# Lesson 8

Formas de encontrar la longitud desconocida (parte 2)

## Warm-up: Verdadero o falso: Ecuaciones con fracciones (10 minutes)

### Narrative

The purpose of this warm-up is to elicit strategies and understandings students have for adding, subtracting, and multiplying fractions and mixed numbers. The series of equations prompt students to use properties of operations (associative and commutative properties in particular) in their reasoning, which will be helpful when students solve geometric problems involving fractional lengths (MP7).

### 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 statement.

### Student Facing

Decide si la afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.

• $$1\frac{1}{5} + 2\frac{2}{5} + 3\frac{3}{5} + 4\frac{4}{5} = 12$$
• $$10 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - \frac{4}{2} = 5$$
• $$1\frac{1}{6} + 2\frac{2}{6} + 3\frac{3}{6} + 4\frac{4}{6} + 5\frac{5}{6} = 15\frac{3}{6}$$
• $$\frac{1}{3} + \frac{2}{3} + \frac{3}{3} = 3 \times \frac{2}{3}$$

### Activity Synthesis

• “¿Qué estrategias les parecieron útiles para sumar o restar estos números con fracciones?” // “What strategies did you find useful for adding or subtracting these numbers with fractions?” (Possible strategies:
• Adding whole numbers separately than fractions
• Noticing that $$1 + 2 + 3 + 4$$ is 10 and using that fact to add or subtract fractions
• Combine fractions that add up to 1 (such as $$\frac{1}{5} + \frac{4}{5}$$ and $$\frac{2}{5} + \frac{3}{5}$$).
• In the second equation, add up the fractions and subtract the sum from 10, instead of subtracting each fraction individually.)
• “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
• “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
• “¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
• “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

## Activity 1: Longitudes desconocidas (20 minutes)

### Narrative

Previously, students reasoned about the perimeter of two-dimensional figures based on given side lengths and known attributes of the figures, including symmetry. In this activity, students find unknown side lengths given the perimeter, some side lengths, and information about the symmetry of the figures. Students have opportunities to practice adding, subtracting, and multiplying numbers with fractions, as not all of the given measurements are whole numbers.

MLR8 Discussion Supports. Synthesis: Create a visual display of the shapes. As students share their strategies, annotate the display to illustrate connections. For example, next to each shape, write expressions and draw the lines of symmetry.
Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 2 of the 4 shapes to work with. Offer feedback that emphasizes effort and time on task, and invite them to try another shape if time allows.
Supports accessibility for: Organization, Attention, Social Emotional Functioning

### Required Materials

Materials to Gather

### 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 students who:
• can clearly articulate how lines of symmetry help them determine unknown side lengths
• can explain how they know that all four sides of Q are equal
• write expressions to show their reasoning

### Student Facing

Estas son cuatro figuras.

• P, R y S tienen, cada una, 1 línea de simetría.
• Q tiene 4 líneas de simetría.

1. Dibuja las líneas de simetría de cada figura.
2. En cada figura, encuentra la longitud de lado desconocida. Muestra tu razonamiento.

### Student Response

If students find the unknown sides lengths for shapes P and Q, but say they need more information to find the unknown length for shapes R and S, consider asking:

• “¿Cómo puedes usar la línea de simetría para marcar más longitudes de lado?” // “How can you use the line of symmetry to label more of the sides?”
• “¿Cuáles lados todavía están sin marcar? ¿Qué relación hay entre estos lados?” // “Which sides are still unlabeled? How are these sides related?”
• “¿Cómo podrías usar una expresión o una ecuación para encontrar las longitudes desconocidas de los lados?” // “How could you use an expression or an equation to help you find the unknown side lengths?”

### Activity Synthesis

• Select students to share their responses and reasoning.
• “¿Cómo les ayudan las líneas de simetría de P, R y S a encontrar las longitudes de lado desconocidas?” // “How do the lines of symmetry in P, R, and S help you find the unknown side lengths?” (The lines of symmetry tell us the lengths of unlabeled sides that mirror labeled sides, making it possible to find the length of the side with a question mark.)
• “¿Y las líneas de simetría de Q?” // “What about the lines of symmetry in Q?” (The vertical line of symmetry tells us the left and right sides have the same length and the horizontal one tells us the top and bottom sides are of equal length, so all sides have the same length.)

## Activity 2: El diseño de Lin (15 minutes)

### Narrative

In this activity, students practice completing a geometric drawing given half of the drawing and a line of symmetry, and reasoning about the perimeter of a line-symmetric figure.

While a precise drawing is not an expectation here, if no students considered using tools and techniques—such as using patty paper or by folding—to complete the drawing precisely, consider asking how it could be done (MP5).

To find the perimeter of the design, students have opportunities to look for and make use of structure (MP7) to expedite their calculation. For instance, instead of adding all 10 segments individually, they may:

• add 5 segments on one side of the line of symmetry and double it: $$(19 + 6 + 25 + 12\frac{3}{4} + 12\frac{3}{4}) \times 2$$
• multiply each segment by 2 and find the sum of those products: $$(2 \times 6) + (2 \times 19) + (2 \times 25) + (2 \times 12\frac{3}{4}) + (2 \times 12\frac{3}{4})$$
• add 19 and 6 to get 25, then multiply 25 and $$12\frac{3}{4}$$ by 4: $$(4 \times 25) + (4 \times 12\frac{3}{4})$$

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give a ruler or a straightedge to each student.
• Display the image for all to see.
• “¿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

• 5–6 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for the different strategies students use to complete the drawing and to find the perimeter of the design (as noted in the Activity Narrative).

### Student Facing

Lin usa 145 pulgadas de cinta decorativa para el contorno de un diseño que es simétrico con respecto a una línea.

Esta es la mitad del diseño. La línea punteada es la línea de simetría.

1. Dibuja el diseño de Lin completo.
2. ¿Lin tiene suficiente cinta para todo el contorno? Muestra tu razonamiento.

Si te queda tiempo: Lin tiene una hoja de papel decorativo que puede recortar para cubrir el interior del diseño. El papel es un rectángulo de 30 pulgadas por 18 pulgadas. Si los ángulos del diseño son ángulos rectos, ¿Lin tiene suficiente papel para cubrir el interior del diseño? Muestra tu razonamiento.

### Activity Synthesis

• Select students to present their completed drawings and share their process of drawing.
• If no one used patty paper, folding, or measurements to draw, consider asking: “Supongan que necesitamos dibujar con precisión la otra mitad del diseño de Lin. ¿Qué estrategias podríamos usar?” // “Suppose we need to draw the other half of Lin’s design precisely. What strategies could we use?”
• Select other students to present their response and reasoning for the last question. Sequence the presentation in the order listed in the Activity Narrative and record the expressions students used to find the perimeter.

## Lesson Synthesis

### Lesson Synthesis

“Hoy usamos las características de las figuras para razonar sobre sus longitudes de lado y su perímetro” // “Today we used attributes of figures to reason about their side lengths and perimeter.“

Display:

“Estas son dos figuras. Supongamos que sabemos que el perímetro de cada figura es 48 unidades. La figura A tiene una línea de simetría y la figura B no tiene líneas de simetría” // “Here are two shapes. Suppose we know the perimeter of each shape is 48 units. Shape A has a line of symmetry and B has none.”

“¿Cómo nos puede ayudar la línea de simetría de A a encontrar las longitudes de lado desconocidas?” // “How can knowing the line of symmetry in A help us find the unknown side lengths?” (The line of symmetry tells us that the longer unlabeled side is 14 and the two shorter sides are equal. We can subtract $$8 + 14 + 14$$ from 48 and divide the result by 2 to get the shorter sides.)

“La figura B no tiene líneas de simetría. ¿Podemos descubrir las longitudes desconocidas?” // “Shape B has no line of symmetry. Can we figure out the unknown lengths?” (No. There isn’t enough information. We'd need to know if some of the sides are the same length.)

If students argue that they can tell that one of the other sides must also be 15 units long, ask: “Sin medir, ¿qué necesitan saber para estar seguros de que uno de los lados que están marcados con un ‘?’ también mide 15 unidades?” // “Without measuring, what would you need to know to be sure one of the labeled sides is also 15 units long?”

“Supongamos que sabemos que B es un paralelogramo. ¿Nos ayudaría esto a encontrar las longitudes desconocidas? ¿Por qué sí o por qué no?” // “Suppose we know that B is a parallelogram. Would that help us find those lengths? Why or why not?” (Yes. Opposite sides of a parallelogram have the same length, so we know the unlabeled sides are 15 and 9.)

## Student Section Summary

### Student Facing

En esta sección usamos características, como longitudes de lado, ángulos, líneas de simetría y lados paralelos, para resolver problemas sobre el perímetro de figuras.

Aprendimos que si una figura tiene ciertas características, podemos usarlas para encontrar su perímetro, incluso cuando no están marcadas todas las longitudes de lado. También aprendimos que si conocemos el perímetro de una figura, podemos encontrar sus longitudes de lado si hay suficiente información sobre las características de la figura.

Por ejemplo, estas son dos figuras:

Si sabemos que el perímetro de cada figura es 48 unidades y que la línea punteada en la figura A es una línea de simetría, podemos encontrar las longitudes de lado que faltan.

La figura B no tiene líneas de simetría, pero si sabemos que sus lados opuestos tienen igual longitud, entonces podemos razonar sobre las tres longitudes de lado que faltan.