# Lesson 15

Más factores, más problemas

## Warm-up: Conversación numérica: Decenas (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplication by 10. These understandings help students develop fluency and will be helpful later in this lesson when students need to be able to represent and solve a problem involving groups of 10.

When students reason why as one factor increases by 1, the product increases by 10, they look for and express the regularity they notice in the expressions (MP8).

### 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

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

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$1\times10$$
• $$2\times10$$
• $$3\times10$$
• $$4\times10$$

### Activity Synthesis

• “¿Qué patrón reconocemos al observar todos los problemas? ¿Por qué ocurre esto?” // “What pattern do we see as we look at all of the problems? Why is that happening?” (The factor that isn’t 10 goes up by 1 each time. The products increase by 10 because I have one more group of 10.)
• “¿Alguien usó otra estrategia?” // “Did anyone use a different strategy”?
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”

## Activity 1: Representemos situaciones con ecuaciones (15 minutes)

### Narrative

The purpose of this activity is for students to represent a situation with a multiplication equation including a symbol for the unknown, and find the number that makes the equation true. Students are able to use an earlier representation to help them solve the problem, however some students may just write the equation and skip-count to find the product. Either is okay. In the synthesis, share different ways students represented the problem beyond the equation. If students used repeated addition, avoid saying ‘multiplication is repeated addition’ because while repeated addition is one way to find the product, it is not the meaning of multiplication.

To add movement to this activity, students can work in groups of 4 to make a poster for one of the problems. After each group is done, they can do a gallery walk to look for things that are the same or different in the posters.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Memory

### Launch

• Groups of 2
• “Vamos a resolver problemas sobre grupos iguales que pueden suceder cuando se prepara o se come una comida” // “We are going to solve some problems about equal groups that you may see when you are making or eating a meal.”
• “¿Qué grupos iguales pueden ver cuando preparan o comen una comida?” // “What are some equal groups that you might see when making or eating a meal?”
• Share responses.
• “Para cada problema, piensen en una posible representación. La idea es que esa representación les ayude a escribir una ecuación que tenga un número desconocido” // “Think about how you could represent these problems in a way that could help you write an equation with an unknown number for each problem.”
• 1 minute: quiet think time
• 1 minute: partner discussion

### Activity

• “Ahora, trabajen de manera independiente en estos problemas” // “Now, independently work on these problems.”
• 5–7 minutes: independent work time
• As you circulate, consider asking:
• “¿Cómo se podría representar esta situación?” // “How could you represent this situation?”
• “¿Qué información no conocemos sobre la situación?” // “What information is missing from the situation?”

### Student Facing

• Escribe una ecuación que corresponda a la situación. Usa un símbolo para representar el número desconocido.
• Encuentra el número que hace que la ecuación sea verdadera. Muestra tu razonamiento.
1. Hay 15 platos. Han puso 5 platos en cada mesa. ¿En cuántas mesas puso platos?

1. ecuación:
2. solución:
2. Lin hizo 6 sándwiches. Lin usó 2 rebanadas de pan para cada sándwich. ¿Cuántas rebanadas de pan usó?

1. ecuación:
2. solución:
3. Han tiene 60 cubos de hielo. Los cubos de hielo vienen en bandejas de 10. ¿Cuántas bandejas de cubos de hielo tiene Han?

1. ecuación:
2. solución:

### Activity Synthesis

• Display samples of student work for each problem next to each other, including a sample of a drawing of equal groups and a sample of a tape diagram.
• “¿Dónde encontramos las partes del problema en el dibujo y en el diagrama?” // “Where do we see the parts of the problem in the drawing and the diagram?” (The number of objects in each group are the dots in the drawing, but the number is written in each part of the diagram.)
• “¿Cómo usaron los factores de cada ecuación para encontrar el producto?” // “How did you use the factors in each equation to find the product?” (The factors told me how many groups there were and how many were in each group.)
• “¿Cómo les ayudan los dibujos y los diagramas a encontrar la solución del problema?” // “How are drawings and diagrams useful for finding the solution to the problem?” (You can count the dots in the drawing. The diagram can be used to count by 10.)

## Activity 2: Salpicón de multiplicación (20 minutes)

### Narrative

The purpose of this activity is for students to practice solving multiplication problems in which the unknown amount can be the number of groups, the number in each group, or the total. The first three problems have the unknown in each of those locations. The sequence of these problems, the context, and the use of the same factors and product encourages students to use a known fact to find the unknown factor in the “how many in each group” problem. Students will make the connection between this problem type and division in a future unit. Students are able to choose the representation they use to represent and solve the problems.

MLR8 Discussion Supports. Monitor and clarify any questions about the context. As students look over the problems, ask, “¿Hay palabras que no conocen o palabras sobre las que tienen preguntas?” // “Are there any words that are unfamiliar or that you have questions about?”

### Launch

• Groups of 2
• “Tómense un minuto para examinar estos problemas. ¿Qué representaciones o estrategias pueden ayudarles a resolver estos problemas?” // “Take a minute to look over these problems. What representations or strategies might be helpful to you as you solve these problems?”
• 1 minute: quiet think time
• Share and record responses.

### Activity

• “Trabajen con su compañero para resolver cada problema” // “Work with your partner to solve each problem.”
• 8-10 minutes: partner work time
• Circulate and consider the following questions to focus students on the structure of the situations:
• “¿Qué información de la situación no conocemos?” // “What information is missing in the situation?”
• “¿Cómo podrían representar esta situación?” // “How could you represent this situation?”

### Student Facing

Resuelve cada problema. Explica o muestra tu razonamiento.

1. Clare tiene 16 calcetines. Los pone en grupos de 2. ¿Cuántos grupos puede armar?
2. Diego tiene 8 grupos de calcetines. Cada grupo tiene 2 calcetines. ¿Cuántos calcetines tiene Diego?
3. Andre tiene 16 calcetines. Los pone en 8 grupos del mismo tamaño. ¿Cuántos calcetines hay en cada grupo?
4. En la tienda hay 9 cajas. En cada caja hay 5 camisas. ¿Cuántas camisas hay en total?
5. En la repisa de una tienda hay 80 suéteres organizados en pilas. Hay 8 suéteres en cada pila. ¿Cuántas pilas de suéteres hay en la repisa?

### Activity Synthesis

• Share student work for each problem and ask students to explain their reasoning. Be sure to share a variety of strategies and representations.
• As students share, consider asking the class:
• “¿Por qué esta estrategia tiene sentido?” // “Why does this strategy make sense?”
• “¿Por qué esta representación tiene sentido?” // “Why does this representation make sense?”
• “¿Alguien resolvió este problema de otra forma?” // “Did anyone solve this problem in a different way?”
• “¿En qué se parecen las representaciones y las estrategias que estamos usando para resolver estos problemas?” // “What do you notice is the same about the representations and strategies that we are using to solve these problems?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy resolvimos problemas de multiplicación usando la estrategia o representación que quisimos” // “Today we solved multiplication problems using any strategy or representation that we wanted.”

“¿Qué estrategia o representación les parece que ayuda más a resolver estos tipos de problemas? ¿Por qué?” // “What strategy or representation do you find most helpful when you are solving these types of problems? Why?” (I like to draw equal groups so I can see how many groups there are and how many are in each group. I think a diagram is nice to draw because I don’t have to draw all the things, but I can still see the groups. I like to use an equation so I can see where the unknown number is.)

“Mencionen algunas cosas que sea importante recordar cuando están resolviendo problemas de multiplicación” // “What are the most important things to remember when you are solving multiplication problems?” (There are always groups that are the same size. You could be looking for the number of groups, how many things are in each group, or the total number of things in all the groups.)

## Student Section Summary

### Student Facing

En esta sección, aprendimos sobre grupos iguales. Hicimos dibujos y diagramas para representar situaciones en las que hay grupos iguales.

situación

Diego tiene 8 grupos de calcetines. Cada grupo de calcetines tiene 2 calcetines.

dibujo

diagrama

Escribimos expresiones y ecuaciones de multiplicación para representar grupos iguales.

expresión

$$8\times2$$

ecuación

$$8\times2=16$$

Aprendimos que los números que se multiplican se llaman factores y el número que es el resultado de la multiplicación se llama producto. En la ecuación $$8\times2=16$$, los números 8 y 2 son los factores y el producto es 16.