# Lesson 20

La propiedad conmutativa

## Warm-up: Conversación numérica: Resta (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.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

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

### Student Response

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### Activity Synthesis

• “¿Cómo les ayudó el valor posicional a encontrar la diferencia en estos problemas?” // “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.)
• Consider asking:
• “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____'s reasoning in a different way?”
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
• “¿Alguien pensó en el problema de otra forma?” // “Did anyone approach the problem in a different way?”
• “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

## Activity 1: Aprendamos más sobre la multiplicación (20 minutes)

### 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.
Advances: Speaking, Representing
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.
• “¿Qué observan? ¿Qué se preguntan?” // “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
• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Activity

• “Consideremos estos dos arreglos con más detalle. Para cada arreglo, escriban una situación que le corresponda” // “Let’s consider these two arrays in more detail. Write an array situation for each array.”
• 3–5 minutes: independent work time
• “Compartan sus situaciones con su compañero. Juntos, consideren en qué se parecen y en qué son diferentes las situaciones que escribieron para el primer arreglo y para el segundo arreglo” // “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
• “¿En qué se parecían y en qué eran diferentes las situaciones que escribieron?” // “What was the same and what was different about the situations you wrote?”
• Share responses.
• “Ahora, escriban una ecuación para cada situación que acaban de inventarse para los arreglos” // “Now, write an equation for each situation you just came up with to match the arrays.”
• 1 minute: independent work time
• “Compartan sus ecuaciones con su compañero y discutan cómo se conecta cada número de sus ecuaciones con sus situaciones y con el arreglo” // “Share your equations with your partner and discuss how each number in your equations connects to your situations and the array.”
• 2 minutes: partner discussion

### Student Facing

¿Qué observas? ¿Qué te preguntas?

Imagen A

Imagen B

1. Para cada arreglo, escribe una situación que le corresponda.

Imagen A

Imagen B

2. ¿En qué se parecen las situaciones? ¿En qué son diferentes las situaciones?

1. Escribe una ecuación para cada situación.

Imagen A

Imagen B

2. ¿Cómo se conecta tu ecuación con la situación y con el arreglo?

Imagen A

Imagen B

### Student Response

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### Activity Synthesis

• “Escribamos las ecuaciones que nos inventamos” // “Let’s write down the equations we came up with.”
• Display $$2\times5=10$$ and $$5\times2=10$$
• “¿Cómo se conecta cada uno de los números de las ecuaciones con la situación que escribieron?” //“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: Retomemos los arreglos (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
• “Hablen con su compañero sobre qué ecuaciones podrían representar este arreglo” // “Talk to your partner about equations that could represent this array.”
• 1 minute: partner discussion

### Activity

• “Escriban dos ecuaciones para este arreglo. Si les ayuda, pueden imaginar que agrupan los puntos como en la actividad anterior. Después, escriban por qué ambas ecuaciones pueden representar el arreglo” // “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. Escribe 2 ecuaciones de multiplicación que representen el arreglo.

2. Explica por qué ambas ecuaciones pueden representar el arreglo.

### Student Response

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### Activity Synthesis

• “Escribamos las ecuaciones que nos inventamos” // “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

• “Compartan con su compañero por qué ambas ecuaciones pueden representar el arreglo. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “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.
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “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 $$4\times3=12$$.

“¿Qué aprendimos hoy cuando pensamos en arreglos y vimos parejas de ecuaciones como esta?” // “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 $$3\times4=4\times3$$.

“Como ​​$$3\times4=12$$ y ​​$$4\times3=12$$, podemos escribir ​​$$3\times4=4\times3$$” // “Since $$3\times4=12$$ and $$4\times3=12$$, we can write $$3\times4=4\times3$$.“

“La idea de que podemos multiplicar dos números en cualquier orden y obtener el mismo producto se llama la propiedad conmutativa” // “The idea that we can multiply two numbers in any order and get the same product is called the commutative property.”

## Cool-down: Reflexión sobre la multiplicación (5 minutes)

### Cool-Down

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## Student Section Summary

### Student Facing

En esta sección, aprendimos cómo se relacionan los grupos iguales con los arreglos y cómo representar arreglos con expresiones y ecuaciones.

dibujo de grupos iguales

arreglo

expresión

$$3\times5$$

ecuación

$$3\times5=15$$

También aprendimos que podemos multiplicar los números en cualquier orden y obtener el mismo producto.

$$3\times5=15$$

$$5\times3=15$$

$$3\times5=5\times3$$