# Lesson 9

Simetría en acción (optional)

## Warm-up: Cuál es diferente: Figuras (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare attributes of two-dimensional figures with attention to the number of sides, symmetry, and presence of parallel and perpendicular lines. The activity enables the teacher to observe the attributes that students notice intuitively and hear the terminologies they feel comfortable using.

### Launch

• Groups of 2
• Display the image.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

### Activity Synthesis

• If no students mention parallel sides as an attribute to consider, ask them about it.
• Consider asking: “Encontremos al menos una razón por la que cada una es diferente” // “Let’s find at least one reason why each one doesn’t belong.”

## Activity 1: Antes y después (15 minutes)

### Narrative

In this activity, students are given the result of folding a shape along one or more lines of symmetry and asked to reason about the original shape. No lines of symmetry are specified, so students must consider all sides of a folded shape as a possible line of symmetry and visualize the missing half accordingly.

The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-shapes on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original shape. Provide access to the materials and tools they might need.

During the activity synthesis, discuss the different ways students approach the second question. Consider preparing cutouts of shapes A–F to facilitate the discussion. (The shapes are provided in the blackline master.)

Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for the task. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Organization, Attention

### Required Materials

Materials to Gather

Materials to Copy

• Before and After

### Launch

• Groups of 2
• Give a ruler or a straightedge to each student.

### Activity

• 5 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for the different strategies students use to identify the original shape of the half-shapes (as noted in the activity narrative).

### Student Facing

1. Mai tiene una hoja de papel. Ella puede obtener dos figuras diferentes al doblarla a lo largo de una línea de simetría. ¿Qué forma tiene la hoja de papel antes de ser doblada?

2. Diego dobló una hoja de papel una vez a lo largo de una línea de simetría y obtuvo este triángulo rectángulo.

¿Qué formas pudo tener la hoja de papel antes de ser doblada? Explica o muestra cómo lo sabes.

### Activity Synthesis

• Invite students to share their responses and strategies.
• When discussing the second question, ask students why B, C, and E are not possible shapes of the original piece of paper even though there’s a line that breaks each figure into two right triangles that match the given triangle. (B and E have no line symmetry. C has lines of symmetry but not diagonally from corner to corner. If you fold there, the triangles would not be on top of one another.)

## Activity 2: Antes y después, edición perímetro (20 minutes)

### Narrative

Previously, students reason about line-symmetric figures that have been folded once along a line of symmetry. In this activity, they encounter figures that have been folded more than once, each time along a line of symmetry, and reason about the perimeter of the original figure. They think about how a given set of expressions could represent the original perimeter of a twice-folded figure, looking for and making use of structure (MP7) as they do so.

MLR8 Discussion Supports. Students who are working toward verbal output may benefit from access to mini-whiteboards, sticky notes, or spare paper to write down and show their responses to their partner.

### Required Materials

Materials to Gather

### Launch

• Groups of 2–4
• Give a ruler or a straightedge to each student.

### Activity

• 3–4 minutes: independent work time for the first set of questions
• Pause for a brief class discussion on possible shapes of the original piece of paper and possible expressions for its perimeter.
• 5 minutes: independent work time for the second question
• 2–3 minutes: group discussion
• Monitor for students who use the drawings or expressions from the first question to help them reason about the second question. Select them to share during synthesis.

### Student Facing

1. Jada dobló una hoja de papel a lo largo de una línea de simetría y obtuvo este rectángulo.

1. ¿Cómo pudo verse el papel antes de ser doblado? Haz uno o más dibujos.
2. Escribe una expresión para el perímetro del papel que no está doblado.
2. Kiran dobló una hoja de papel dos veces, cada vez a lo largo de una línea de simetría, y obtuvo el mismo rectángulo que Jada.

Muestra que cada expresión podría representar el perímetro del papel que Kiran dobló.

1. $$(4 \times 182) + (4 \times 105)$$
2. $$(2 \times 182) + (8 \times 105)$$
3. $$(8 \times 182) + (2 \times 105)$$

### Activity Synthesis

• Select students to share their responses and reasoning to the second question.
• Consider asking: “Vimos tres hojas distintas que corresponden a los distintos perímetros, ¿se les ocurre otra forma para la hoja de papel original?” // “Aside from the three figures whose perimeters are represented here, are there other possible figures that the original piece of paper could have?” (No)
• “¿Cómo lo saben?” // “How do you know?” (The folded rectangle has two pairs of sides of the same length. There are only three possible pairs of lines of symmetry: both along the 182 mm side, both along the 105 mm side, and once along each 182 and 105 mm side. All three are already represented by the given expressions.)
• “Si las distintas figuras originales se pueden doblar y formar la misma figura, ¿quiere decir esto que las figuras originales tienen el mismo perímetro?” // “If different original figures can be folded into the same figure, does that mean the original figures have the same perimeter?” (No)

## Lesson Synthesis

### Lesson Synthesis

“Hoy practicamos cómo visualizar figuras que han sido dobladas a lo largo de una línea de simetría y razonamos sobre el perímetro de las figuras originales” // “Today we practiced visualizing shapes that have been folded along a line of symmetry and reasoning about the perimeter of the original shapes.“

Display:

“Supongamos que este triángulo rectángulo fue el resultado de doblar una figura una vez a lo largo de una línea de simetría. ¿Qué estrategias podemos usar para decidir qué figuras posibles se tenían antes de doblar?” // “Suppose this right triangle is a result of folding once along a line of symmetry. What strategies could we use to determine the possible shapes before they were folded?” (Reflect the triangle along each of the sides—mentally, using tracing paper, or cutting out two copies of the triangle and arranging them so they mirror each other.)

“Para encontrar el perímetro de la figura original, ¿podemos simplemente duplicar el perímetro de la figura doblada? ¿Por qué sí o por qué no?” // “To find the perimeter of the original shape, could we just double the perimeter of the folded shape? Why or why not?” (No, because there is one side—along the folding line—that is not part of the perimeter of the original shape.)

“¿Cuáles podrían ser los perímetros de las figuras originales que se doblaron para obtener este triángulo?” //  “What could be the perimeters of the original shapes that fold into this triangle?” (The perimeter varies depending on line of symmetry or line of folding:

• If folded along the longest side: it will be 28, or $$(2 \times 8) + (2 \times 6)$$.
• If folded along the side that is 8 units long, it will be 32, or $$(2 \times 6) + (2 \times 10)$$.
• If folded along the shortest side, it will be 36, or $$(2 \times 8) + (2 \times 10)$$.)