# Lesson 4

Simetría de figuras (parte 1)

## Warm-up: Observa y pregúntate: Veo doble (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea of mirror images that match exactly, which will be useful when students generate a definition for symmetry in a later activity. Students may notice and wonder about possible real-life objects the block structure represents, but focus the discussion on describing how the two halves are reflections of each other.

Elicit the term “symmetry” or language that describes attributes of line-symmetric figures.

### 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

### Activity

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Activity Synthesis

• Highlight the language students used to describe the symmetry in the design such as:
• mirror images
• the same parts on both sides of the image
• “¿Cómo podemos comprobar si la parte de la izquierda y la parte de la derecha son iguales?” // “How might we check to see if the part on the left and that on the right are the same?” (We could place each half of the image on top of the other to see if they match.)

## Activity 1: Coincidencias perfectas (20 minutes)

### Narrative

This activity uses the idea of folding to introduce students to line symmetry. Students analyze examples of figures that have a line of symmetry and those that don’t, and use their observations to formulate a definition of line of symmetry, which they then refine with their peers (MP6). Students later identify and draw lines of symmetry in various figures.

Some students may wish to use tools to help them define and find lines of symmetry. Provide access to patty paper, rulers, protractors, scissors, and copies of the figures (provided in the blackline master), if requested.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

Representation: Develop Language and Symbols. Support student understanding of the first question by acting out Lin’s process with a class set of cutout figures. Invite students to examine and manipulate these figures as they work on this question.
Supports accessibility for: Visual Spatial Processing, Attention

### Required Materials

Materials to Gather

Materials to Copy

• Perfect Matches

### Required Preparation

• Make copies of the set of figures in the second question available for cutting and for demonstration during the lesson synthesis.

### Launch

• Groups of 2–4
• Give a ruler or straightedge to each student.
• Provide access to patty paper, protractors, scissors, and copies of the shapes in the second question.
• Display or sketch these parallelograms.

“¿En qué se parecen las dos figuras? ¿En qué son diferentes?” // “How are the two figures alike? How are they different?”

• 1 minute: quiet think time
• Discuss responses. (Students may say:
• Both parallelograms show a dashed line through opposite corners.
• The dashed line creates two identical triangles.
• The triangles in the first shape would match up exactly if the shape is folded along the line, but not so with those in the second shape.)

### Activity

• 3 minutes: independent work time on the first question

MLR1 Stronger and Clearer Each Time

• “Compartan con su compañero cómo completaron la frase ‘una línea de simetría es . . .’. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your response to ‘a line of symmetry is…’ with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 2 minutes: structured partner discussion
• Repeat with 2–3 different partners.
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time
• 3 minutes: independent work time on the remaining questions

### Student Facing

1. Lin tenía unos pedazos de papel con formas diferentes. Ella dobló cada pedazo de papel una vez y formó dos partes más pequeñas.

Después, clasificó los papeles en dos categorías según las líneas de los dobleces.

Estudia las figuras de cada categoría. ¿Qué crees que significa una línea de simetría?

Completa esta frase:

Una línea de simetría es . . .

2. ¿Las siguientes figuras tienen una línea de simetría? Si la tienen, dibuja la línea. Si no la tienen, explica cómo lo sabes.

3. ¿Hay figuras que tienen más de una línea de simetría? Si piensas que sí, dibuja todas las líneas de simetría.

### Student Response

If students identify a line of symmetry that would not result in two identical halves of the original figure, consider asking:

• “¿Cómo decidiste dónde dibujar tu línea? ¿Cómo sabes que es una línea de simetría?” // “How did you decide where to draw your line? How do you know that it's a line of symmetry?”
• “¿Cómo puedes usar una herramienta para mostrar que tu línea es una línea de simetría?” // “How can you use a tool to prove that your line is a line of symmetry?”

### Activity Synthesis

• Ask groups to share their definitions of line of symmetry and record key phrases. Invite the class to comment.
• Discuss responses to the last two questions. Ask students to share the lines of symmetry they drew.
• “¿Cómo podemos comprobar que la línea que dibujamos es una línea de simetría? ¿Cómo podemos saber si las dos mitades que parecen ser idénticas en realidad lo son?” // “How might we check that the line we drew is a line of symmetry? How can we tell if the two halves that appear to be identical are indeed the same?” (Some ways:
• Cut out the figure and fold it along the line we drew.
• Trace one half of the figure on patty paper and turn it over to see if it matches the other half.
• Measure the segments and angles.)
• “Si una figura se puede doblar por una línea para formar dos partes que son un reflejo de espejo la una de la otra y que coinciden exactamente, decimos que la figura es simétrica o que es simétrica con respecto a una línea” // “If a figure can be folded along a line to create two parts that are a mirror reflection of one another and would match up exactly, we say that the figure has symmetry or has line symmetry.”
• “La línea que divide la figura en dos partes que son un reflejo la una de la otra y que coinciden exactamente se llama una línea de simetría” // “The line that splits a figure into two mirroring and matching parts is called a line of symmetry.”

## Activity 2: En búsqueda de la simetría (15 minutes)

### Narrative

In this activity, students practice identifying two-dimensional figures with line symmetry. They sort a set of figures based on the number of lines of symmetry that the figures have.

Continue to provide access to patty paper, rulers, and protractors. Students who use these tools to show that a shape has or does not have a line of symmetry use tools strategically (MP5). Consider allowing students to fold the cards, if needed.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “Observé _____, entonces agrupé . . .” // “I noticed _____ , so I matched . . .” Encourage students to challenge each other when they disagree.

### Required Materials

Materials to Gather

Materials to Copy

### Required Preparation

• Sort the shape cards from the previous lessons into three groups of 12 cards (A–L, M–X, and Y–JJ).

### Launch

• Groups of 2
• Give each group a set of 12 shape cards (A–L, M–X, or Y–JJ).

### Activity

• “Con un compañero, clasifiquen las figuras según el número de líneas de simetría que tienen” // “Work together with a partner to sort the figures by the number of lines of symmetry.”
• 5 minutes: group work time

### Student Facing

Tu profesor le dará a tu grupo unas tarjetas.

1. Clasifiquen las figuras de las tarjetas según el número de líneas de simetría que tienen.
0 líneas de simetría 1 línea de simetría 2 líneas de simetría 3 líneas de simetría

2. Busquen otro grupo que tenga las mismas tarjetas. Comparen cómo clasificaron las figuras. ¿Estuvieron de acuerdo en cómo clasificaron las figuras? Si no es así, discutan sobre los desacuerdos.

### Student Response

If students draw fewer or more lines of symmetry than the figure actually has, consider asking:

• “¿Cómo decidiste dónde dibujar tu línea o tus líneas?” // “How did you decide where to draw your line(s)?”
• “¿Cómo puedes comprobar que cada línea es una línea de simetría?” // “How can you prove that each line is a line of symmetry?”
• “¿Ya encontraste todas las líneas de simetría de esta figura? ¿Cómo lo sabes?” // “Have you found all the lines of symmetry for this figure? How do you know?”

### Activity Synthesis

• “¿Cómo decidieron si una figura tenía una línea de simetría? ¿Qué buscaron?” // “How did you decide if a figure had a line of symmetry? What did you look for?” (Look for matching parts on opposite sides of a figure—parts that have the same size and mirror each other.)
• “¿Supieron inmediatamente que algunas figuras no tenían líneas de simetría? ¿Qué característica de las figuras hizo que eso fuera evidente?” // “Were there figures that you could immediately tell had no lines of symmetry? What was it about the figures that made it clear?” (The parts are different from one another. No folding lines could make two parts mirror each other.)

## Activity 3: Solo sigue doblando [OPTIONAL] (10 minutes)

### Narrative

This optional activity gives students an opportunity to reason about lines of symmetry. Students see that some shapes can be folded in half again and again. Some students may notice that certain shapes, like a square, can be divided into two equal halves in perpetuity (MP8). Each new half created by a line of symmetry will continue to be line-symmetric.

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• 4–5 minutes: independent work time
• 1–2 minutes: group discussion

### Student Facing

Priya dobla hojas de papel de distintas formas por sus líneas de simetría. Ella dobla y dobla cada una hasta que la figura doblada ya no tiene más líneas de simetría.

1. ¿Cuántas veces puede doblar cada figura hasta que ya no puede doblar más?
2. ¿Qué observas acerca de cada figura doblada cuando ya no se puede doblar más?

### Activity Synthesis

• “Las figuras A, B y C tienen más de una línea de simetría. ¿La línea por la que empezamos a doblar influye en el número de veces que se puede doblar la figura hasta que ya no tenga líneas de simetría?” // “A, B, and C have more than one line of symmetry. Does the folding line we start with affect how many times the shape could be folded before there is no more line symmetry?” (In these shapes, no, the starting fold doesn’t determine how many folds can be made afterward.)
• 1 minute: quiet think time
• “¿Cómo pueden saber cuándo una figura doblada ya no tiene líneas de simetría?” // “How can you tell when a folded shape no longer has line symmetry?” (No segments are the same length and no angles are the same size.)
• “¿Qué ideas nuevas sobre las líneas de simetría aprendieron?” // “What new ideas about lines of symmetry did you learn?” (Sample responses:
• Folding a shape along a line of symmetry may create new shapes that also have lines of symmetry. These lines may create more shapes with symmetry.
• For some shapes, this process could go on even if we cannot physically fold the paper any longer.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy encontramos líneas de simetría de figuras planas” // “Today we found lines of symmetry in flat figures.”

Display the two parallelograms from the first activity.

“En ambas figuras, hay una línea que forma dos triángulos idénticos. ¿Por qué la primera figura es simétrica con respecto a la línea, pero la otra no?” // “In both figures, there’s a line that creates two identical triangles. Why does the first figure have line symmetry but the other doesn’t?“ (The triangles in the first parallelogram match each other exactly when folded along the line, which is not the case in the second parallelogram.)

“No todas las líneas que dividen una figura en dos mitades idénticas son líneas de simetría” // “Not all lines that divide a figure into two identical halves are lines of symmetry.”