# Lesson 20

The Commutative Property

## Warm-up: Number Talk: Subtraction (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for subtracting within 100. It also provides an opportunity to observe student strategies as they work toward becoming fluent in addition within 1,000.

When students use strategies based on place value to subtract, they look for and make use of structure (MP7).

### 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.

• $$70-10$$
• $$68-10$$
• $$70-12$$
• $$68-12$$

### Activity Synthesis

• “How was place value helpful as you found the difference in these problems?” (I subtracted the tens first then adjusted my answer each time. I subtracted tens from tens and ones from ones.)
• “Who can restate _____'s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”

### Narrative

The purpose of this activity is to introduce the commutative property. Students write array situations for a pair of arrays and discuss similarities and differences. While the situations will have the same total number of objects, how the objects are grouped should be different. Then, students write equations to go with the arrays and situations, and make connections between the representations (MP2). Students notice that, while the order of the factors in the multiplication equation changes, the product does not change (MP7).

MLR8 Discussion Supports. Synthesis: Create a visual display of the equations and corresponding arrays. As students describe their connections between the equations and the situations, annotate the display to illustrate the connections. For example, below each number, write either rows, columns, or total.
Representation: Access for Perception. Students may benefit from the opportunity to observe a demonstration that shows the grouping of dots in the arrays. For example, prepare a display of Image A and Image B showing only the dots. Then, invite students to watch as you circle the groups accordingly.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?” (Students may notice: Both groups of dots are arranged as arrays. They both have 10 dots. One array has groups of 2. One array has groups of 5. Students may wonder: Why are the dots grouped differently? Are the arrays the same?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

### Activity

• “Let’s consider these two arrays in more detail. Write an array situation for each array.”
• 3–5 minutes: independent work time
• “Share your situations with your partner. Together, consider how the situations you wrote for the first array are the same and different from the situations you wrote for the second array.”
• 3 minutes: partner work time
• “What was the same and what was different about the situations you wrote?”
• Share responses.
• “Now, write an equation for each situation you just came up with to match the arrays.”
• 1 minute: independent work time
• 2 minutes: partner discussion

### Student Facing

What do you notice? What do you wonder?

Image A

Image B

1. Write an array situation for each array.

Image A

Image B

2. How are the situations the same? How are the situations different?

1. Write an equation for each situation.

Image A

Image B

2. How does your equation connect to the situation and array?

Image A

Image B

### Activity Synthesis

• “Let’s write down the equations we came up with.”
• Display $$2\times5=10$$ and $$5\times2=10$$
• “How do each of the numbers in the equations connect to the situation you wrote?” (5 is the number of columns in one situation, but it's how many dots are in each row in the other situation. 2 is the number of groups in one situation, but it's how many are in each group in the other situation. 10 is the total number of objects in both situations.)

## Activity 2: Revisit Arrays (15 minutes)

### Narrative

The purpose of this activity is to reinforce the idea of the commutative property. In this activity, students write two equations to match an array to show again that reversing the order of the factors does not change the product. If students do not immediately see how they might write different equations for the array, encourage them to consider different ways of grouping the dots in the array, similar to the previous activity. Students use the vocabulary they have learned for describing arrays and multiplication to explain why both equations match an array with their partner. The Stronger and Clearer Each Time routine allows students to receive feedback and revise their explanation for clarity (MP3, MP6).

If students finish early, consider drawing another array. Have students write two equations for the array and consider how they can think of the rows or columns as equal groups.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

### Launch

• Groups of 2
• “Talk to your partner about equations that could represent this array.”
• 1 minute: partner discussion

### Activity

• “Write two equations for this array. If it helps you, you can imagine grouping the dots as in the previous activity. Then write down why both equations can represent the array.”
• 5 minutes: independent work time

### Student Facing

1. Write 2 multiplication equations that represent the array.

2. Explain why both equations can represent the array.

### Activity Synthesis

• “Let’s write down the equations we came up with.”
• Display the equations $$3\times6=18$$ and $$6\times3=18$$.

MLR1 Stronger and Clearer Each Time

• “Share why both equations can represent the array 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.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• Have students share the revisions they made to their initial draft.

## Lesson Synthesis

### Lesson Synthesis

Display a 3 by 4 array and the equations $$3\times4=12$$ and $$4\times3=12$$.

“What did we learn from thinking about arrays and seeing pairs of equations like this today?” (The order of the factors does not change the product or the total number of objects in the array or situation. Connecting the numbers in your equations to arrays and situations helps clarify what each number means.)

Display $$3\times4=4\times3$$.

“Since $$3\times4=12$$ and $$4\times3=12$$, we can write $$3\times4=4\times3$$.“

“The idea that we can multiply two numbers in any order and get the same product is called the commutative property.”

## Student Section Summary

### Student Facing

In this section, we learned how equal groups are related to arrays and how to represent arrays with expressions and equations.

drawing of equal groups

array

expression

$$3\times5$$

equation

$$3\times5=15$$

We also learned that we can multiply numbers in any order and get the same product.

$$3\times5=15$$

$$5\times3=15$$

$$3\times5=5\times3$$