# Lesson 9

Patrones en la tabla de multiplicar

## Warm-up: Observa y pregúntate: Tabla de multiplicar (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that the product of two factors on the multiplication table is found where the row and column of each factor intersect. While students may notice and wonder many things about these products, the patterns in the multiplication table and how the table is structured are the important discussion points.

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Activity Synthesis

• If not mentioned in students’ responses, explain: “En una tabla de multiplicar se usan filas y columnas para mostrar productos de dos números. Los números que están en la columna de más a la izquierda y en la fila de más arriba son factores” // “A multiplication table uses rows and columns to show products of two numbers. The numbers in the leftmost column and the top row are factors.”
• “Cada número de la tabla (de la parte que no está sombreada) es el resultado de multiplicar los dos factores que están en la misma fila y columna que ese número” // “Each number in the (non-shaded part of the) table is the result of multiplying the two factors in the same row and column as that number.”
• “¿Qué patrones ven en la tabla de multiplicar y por qué funcionan?” // “What are some patterns that you see in the multiplication table and why do they work?” (As we move right on the 3s row or down in the 3s column, the products increase by 3, because we are adding groups of 3. The number 15 appears in two places because we can find $$3 \times 5$$ or $$5 \times 3$$ to get 15. We see 12 in two places in the table because we can get 12 by counting by 3 like 3, 6, 9 12 or counting by 4 like 4, 8, 12.)
• “Encuentren todas las posiciones en las que aparece 20. ¿Cuáles parejas de factores al ser multiplicadas dan 20?” //  “Find all the places where 20 appears. Which pairs of factors multiply to 20?” (4 and 5)

## Activity 1: Productos en la tabla (20 minutes)

### Narrative

The purpose of this activity is for students to apply multiplication strategies based on properties of operations to find products in a multiplication table. While students may use various strategies based on properties of operations, look for opportunities to highlight strategies based on the commutative property. Students consider how known products that are already in the table can help find an unknown product in the multiplication table.

When students use a multiplication fact that they know to determine a multiplication fact that they don’t know, they look for and make use of structure (MP7).

MLR2 Collect and Display. Circulate, listen for and collect the language students use as they find the missing products on the table and describe the strategies they used. On a visible display, record words and phrases such as: add one more group, the same factors, switch the order, take one group away, double. Invite students to borrow language from the display as needed, and update it throughout the lesson.

### Launch

• Groups of 2
• “En esta actividad vamos a trabajar con otra tabla de multiplicar. ¿Qué diferencias hay entre esta tabla y la primera tabla que vimos?” //  “We’ll work with another multiplication table in this activity. How is this table different from the first table we saw?” (It has more products than the first table. It doesn’t have all of the products in it. Some of the boxes have letters in them.)
• 1 minute: quiet think time
• Share responses.

### Activity

• “Usen los números de la tabla como ayuda para encontrar los números que deberían ir en lugar de las letras de la A a la G. Piensen en cómo podrían ayudarles los números que ya están en la tabla” //  “Use the numbers in the table to help you find the numbers that should replace the letters A–G. Think about how the numbers that are already in the table might help.”
• “Después, encuentren los números que deberían ir en otras tres casillas vacías de la tabla. Prepárense para explicar su razonamiento” // “Afterwards, find numbers that should go in three other empty cells in the table. Be prepared to explain your reasoning.”
• 5–7 minutes: independent work time
• “Compartan con su compañero cómo encontraron los números que faltan en la tabla” // “Share with your partner how you found the missing numbers in the table.”
• 3–5 minutes: partner discussion
• Monitor for students who:
• use $$7\times2$$, which is in the table, to find $$2\times7$$ or A
• add one more group of 4 to 20 to find C
• use a product from the 9s row to find a product in the 9s column

### Student Facing

Esta es una tabla de multiplicar que no se ha completado totalmente.

1. Usa los productos de la tabla para ayudarte a encontrar los números que deberían ir en lugar de las letras de la A a la G. Prepárate para explicar tu razonamiento.

2. Encuentra los números que deberían ir en otras tres casillas vacías de la tabla. Usa:

1. 7 como un factor
2. 9 como un factor
3. 10 como un factor
Prepárate para explicar tu razonamiento.

### Activity Synthesis

• Select previously identified students to share how they used the numbers that were in the table to find unknown products. If possible, display and annotate the table to illustrate students’ reasoning.

## Activity 2: Si sé que..., entonces sé que...: Multiplicación (15 minutes)

### Narrative

The purpose of this activity is for students to articulate how they use known products to find unknown products, using a structure similar to that used in an earlier lesson. Students may describe strategies that are based on any property of operations. The focus should be on the description of the strategy (such as “multiplying two numbers in any order gives the same product”) rather than remembering the property on which the strategy is based (such as “commutative property”).

Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Memory, Conceptual Processing

• Groups of 2

### Activity

• “Trabajen individualmente. En la columna de la derecha, escriban al menos dos hechos de multiplicación que pueden descifrar porque conocen el hecho de multiplicación dado en la columna de la izquierda” // “In the right column, work independently to write down at least two multiplication facts you can figure out because you know the given multiplication fact in the left column.”
• 3–5 minutes: independent work time
• “Ahora, compartan con su compañero los hechos que encontraron. Anoten todos los hechos que encontró su compañero y que ustedes no encontraron. Asegúrense de explicar su razonamiento” //  “Now, share the facts that you found with your partner. Record any facts that your partner found that you didn’t find. Be sure to explain your reasoning.”
• 3–5 minutes: partner work time

### Student Facing

1. En cada fila, escribe al menos dos hechos de multiplicación que puedes descifrar porque conoces el hecho de multiplicación dado en la columna de la izquierda. Prepárate para compartir tu razonamiento.
Si sé que . . . , entonces también sé que . . .
$$2 \times 4$$ $$4 \times 2$$, $$4 \times 4$$, $$2 \times 8$$
$$3 \times 5$$
$$4 \times 10$$
$$7 \times 2$$
$$5 \times 8$$
2. Si te queda tiempo, completa el resto de la tabla de multiplicar. Usa los hechos de multiplicación que conoces para encontrar aquellos que no conoces.

### Activity Synthesis

• For each given product, invite 1–2 students to share the products they found and how they were related to the given product.

## Lesson Synthesis

### Lesson Synthesis

“Hoy usamos productos que nos sabíamos para encontrar productos que no nos sabíamos” // “Today we used products that we knew to find products that we didn’t know.”

“¿Qué patrones les parecieron útiles?” // “What patterns did you find helpful?” (We can write the factors in any order, the result is still the same, like $$3 \times 6$$ has the same value as $$6 \times 3$$. If we know $$3 \times 5$$ is 15 and 6 is $$2 \times 3$$, then $$6 \times 5$$ is twice $$3 \times 5$$ or $$2 \times (3 \times 5)$$, or twice 15, which is 30. We can find the value of $$8 \times 2$$ by thinking of 8 as $$3 + 5$$ and then finding $$3 \times 2$$ and $$5 \times 2$$. When 2, 4, 6, 8, and 10 is a factor, the product is even. When 5 is a factor, the product alternates between 5 and 10. When 10 is a factor, the product ends in 0.)

Record the patterns students noticed.