# Lesson 18

Representemos arreglos con expresiones

## Warm-up: Cuántos ves: Un arreglo de figuras (10 minutes)

### Narrative

The purpose of this How Many Do You See is for students to subitize or use grouping strategies to describe the images they see. When students use the structure of the array to figure out how many objects are shown, they look for and make use of structure (MP7).

### Launch

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Flash the image.
• 30 seconds: quiet think time

### Activity

• Display the image.
• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

### Student Facing

¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?

### Activity Synthesis

• “¿Cómo les ayudó ver grupos iguales a saber cuántos triángulos había en el arreglo?” // “How did seeing equal groups help you know how many triangles there were in the array?”
• “¿Alguien puede expresar la manera en la que _____ vio los triángulos de otra forma?" //  “Who can restate the way _____ saw the triangles in different words?”
• “¿Alguien vio los triángulos de la misma manera, pero puede explicarlo de otra forma?” // “Did anyone see the triangles the same way but would explain it differently?”

## Activity 1: Representemos situaciones con arreglos (20 minutes)

### Narrative

The purpose of this activity is for students to represent multiplication situations with arrays and multiplication expressions. Students should have the option to use math tools, such as counters or connecting cubes, to create the arrays before they draw them. Connecting situations, arrays, and expressions reinforces the idea that multiplication can be used to express the total number of objects in equal groups (MP2).

MLR2 Collect and Display. Amplify language used to describe arrays. On a visible display, record words, phrases and expressions such as: row, column, each, for every, 3 by 5, 5 by 3, $$5\times3$$, and $$3\times5$$. Include diagrams and annotations. Invite students to borrow language from the display as needed, and update it throughout the lesson.
Representation: Internalize Comprehension. Make connections between representations visible. Invite students to use gestures or drawings as they verbally describe correspondences between their arrays and expressions. For example, “The 4 in my expression $$4\times5$$, shows the number of rows, and the 5 shows that there are 5 cars in each row.”
Supports accessibility for: Conceptual Processing, Memory

### Required Materials

Materials to Gather

### Required Preparation

• Each group of 2 will need 20 connecting cubes or counters.

### Launch

• Groups of 2
• Display the first situation.
• “Tómense un minuto para representar esta situación con un arreglo. Pueden usar objetos o dibujos” // “Take a minute to represent this situation with an array. You can use drawings or objects.”
• 2 minutes: independent work time
• “Discutan sus ideas con su compañero” // “Discuss your ideas with your partner.”
• 2 minutes: partner discussion
• Share responses. Emphasize ways students used equal groups to create their arrays.

### Activity

• “Con su compañero, representen con un arreglo las siguientes tres situaciones. Prepárense para compartir cómo ven grupos iguales en su arreglo” // “Work with your partner to represent the next three situations with an array. Be prepared to share how you see equal groups in your array.”
• 5 minutes: partner work time
• Have students share an array for problems 2–4. Try to show both drawings and arrays made of objects.
• “¿Cómo ven grupos iguales en sus arreglos?" // “How do you see equal groups in your arrays?” (The rows can show the number of groups, and then you put however many are in each group across the row.)
• Display the first situation again.
• “Retomemos la primera situación. ¿Qué expresión de multiplicación representaría esta situación?” // “Let’s revisit the first situation. What multiplication expression would represent this situation?”
• 30 seconds: quiet think time
• Share responses. Emphasize how students use equal groups to write the expression.
• “Con su compañero, escriban expresiones de multiplicación para las otras tres situaciones” // “Work with your partner to write multiplication expressions for the other three situations.”
• 2 minutes: partner work time

### Student Facing

Usa objetos o dibujos para representar cada una de las situaciones con un arreglo.

1. Hay 3 filas de sillas. Cada fila tiene 5 sillas.

2. Hay 4 filas de automóviles. Cada fila tiene 5 automóviles.

3. Hay 2 filas de huevos. Cada fila tiene 6 huevos.

4. Hay 2 equipos de estudiantes en fila. Cada equipo tiene 10 estudiantes.

Escribe una expresión de multiplicación que represente cada situación.

### Activity Synthesis

• “¿Cómo usaron grupos iguales para escribir su expresión de multiplicación?” // “How did you use equal groups to write your multiplication expression?” (I thought about how many groups were in the situation and then how many things were in each group. For example, I know that 2 teams of 10 students is 2 groups of 10, so I can write $$2\times10$$.)

## Activity 2: Conectemos arreglos con expresiones (15 minutes)

### Narrative

The purpose of this activity is for students to apply their knowledge from previous activities to draw arrays to match multiplication expressions. Have connecting cubes or counters available for students who need them. In the launch, students use their bodies to make an array for the expression $$4\times6$$. Feel free to adjust this expression to better fit the number of students in your class.

### Launch

• Groups of 2
• Display $$4\times6$$
• “En grupo, con toda la clase, usen sus cuerpos para crear un arreglo que muestre esta expresión de multiplicación” // “Work together as a class to use your bodies to create an array for this multiplication expression.”
• If the number of students does not exactly match the product, ask extra students to monitor the array and be prepared to explain where they see parts of the expression in the array.
• 3–5 minutes: whole-class work time

### Activity

• “Con su compañero, dibujen un arreglo para cada expresión de multiplicación. Prepárense para compartir las conexiones que observen entre las expresiones de multiplicación y los arreglos” // “Work with your partner to draw an array for each multiplication expression. Be ready to share connections you notice between the multiplication expressions and arrays.”
• 2–3 minutes: partner work time

### Student Facing

Dibuja un arreglo para cada expresión de multiplicación. Prepárate para compartir tu razonamiento.

1. $$2\times3$$
2. $$5\times2$$
3. $$4\times4$$

### Student Response

If students draw equal groups that are not arrays, consider asking:

• “¿Cómo usaste la expresión para hacer tu dibujo?” // “How did you use the expression to create your drawing?”
• “¿Cómo podríamos reorganizar tu dibujo en un arreglo?” // “How could we rearrange your drawing into an array?”

### Activity Synthesis

• Share an array or two for each expression.
• “¿Qué conexiones vieron con su compañero entre las expresiones de multiplicación y los arreglos?” // “What connections did you and your partner see between the multiplication expressions and arrays?” (The factors tell us how many things are in each row and column. For $$2\times3$$, we drew an array with 2 columns that have 3 things in each column. For $$2\times3$$, we drew an array with 2 rows that have 3 things in each row.)

## Lesson Synthesis

### Lesson Synthesis

Display a situation from the first activity, and an array and an expression that represents the situation.

“Aprendimos que la multiplicación es la manera en la que expresamos el número total de objetos que hay en grupos iguales” // “We learned that multiplication is how we express the total number of objects in equal groups.”

“¿Cómo les ayudaron sus conocimientos sobre grupos iguales a escribir expresiones para las situaciones de multiplicación?” // “How did your knowledge of equal groups help you create arrays and write expressions for multiplication situations?” (I thought about how many groups there were and drew the groups as each row [or column]. Then the number of groups tells me how many rows [or columns] there are. The array and the expression represent the total number of objects in the problem.)