# Lesson 8

Different Ways to Decompose

## Warm-up: Number Talk: Multiples of 10 (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit the ways students look to add or subtract based on place value. When students describe ways to add or subtract by adding or subtracting tens and tens, they make use of the base-ten structure of the numbers. When they describe ways to use the value of the sums to find the value of the differences, they look for and make use of the structure of expression and the relationship between addition and subtraction (MP7). Both of these understandings help students develop fluency with addition and subtraction within 100.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$18 + 10 + 10$$
• $$18 + 20 + 10$$
• $$38 - 20$$
• $$48 - 30$$

### Activity Synthesis

• “How are the addition expressions related to the subtraction expressions?” (The second expression is the opposite of the last expression. They are in the same fact family. The first expression helped me solve the third expression because I know $$18 + 20 = 38$$, so $$38 - 20$$ must be 18.)

## Activity 1: Aren’t You Missing Something? (15 minutes)

### Narrative

The purpose of this activity is for students to analyze two different subtraction methods that are based on place value and connect the methods to equations. In previous lessons, students analyzed base-ten drawings like Lin’s where a student recognizes a ten needs to be decomposed before they draw the blocks. Clare’s drawing represents a student who mentally subtracts tens from tens before drawing and then considers decomposing units. Students discuss how each method works and deepen their understanding of the properties of operations (MP7).

Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community. For example, encourage students to take turns sharing their ideas about Lin and Clare’s methods and give feedback based on their responses.
Supports accessibility for: Social-Emotional Functioning

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Lin and Clare made base-ten diagrams to find the value of $$71-56.$$
• “What do you notice about their work? What do you wonder?” (Lin drew 6 tens and decomposed one ten. Lin crossed out 5 tens and 6 ones. Clare just drew 2 tens and decomposed 1 of them. Clare crossed out 6 ones and the ten she decomposed. Did Clare find $$71-56$$? Why did Clare only draw 2 tens?)
• 12 minutes: partner discussion
• Share responses.
• “Do you think Clare found the value of $$71-56$$? Why or why not?” (Yes, her diagram shows 15 left. No, she showed $$21-6$$. She didn’t represent all the tens.)
• 1 minute: partner discussion
• Share responses.

### Activity

• “Lin and Clare used equations to show their thinking. Work with your partner to match the equations to Lin’s work and Clare’s work. Then discuss how the methods are the same and how they are different.”
• 5 minutes: partner work time

### Student Facing

Lin and Clare made base-ten diagrams to find the value of $$71 - 56$$.

Lin

Clare

1. What do you notice about their work? What do you wonder?
2. Lin and Clare each wrote equations to show their thinking. Explain how you know which group of equations matches Lin’s work and which matches Clare’s work.

A

B

$$71 - 50 = 21$$
$$21 = 10 + 11$$
$$11 - 6 = 5$$
$$10 + 5 = 15$$

$$71 = 60 + 11$$
$$11 - 6 = 5$$
$$60 - 50 = 10$$
$$10 + 5 = 15$$

3. How are Lin and Clare’s methods the same? How are they different?

### Activity Synthesis

• Invite students to share the group of equations that match Lin's work and Clare’s work.
• “How are these methods the same?” (They both decomposed a ten. They both showed subtracting 6 in the diagram. They both subtracted 5 tens.)
• “How are the methods different?” (Clare took away 5 tens before she drew and before she decomposed. Then she took away the ones. Lin drew tens and decomposed first, then took away ones, and then tens. They did the same things, just in a different order and drew in different ways.)
• “Did Clare find the value of $$71-56$$? What did you learn about her method?” (Yes, she did because she did take away tens. I learned you can take away tens first and it doesn’t change the difference.)

## Activity 2: Different Ways to Decompose (20 minutes)

### Narrative

The purpose of this activity is to analyze a subtraction method that is based on place value and connect it to equations. Students analyze the method and explain why they think it best matches one of the methods they saw in the previous activity. Then they practice subtraction using any method that makes sense to them. Monitor for different methods to share in the synthesis, including students who show ways to subtract ones from ones first and those that subtract tens from tens first.

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “exchanged ten or traded ten” for “decomposed ten”.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Andre found the value of $$65-28$$. Take a minute to look at his work.”
• 1 minute: quiet think time
• “Do you think it’s more like Clare or Lin’s method? Discuss with your partner.” (It’s more like Lin’s because he drew all the tens first. It’s more like Clare’s because he took away tens first, he just drew them out.)
• 2-3 minutes: partner discussion
• Share responses.

### Activity

• “Find the value of each difference. Use any method that makes sense to you. Then share your thinking with your partner.”
• 5 minutes: independent work time
• 2-3 minutes: partner work time

### Student Facing

Andre found the value of $$65 - 28$$. He made a base-ten diagram and wrote equations to show his thinking.

$$65 - 28$$
$$65 - 20 = 45$$
$$45 = 30 + 15$$
$$15 - 8 = 7$$
$$30 + 7 = 37$$

1. Do you think Andre’s method is more like Clare’s or Lin’s method? Explain.
2. Find the value of each difference. Show your thinking.

1. $$34 - 18$$

2. $$82 - 37$$
3. $$71 - 53$$

### Student Response

• Students may use a method that shows they decompose and subtract by place, but they write a value that does not match the difference. Consider asking:
• “What did you do first to find the value of _____? Can you show me in your drawing/equations?”
• “What did you do next?”
• “How could you use labels or equations to help keep track of your steps?”

### Activity Synthesis

• Invite 12 students to share their method for each difference.
• Consider asking or inviting peers to ask questions of students who share:
• “What did you do first?”
• “Why did you choose this representation?”
• “How is your method like _____’s method?”

## Lesson Synthesis

### Lesson Synthesis

“Today we made sense of and compared different methods for subtracting two-digit numbers.”

Display Lin, Clare, and Andre’s methods or different examples of student work from the last activity.

“How were the methods you saw today the same? How were they different?”

“Which methods make the most sense to you? Explain.”