# Lesson 10

Resolvamos problemas sobre perímetros y áreas

## Warm-up: Verdadero o falso: Dividamos en partes (10 minutes)

### Narrative

The purpose of this True or False is to elicit strategies and understandings students have for dividing within 100. It also prompts them to rely on properties of operations and familiar division facts to facilitate division.

When students think about how to decompose larger dividends using facts about 10 to make the division easier, they look for and make use of structure (MP7).

### Launch

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar cómo razonaste.

• $$60 \div 6 = 10$$
• $$72 \div 6 = (60 \div 6) + (12 \div 6)$$
• $$78 \div 6 = (60 \div 10) + (18 \div 6)$$
• $$96 \div 8 = (80 \div 8)-(16 \div 8)$$

### Activity Synthesis

• “¿Cómo pueden explicar su respuesta sin encontrar el valor de ambos lados?” // “How can you explain your answer without finding the value of both sides?”

## Activity 1: Encerremos la huerta (15 minutes)

### Narrative

The purpose of this activity is for students to differentiate methods for finding perimeter from those for finding area. While addition and multiplication are both involved in various ways, students need to understand the problem situation and think about whether the operations performed will provide the desired information. As in earlier problems, students can find perimeter in various ways. The emphasis should be on how understanding the problem situation and the information given should inform the solution method.

When students analyze claims about how to use addition and multiplication to find the perimeter of a rectangle they construct viable arguments (MP3).

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “Who do you agree with? Explain or show your reasoning.” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

### Launch

• Groups of 2 and 4
• Display the situation: “Andre quiere saber cuánta cuerda se necesita para encerrar la nueva huerta rectangular de la escuela. El largo de la huerta es 30 pies. El ancho de la huerta es 8 pies” // “Andre wants to know how much rope is needed to rope off the new rectangular school garden. The length of the garden is 30 feet. The width of the garden is 8 feet.”
• “Tómense un minuto para leer esta situación sobre Andre y la huerta de la escuela. ¿Cómo averiguarían cuántos pies de cuerda se necesitan?” // “Take a minute to read this situation about Andre and the school garden. How would you figure out how many feet of rope is needed?”
• 1 minute: quiet think time

### Activity

• “Con su compañero, examinen las ideas de Clare y de Diego, y respondan las preguntas” // “With your partner, look at Clare and Diego’s thinking and answer the questions.”
• 3–5 minutes: partner work time

### Student Facing

Andre quiere saber cuánta cuerda se necesita para encerrar la nueva huerta rectangular de la escuela. El largo de la huerta es 30 pies. El ancho de la huerta es 8 pies.

• Clare dice que puede usar la multiplicación para encontrar la longitud de cuerda que Andre necesita.
• Diego dice que puede usar la suma para encontrar la longitud de cuerda que Andre necesita.

¿Con quién estás de acuerdo? Explica o muestra cómo razonaste.

### Activity Synthesis

• Invite students to share who they agreed with and why. Record their reasoning for all to see.
• “¿Cómo supieron que al multiplicar $$8 \times 30$$ no se obtendría la cantidad total necesaria de cuerda?” // “How did you know that multiplying $$8 \times 30$$ would not give you the total amount of rope needed?” (Multiplying $$8 \times 30$$ would give us the area of the rectangle, not the distance around the rectangle.)
• “¿Cómo supieron que la estrategia de Diego iba a funcionar?” // “How did you know that Diego’s strategy would work?” (Diego is finding the perimeter by adding the side lengths of the garden.)
• “¿En qué casos puede ser apropiado usar la multiplicación para encontrar el perímetro?” // “When might it be appropriate to use multiplication to find perimeter?” (When there are two or more sides that are the same length. When we know half of the perimeter, we can double that number to find the whole perimeter.)

## Activity 2: Falta de información: Una huerta y un patio de recreo (20 minutes)

### Narrative

This info gap activity gives students a chance to understand that given the area and one side length of a rectangle, the perimeter can be found, and that given the perimeter and one side length of a rectangle, the area can be found. In both cases, students need to find the missing side length to solve the problem. There are several ways students might find the missing side length and then the perimeter or area once the missing side length is known.

This activity uses MLR4 Information Gap.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is an image of the cards for reference:

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were needed to solve the problem. Display the sentence frame, “La próxima vez que use el área de un rectángulo para encontrar el perímetro, buscaré . . .” // “The next time I use the area of a rectangle to find the perimeter, I will look for . . . .“
Supports accessibility for: Memory, Visual-Spatial Processing

### Required Materials

Materials to Copy

• Info Gap: A Garden and a Playground, Spanish

### Required Preparation

• Each group of 2 will need a copy of the 2 data and problem card sets. Keep set 1 separate from set 2.

### Launch

• Groups of 2

MLR4 Information Gap

• Display the task statement, which shows a diagram of the info gap structure.
• 1 minute: quiet think time
• Read the steps of the routine aloud.
• “Les voy a dar una tarjeta de problema o una tarjeta de datos. Lean su tarjeta en silencio. No se la muestren ni se la lean a su compañero” // “I will give you either a problem card or a data card. Silently read your card. Do not read or show your card to your partner.”
• Distribute cards.
• 1–2 minutes: quiet think time
• Remind students that after the person with the problem card asks for a piece of information the person with the data card should respond with “¿Por qué necesitas saber ________?” // “Why do you need to know (restate the information requested)?”

### Activity

• 3–5 minutes: partner work time
• After students solve the first problem, distribute the next set of cards. Students switch roles and repeat the process with Problem Card 2 and Data Card 2.

### Student Facing

Tu profesor te dará una tarjeta de problema o una tarjeta de datos. No se la muestres ni se la leas a tu compañero.

Haz una pausa aquí para que tu profesor pueda revisar tu trabajo.

Pídele al profesor un nuevo grupo de tarjetas. Intercambia roles con tu compañero y repite la actividad.

### Activity Synthesis

• Share and record responses.
• Display the info gap cards.
• “¿Qué necesitan saber para encontrar el perímetro o el área de estos rectángulos?” // “What do you need to know in order to find the perimeter or area of these rectangles?” (The missing side length.)
• “¿Cómo usaron el área y un lado conocido para encontrar la longitud de los otros lados?” // “How did you use the area and one side you knew to find the missing side length?”
• “¿Cómo usaron el perímetro y un lado conocido para encontrar la longitud de los otros lados?” // “How did you use the perimeter and one side to find the missing side length?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy vimos algunos problemas en los que se nos pedía pensar sobre área y perímetro juntos” // “Today we saw some problems that asked us to think about area and perimeter together.”

“¿En qué se parecen el área y el perímetro?” // “How are perimeter and area alike?” (They are both measurements of shapes. We need side lengths to find both the area and perimeter of rectangles.)

“¿En qué son diferentes?” // “How are they different?” (Perimeter is about distance, so it is measured in length units. Area is about the amount of space within a shape, so it is measured in square units.)

If the different types of units used to measure area and perimeter don’t come up, ask, “¿En qué son diferentes las unidades que usamos para medir el área y las que usamos para medir el perímetro? ¿Por qué?” // “How are the units we use to measure area and perimeter different? Why?”

Consider recording students’ ideas in two columns labeled “parecidas” // “alike” and “diferentes” // “different.”