or a division equation can be used to represent the same relationship.
When students express place value relationships with multiplication and division they observe structure in the place values (MP7). When they help one another improve their explanations, they critique each other's reasoning (MP3).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.
- Groups of 2
MLR1 Stronger and Clearer Each Time
- 5 minutes: independent work
- “Share your response to the second problem with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
- 3–5 minutes: structured partner discussion
- Repeat with 2–3 different partners.
- If needed, display question starters and prompts for feedback.
- “Can you give an example to help show . . . ?”
- “Can you use the word _____ in your explanation?”
- “Revise your initial draft based on the feedback you got from your partners.”
- 2–3 minutes: independent work time
- Monitor for students who write a multiplication and division equation to represent the relationship between the values of the 8 in two different numbers.
Complete the table with the value of the 8 in each number.
number value of the 8 180,000 108,000 100,800 100,080 100,008
- Describe the relationship between the value of the 8 in each number.
- Write a multiplication or division equation to represent the relationship between the values of the 8 in two different numbers in the table.
- Select 1-2 students to share a multiplication and a division equation.
Activity 2: More and More Money (20 minutes)
In this activity, students use the context of money to deepen their understanding of the relationship between the value of digits in different places—by counting equal groups of tens, hundreds, thousands, and ten-thousands. Writing the value of each stack of bills reinforces the “ten times” relationship between the place values, which in turns supports students in writing multiplication and division equations (MP7).
If play money is available, consider creating a counting collection and asking students to organize each stack and write equations to represent the stacks of bills. Ultimately, students would evaluate the expressions used by different groups and discuss the reasoning behind each equation.
Supports accessibility for: Conceptual Processing, Organization, Attention
- Groups of 2
- “Take a minute to read over the directions to the activity and explain them to a partner.”
- 5 minutes: independent work
- 10 minutes: partner work
- As students work, monitor for the different ways they describe the relationship between the stack of bills.
- Each stack has the same number of bills but have different values.
- The value of some stacks can be multiplied by ten to have the same value as another stack.
Diego’s class is counting collections of play money during a math class. There are four types of bills: tens, hundreds, thousands, and ten-thousands.
Diego found 9 of each type of bill. He organized each type into a stack, creating four stacks.
How much money is in each stack of bills?
- Describe the relationship between the values of each stack of bills.
- How is the value of the stack of thousands related to the value of the stack of ten-thousands? Write an equation for that relationship.
Clare had 21 bills of each type. How much money is in each stack of bills Clare has?
- What is the value of the 2 in each stack of bills?
- How is the value of the 2 in the stack of thousands related to the value of 2 in the stack of ten-thousands? Write an equation for that relationship.
Advancing Student Thinking
Students may be able to write an equation to represent the relationship between digits in the hundreds and thousands but get stuck when writing equations that represent the relationships between digits in larger place values. Consider asking students to describe the relationship between 1,000 and 10,000 by asking:
- “How many groups of 1,000 are needed to create a group of 10,000?”
- “How might you use this reasoning to think about the relationship between 2,000 and 20,000?”
- See lesson synthesis.
“Today we wrote multiplication and division equations to represent the relationship between the digits in different places in multi-digit numbers.”
- \(2,\!000 \times 10 = 200\)
- \(2,\!000 \times 10 = 20,\!000\)
- \(20,\!000 \div 10 = 2,\!000\)
- \(20,\!000 \times 10 = 200,\!000\)
- \(200,\!000 \div 10 = 200\)
“Which of these equations represent the relationship between the digit 2 in the stacks of hundreds, thousands, and ten-thousands?” (B, C, D)
“Can you write a new equation that correctly describes the relationship between the digit 2 in two of the stacks?”
Cool-down: Same Digit, Different Place (5 minutes)
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