# Lesson 7

Mismo perímetro, figuras diferentes

## Warm-up: Verdadero o falso: Sumas de cuatro números (10 minutes)

### Narrative

The purpose of this True or False is to elicit strategies and understandings students have for adding multi-digit numbers. It prompts students to rely on their understanding of the properties of operations and place value. The strategies used here will be helpful as students find the perimeter of shapes with repeated side lengths later in the lesson.

### Launch

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

### Activity

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

### Student Facing

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

• $$123 + 75 + 123 + 75 = 100 + 100 + 70 + 70 + 5 + 5 + 3 + 3$$
• $$123 + 75 + 123 + 75= (2 \times 123) + (2 \times 75)$$
• $$123 + 75 + 123 + 75 = 208 + 208$$
• $$123 + 75 + 123 + 75 = 246 + 150$$

### 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: Todo tipo de figuras (15 minutes)

### Narrative

The purpose of this activity is for students to understand that many different shapes can have the same perimeter. Students start to focus more specifically on shapes with repeated side lengths, so they can leverage the efficient addition strategies elicited in the warm-up (MP7).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches for finding the perimeter of one of the shapes. Ask, “¿Por qué se obtiene el mismo perímetro al usar cada método?” // “How did the same perimeter show up in each method?” and “¿Por qué las distintas estrategias nos llevan al mismo resultado?” // “Why did the different approaches lead to the same outcome?”
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, Visual-Spatial Processing

### Launch

• Groups of 2
• Display the shapes.
• “¿Cuál figura piensan que tiene el perímetro más largo y cuál tiene el más corto?” // “Which shape do you think has the longest perimeter and which has the shortest?” (I think shape J has the longest—it looks like a really long shape. I think D has the shortest. It is small.)
• 1–2 minutes: partner discussion
• Share responses.

### Activity

• “Con su compañero, encuentren el perímetro de 3 figuras” // “Work with your partner to find the perimeter of 3 shapes.”
• “Después, individualmente, encuentren al menos una figura que tenga el mismo perímetro que una de las figuras que escogieron” // “Then, work independently to find at least one shape that has the same perimeter as a shape you chose.”
• 5 minutes: partner work time
• 2–3 minutes: independent work time
• Monitor for students who use sides of the same length or symmetry of the shape to find the perimeter in efficient ways.

### Student Facing

1. Escoge 3 figuras y encuentra el perímetro de cada una. Explica o muestra tu razonamiento.
2. Encuentra una figura que tenga el mismo perímetro que una de las figuras que escogiste antes. Prepárate para explicar tu razonamiento.

### Student Response

If students count the individual units around each shape, consider asking:
• “¿Cómo encontraste el perímetro de esta figura?” // “How did you find the perimeter of this shape?”
• “¿Qué otras estrategias puedes usar para no tener que contar las unidades una por una?” // “What other strategies could you use so you wouldn’t have to count one unit at a time?”

### Activity Synthesis

• Invite students to share a variety of methods for finding the perimeter of the different shapes. Ask students who found the perimeter of shapes in efficient ways to share their reasoning.
• Consider asking: “¿Alguien encontró el perímetro de otra manera?” // “Did anyone find the perimeter in a different way?”
• “Las figuras A y G se ven muy diferentes, pero sus perímetros tienen la misma longitud. ¿Cómo es posible?” // “Shapes A and G look very different but have the same length for their perimeter. How could that be?” (The shapes may look different but the distance around each is the same number of units. Different shapes can have the same perimeter.)

## Activity 2: Dibuja una propia (20 minutes)

### Narrative

The purpose of this activity is for students to draw shapes with specific perimeters. Students may create any shape that uses horizontal and vertical lines. Since diagonal lines that connect the dots are not one length unit, students cannot find the perimeter of shapes that include diagonal sides. Encourage students to be creative in drawing their shapes to reinforce the idea that different shapes can have the same perimeter.

• Groups of 2

### Activity

• “Individualmente, dibujen dos figuras que tengan cada uno de los perímetros. Prepárense para explicar cómo dibujaron sus figuras” // “Work independently to draw two shapes that have each perimeter. Be prepared to explain how you drew your shapes.”
• 8–10 minutes: independent work time
• Monitor for students who:
• draw shapes other than rectangles or squares
• can explain their method for drawing shapes with a specific perimeter, such as drawing sides of the same length first or drawing sides one at a time around the shape
• “Compartan con su compañero las figuras que dibujaron. Piensen en qué se parecen y en qué son diferentes sus figuras. Luego, trabajen juntos en el último problema” // “Share the shapes you drew with your partner. Think about how your shapes are alike and how they are different. Afterwards, work on the last problem together.”
• 5 minutes: partner work time

### Student Facing

1. En cada caso, dibuja 2 figuras que tengan ese perímetro.

1. Con tu compañero, escoge una longitud. Después, dibuja una figura que tenga ese perímetro.

2. Compartan las figuras que dibujaron y discutan en qué se parecen y en qué son diferentes.

### Activity Synthesis

• Select previously identified students to share their strategies for drawing shapes for one of the perimeters in the first problem.
• “Para dibujar la figura que tuviera cierto perímetro, ¿qué perímetro escogieron con su compañero? ¿Por qué escogieron ese número?” // “What perimeter did you and your partner choose to draw a shape for? Why did you pick that number?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy aprendimos que figuras diferentes pueden tener el mismo perímetro” // “Today we learned that different shapes can have the same perimeter.”

“¿Cómo le explicarían a alguien que esto es posible?” // “How would you explain to someone how this is possible?” (The perimeter is the total length of all the sides of a shape, and there are different ways to add numbers to get the same sum.)

Consider using a string of interconnected paper clips to form different shapes. The shapes would have the same perimeter because the length of the string (or the number of paper clips) hasn’t changed.