# Lesson 10

Maneras de encontrar medidas de ángulos (optional)

## Warm-up: Cuántos ves: Simetrías de una estrella (10 minutes)

### Narrative

This warm-up encourages students to look for and make use of structure in an image to identify the lines of symmetry it has (MP7). Students could try to find all the segments or angles that are the same size as a way to identify lines of symmetry, but keeping track of all the pieces can be rather impractical. Instead, students could pay attention to the composition of the image— a five-sided figure on the outside and a five-point star on the inside, where all of the five parts are the same—and use that insight to determine the number of lines of symmetry.

### Launch

• Groups of 2
• “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
• Display the image.
• 1 minute: quiet think time

### Activity

• Display the image.
• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

### Student Facing

¿Cuántas líneas de simetría ves? ¿Cómo lo sabes?, ¿qué ves?

### Activity Synthesis

• Invite students to share how they identified the lines of symmetry. Display the original image for students to mark up and use in their explanation.
• Some students may have double-counted and say that there are 10 lines of symmetry. Invite others to respond to that claim.
• “¿Alguien puede expresar con otras palabras la forma en la que ______ vio las líneas de simetría?” // “Who can restate the way _____ saw the lines of symmetry in different words?”
• “¿Alguien vio las líneas de simetría de la misma manera, pero lo explicaría de otra forma?” // “Did anyone see the lines of symmetry the same way but would explain it differently?”
• “¿Alguien quiere compartir otra observación sobre la manera en la que _____ vio las líneas de simetría?” // “Does anyone want to add an observation to the way _____ saw the lines of symmetry?”

## Activity 1: Antes y después, edición ángulo (25 minutes)

### Narrative

This activity serves several goals. Students continue to practice visualizing and drawing a complete shape given a line of symmetry and one half of the shape. As they do so, they practice drawing angles of certain measurements. Students also use symmetry to reason about unknown angle measurements in two-dimensional figures.

For the drawing portion of the activity, assign a different shape for each group member to start with and ask students to draw as precisely as possible (MP6). Provide access to protractors and patty paper (MP5). Most students are likely to find Andre’s shape most challenging to draw. Differentiate the starting drawing for each student as needed.

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.

### Required Materials

Materials to Gather

### Launch

• Read the opening paragraph of the activity statement as a class. Display the four images. Clarify the context as needed before students begin the activity.
• Groups of 4
• Ask each group member to start with the drawing for a different student (one member starts with Noah’s, another with Clare’s, and so on), but try to complete at least 2 of the 4 drawings.
• Give a protractor and a ruler to each student.
• Provide access to patty paper, scrap paper, and scissors.

### Activity

• 7–8 minutes: independent work time
• 3–4 minutes: group discussion
• Monitor for the different ways students use tools to draw precisely.
• Pause for a brief class discussion. Invite students to share their completed drawings before students proceed to the second question.

### Student Facing

Noah, Clare, Andre y Elena tienen, cada uno, una hoja de papel que tiene una línea de simetría. Cuando doblaron su papel a lo largo de la línea de simetría, todos obtuvieron la misma figura. La línea punteada representa la línea por donde se dobla.

1. Dibuja la forma del papel que cada estudiante recibió, antes de ser doblado. Sé tan preciso como puedas.
2. Sin medir, encuentra la medida de todos los ángulos de la figura (la forma del papel sin dobleces) que dibujaste.

### Advancing Student Thinking

If students identify some but not all of the interior angles of the unfolded paper, consider asking:

• “¿Cómo encontraste las medidas de los ángulos del papel que no está doblado sin medirlas?” // “How did you find the angle measurements of the unfolded paper without measuring?”
• “¿Encontraste todos los ángulos dentro de la forma del papel que no está doblado? ¿Cómo lo sabes?” // “Have you found all the angles within the shape of the unfolded paper? How do you know?”
• “¿Cómo puedes usar los ángulos de las formas originales para encontrar los nuevos ángulos de la forma del papel que no está doblado?” // “How can you use the angles in the original shapes to find the new angles in the shape of the unfolded paper?”

### Activity Synthesis

• Display the drawings that students completed and select students to share how they found the angles inside each shape.
• Highlight that lines of symmetry can be used to identify angles that have the same size as a given angle, or angles that are twice the size of a given angle.

## Activity 2: Pez con ángulos (15 minutes)

### Narrative

In this activity, students apply their understanding of symmetry and knowledge of angles to find angle measurements in a more complex line-symmetric figure. Students use what they know about the measurement of a straight angle and the measurement of a full rotation of a ray around a point to find the unknown angles. As needed, review the measurements of benchmark angles ($$90^\circ$$$$180^\circ$$, $$360^\circ$$) before the activity.

Representation: Internalize Comprehension. Activate or supply background knowledge. Ask, “¿Qué puede ser útil para encontrar la medida de un ángulo desconocido?” // “What might be useful when finding the size of a missing angle?” Prompt students to think beyond a protractor. Then, provide students with colored pencils, and invite them to shade angles that add up to 180 in one color and angles that add up to 360 in another.
Supports accessibility for: Conceptual Processing, Memory, Attention

### Launch

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time
• 1 minute: discuss observations and questions

### Activity

• 5 minutes: independent work time
• 3 minutes: partner discussion

### Student Facing

Este es un diagrama de un pez de origami que tiene una línea de simetría.

1. Dibuja la línea de simetría.
2. Sin medir, encuentra la medida de los ángulos marcados de la $$a$$ a la $$f$$. Prepárate para explicar tu razonamiento.

### Advancing Student Thinking

Students may find the measurement of some angles using the figure's symmetry (angles b, c, and f), but not of angles that require reasoning about a full turn or half turn around a point (angles a, d, and e). Consider asking:

• “¿Qué hiciste para encontrar la medida de estos ángulos? ¿Funcionará esa estrategia para los otros ángulos? ¿Por qué sí o por qué no?” // “What did you do to find the measurement of these angles? Will that strategy work for the other angles? Why or why not?”
• “¿Cuáles ángulos comparten el mismo vértice y los mismos rayos con el ángulo desconocido? ¿Cómo podrías usarlos para encontrar el ángulo desconocido?” // “What angles share the same vertex and rays as the angle that is unknown? How could you use them to find the the unknown angle?”
• “¿Qué sabes sobre la medida de (un giro completo o medio giro) alrededor de un punto? ¿Cómo te puede ayudar eso a encontrar un ángulo desconocido?” // “What do you know about the measure of (a full turn or a half turn) around a point? How could that help you find an unknown angle?”

### Activity Synthesis

• Invite students to share their responses and reasoning.
• Highlight that:
• Angles $$a$$ and $$d$$ are each part of a straight angle, so each can be found by subtracting the adjacent angle measure from 180.
• Angle $$e$$ and the two angles one either side of it make a full turn or $$360^\circ$$.
• Consider asking: “¿Qué otros ángulos pueden encontrar dentro o alrededor del diagrama del pez?” // “Which other angle measurements in or around the fish diagram can you find?” If time permits, encourage students to find as many as they can.

## Lesson Synthesis

### Lesson Synthesis

“Hoy vimos que las líneas de simetría pueden ser útiles para encontrar medidas de ángulo desconocidas” // “Today we saw that lines of symmetry can be handy for finding unknown angle measurements.”

Display:

“Estas son dos figuras en forma de V. Una es simétrica con respecto a una línea y la otra no. En cada diagrama, se conoce una medida de ángulo” // “Here are two V-shaped figures—one has line symmetry and the other does not. In each diagram, one angle measurement is known.”

“¿Pueden encontrar la medida de cada ángulo que está marcado con un signo de interrogación? ¿Por qué sí o por qué no?” // “Can you find the size of each angle marked with a question mark? Why or why not?” (Yes for the first one, but no for the second. The first one has a vertical line of symmetry, so the two unknown angles are the same size. In the second figure, the two angles are different sizes.)

“¿Cómo encontrarían las medidas de los ángulos de la primera figura?” // “How would you find the angle measurements in the first figure?” (The two unknown angles plus $$300^\circ$$ make $$360^\circ$$. Since the two angles are the same size, each one is $$30^\circ$$.)
“¿Qué sabemos sobre los dos ángulos de la segunda figura?” // “What do we know about the two angles in the second figure?” (They also add up to $$60^\circ$$. One is less than $$30^\circ$$ and the other is greater than $$30^\circ$$.)