# Lesson 8

Encontremos el perímetro

## Warm-up: Conversación numérica: Dividendo que disminuye (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for dividing within 100. These understandings help students develop fluency and are helpful as students use division to solve problems involving perimeter.

### 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.

• $$90 \div 9$$
• $$81 \div 9$$
• $$45 \div 9$$
• $$54 \div 9$$

### Activity Synthesis

• “Al encontrar el valor de las otras expresiones, ¿cómo puede ayudarlos saber cuánto es $$90 \div 9$$?” // “How could knowing $$90 \div 9$$ help you find the value of the other expressions?” (Once I knew $$90\div9$$ I was able to take away a group of 9 to find $$81\div9$$. I was able to find $$45\div9$$ by splitting the value of $$90 \div 9$$ because 45 is half of 90.)

## Activity 1: Maneras de encontrar el perímetro (20 minutes)

### Narrative

The purpose of this activity is for students to practice finding the perimeter of shapes that have labeled side lengths. The synthesis focuses on methods students have for efficiently finding the perimeter of shapes with some or all side lengths having equal length. As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

Monitor and select students who find the perimeter of the hexagon by:

• adding the individual side lengths around the shape
• adding the two 8–inch side lengths together and the four 4–inch side lengths together and then adding those sums together
• multiplying like side lengths, then adding, such as $$2 \times 8$$ for the long sides and $$4 \times 4$$ for the short sides and then adding those products together
• using symmetry to split the shape in half horizontally and adding $$4 + 8 + 4 = 16$$ for the top half of the shape and then doubling that for the sides on the bottom half of the shape
MLR8 Discussion Supports. Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (Students may notice: One rectangle has numbers on the sides. One rectangle has tick marks on the sides. The rectangles are the same size. Students may wonder: Why are the sides of the rectangles marked differently? Could we find the distance around the rectangle with the numbers on the sides?)
• 1 minute: quiet think time
• 1 minute: partner discussion
• Share and record responses.

### Activity

• “Encuentren el perímetro de cada figura con su compañero” // “Work with your partner to find the perimeter of each shape.”
• 5–7 minutes: partner work time
• As students work, consider asking:
• “¿Cómo nos ayuda tener lados de la misma longitud a encontrar el perímetro?” // “How does having sides of the same length help us find the perimeter?”
• “¿Pueden multiplicar para encontrar el perímetro?” // “Can you multiply to find the perimeter?”

### Student Facing

¿Qué observas? ¿Qué te preguntas?

Encuentra el perímetro de cada figura. Explica o muestra tu razonamiento.

1.

2.

3.
4.

5.

### Activity Synthesis

• Ask previously identified students to share their strategies for finding the perimeter of the hexagon. Arrange the presentations in the order listed in the activity narrative.
• Give students a chance to ask questions about each strategy as it is shared.
• “¿Cómo nos ayuda tener lados de la misma longitud a encontrar el perímetro?” // “How does having sides of the same length help us find the perimeter?”
• “¿Por qué esta estrategia funciona con esta figura?” // “Why does this strategy work with this shape?”
• “¿Alguien encontró el perímetro de esta figura de otra manera?” // “Did anyone else find the perimeter of this shape in a different way?”
• “¿Fue más fácil encontrar el perímetro de unas figuras que de otras en esta actividad? ¿Por qué?” // “Was it easier to find the perimeter of some shapes in this activity than others? Why?” (Yes, some of the shapes had several sides that are the same length, so we could multiply. In a rectangle, we can add two sides and then double the result to find the whole perimeter.)

## Activity 2: Falta algo (15 minutes)

### Narrative

The purpose of this activity is for students to find the perimeter of shapes when some of the side lengths are not given. Students use their knowledge of shapes to reason about the length of the missing sides before they find the perimeter of the shape (MP7).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most useful to solve the problem. Display the sentence frame: “La próxima vez que encuentre el perímetro de una figura en la que algunas de las longitudes de los lados no estén dadas, prestaré atención a . . .” // “The next time I find the perimeter of a shape where some side lengths are not given, I will pay attention to . . . .“
Supports accessibility for: Memory, Visual-Spatial Processing

### Launch

• Groups of 2
• Display the rectangle from the first problem.
• “Encuentren el perímetro de este rectángulo” // “Find the perimeter of this rectangle.“
• 1–2 minutes: independent work time
• “Discutan con su compañero cómo encontraron el perímetro de este rectángulo a pesar de que algunas de las longitudes de sus lados no estaban marcadas” // “Discuss with your partner how you found the perimeter of this rectangle even though some of the side lengths were not labeled.” (Since the shape is a rectangle, we know opposite sides of a rectangle are the same length.)
• 1 minute: partner discussion
• Share and record responses.
• Give each group tools for creating a visual display.

### Activity

• “Encuentren el perímetro de las otras dos figuras con su compañero. Asegúrense de escribir su razonamiento para compartirlo con toda la clase” // “Work with your partner to find the perimeter of the other two shapes. Be sure to record your reasoning to share with the class.”
• 6–8 minutes: partner work time
• Consider asking: “¿Cómo supieron cuál era la longitud de ese lado?” // “How did you know the length of that side?”

MLR7 Compare and Connect

• “Creen una presentación visual que muestre cómo pensaron en el segundo problema. Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about the second problem. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
• 3–5 minutes: partner work time
• 5 minutes: gallery walk

### Student Facing

1. Encuentra el perímetro de este rectángulo. Explica o muestra tu razonamiento.

2. Todos los lados cortos de esta figura tienen la misma longitud y todos los ángulos son ángulos rectos. Encuentra el perímetro. Explica o muestra tu razonamiento.

3. Todos los lados del octágono tienen la misma longitud. Encuentra el perímetro. Explica o muestra tu razonamiento.

### Activity Synthesis

• “Tuvimos que encontrar muchas longitudes desconocidas de lados de esta figura antes de poder encontrar el perímetro” // “We had to find a lot of missing side lengths in this shape before we could find the perimeter.”
• “Cuando fueron a ver las presentaciones de los demás, ¿qué observaron acerca de cómo encontraron ellos las longitudes de lado desconocidas?” // “As you visited the displays, what did you notice about how others found the missing side lengths?” (I noticed some groups counted the number of short sides and multiplied by 40. I noticed some put the short side lengths into smaller groups before finding their combined lengths.)
• “¿Alguien encontró las longitudes de lado desconocidas de una manera diferente a como ustedes y su compañero lo hicieron?” // “Did anyone find the missing side lengths in a different way than you and your partner?”
• “¿Alguien encontró el perímetro de una manera diferente a como ustedes y su compañero lo hicieron?” // “Did anyone find the perimeter in a different way than you and your partner?”

## Lesson Synthesis

### Lesson Synthesis

“Cuando encuentran el perímetro de una figura, siempre se pueden sumar las longitudes de los lados de una en una. ¿Qué otros métodos conocen para encontrar el perímetro de las figuras?” // “When you are finding the perimeter of a shape, you can always add the lengths of the sides one at a time. What other methods do you have for finding the perimeter of shapes?” (We can look for side lengths that are the same and group them together. In a square, we can multiply one side length by 4 since they are all the same length. In a rectangle, we can add a long side to a short side and then double that for the whole perimeter.)

Display a rhombus with side lengths that are the same length, but only one side labeled 7 in, such as:

“¿Cómo podemos encontrar el perímetro de este rombo si solo está marcado un lado?” // “How can we find the perimeter of this rhombus if only one side is labeled?” (We know that a rhombus has four equal sides, so we can find $$4 \times 7$$, which is 28.)