# Lesson 11

Usemos factores para encontrar fracciones equivalentes

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

### Narrative

This warm-up prompts students to carefully analyze and compare representations of fractions. To make comparisons, students need to draw on their knowledge about fractional parts, the size of fractions, and equivalent fractions.

### 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 cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

1. $$\frac{1}{4}$$

### Activity Synthesis

• “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: Al contrario (20 minutes)

### Narrative

In this activity, students see that they can find equivalent fractions by dividing the numerator and denominator by a common factor. They connect this strategy to the process of grouping unit fractions on a number line into larger equal-size parts. The result is a fewer number of parts, and smaller numbers for the numerator and denominator of equivalent fractions.

MLR8 Discussion Supports. Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain the strategies used in the given examples. Invite groups to rehearse what they will say when they share with the whole class.

• Groups of 2

### Activity

• “Resuelvan los primeros tres problemas con su compañero” // “Work with your partner to answer the first three problems.”
• “Prepárense para explicar cómo piensan que se relacionan las estrategias de Andre y Kiran” // “Be prepared to explain how you think Andre’s and Kiran’s strategies are related.”
• 7–8 minutes: partner work time
• “Tómense unos minutos para responder individualmente el último problema” // “Take a few minutes to answer the last problem independently.”
• 2–3 minutes: independent work time for the last problem

### Student Facing

1. Andre dibujó una recta numérica y marcó un punto en ella. Escribe debajo del punto la fracción que corresponde.

2. Para encontrar otras fracciones que corresponden al punto, Andre hizo otras rectas numéricas. Dibujó marcas más oscuras en algunas de las marcas que ya había.

En cada recta numérica, escribe el número que corresponde debajo de las marcas más oscuras que hizo Andre.

3. Kiran escribió las mismas fracciones para los puntos, pero usó una estrategia diferente, como se muestra a continuación. Analiza su razonamiento.

$$\frac{8 \ \div \ 4}{12 \ \div \ 4}=\frac{2}{3}$$

$$\frac{8 \ \div \ 2}{12 \ \div \ 2}=\frac{4}{6}$$

¿Cómo crees que se relacionan las estrategias de Andre y Kiran?

4. Intenta usar la estrategia de Kiran para encontrar una o más fracciones que sean equivalentes a $$\frac{10}{12}$$ y $$\frac{18}{12}$$.

### Activity Synthesis

• “¿Qué hizo Andre con las rectas numéricas? ¿Cómo le ayudó esto a encontrar fracciones equivalentes?” // “What did Andre do with the number lines? How would it help him find equivalent fractions?” (Andre grouped the 12 parts into equal groups of different sizes—2s, 4s—to make bigger parts. Then, he counted the number of those new parts.)
• “¿Qué hizo Kiran? ¿Cómo se relaciona su estrategia con la de Andre?” // “What did Kiran do? How is his strategy related to Andre’s?” (Kiran divided 12 by 4 and then by 2, similar to how Andre put 12 parts into groups of 4 and then of 2.)
•  “Observen que ambas fracciones equivalentes, $$\frac{2}{3}$$ y $$\frac{4}{6}$$, tienen números más pequeños para el numerador y el denominador que la fracción original. ¿Pueden usar la recta numérica de Andre para mostrar por qué sucede esto?” //  “Notice that the equivalent fractions $$\frac{2}{3}$$ and $$\frac{4}{6}$$ both have smaller numbers for the numerator and denominator than the original fraction. Can you use Andre’s number line to show why this might be?” (The size of the parts are bigger, so there are fewer parts in 1 whole.)

• “¿Cuántas fracciones equivalentes pueden encontrar para $$\frac{10}{12}$$ usando la estrategia de Kiran?” // “How many equivalent fractions can you find for $$\frac{10}{12}$$ using Kiran’s way?” (One) “Cuántas pueden encontrar para $$\frac{18}{12}$$?” // “How many can you find for $$\frac{18}{12}$$?” (Three)
• “¿Cuál puede ser una razón por la que se pueden encontrar más fracciones equivalentes para $$\frac{18}{12}$$ que para $$\frac{10}{12}$$?” // “What might be a reason that you could find more equivalent fractions for $$\frac{18}{12}$$ than for $$\frac{10}{12}$$?” (18 and 12 have more factors in common than 10 and 12.)
• “¿Cómo mostrarían $$\frac{9 \ \div \ 3}{12 \ \div \ 3}=\frac{3}{4}$$ en la recta numérica?” // “How would you show $$\frac{9 \ \div \ 3}{12 \ \div \ 3}=\frac{3}{4}$$ on the number line?” (Put the original 12 parts into groups of 3 to get 4 parts, each being a fourth. Mark 3 of those 4 parts to show $$\frac{3}{4}$$.)

## Activity 2: ¿Cómo las encontrarías? (15 minutes)

### Narrative

In this activity, students generate equivalent fractions by applying the numerical strategies they learned. (Students might opt to use other strategies, but most of the given fractions have numbers that would make visual representation and reasoning inconvenient.) Depending on the given fractions, students need to decide whether it makes sense to multiply or divide the numerator and denominator by a common number.

Engagement: Provide Access by Recruiting Interest. Synthesis: Leverage choice around perceived challenge. Invite students to select at least 3 of the 5 given fractions for this activity.
Supports accessibility for: Organization, Attention, Social-Emotional Functioning

• Groups of 2

### Activity

• “Trabajen en la actividad individualmente. Después, compartan sus respuestas con su compañero y revisen el trabajo de cada uno” // “Work on the activity independently. Then, share your responses with your partner and check each other’s work.”
• 8–10 minutes: independent work time
• 3–5 minutes: partner discussion

### Student Facing

Encuentra al menos dos fracciones que sean equivalentes a cada fracción. Muestra tu razonamiento.

1. $$\frac{16}{8}$$
2. $$\frac{40}{10}$$
3. $$\frac{7}{6}$$
4. $$\frac{90}{100}$$
5. $$\frac{5}{4}$$

### Student Response

If students attempt to partition the number line into 30, 60, or 90 parts, consider asking: “¿Cómo puede ayudarnos aquí usar los patrones de la actividad anterior?” // “How can we use the patterns from the previous activity to help us here?”

### Activity Synthesis

• See lesson synthesis.

## Activity 3: Clasificación de tarjetas: Fracciones por montones [OPTIONAL] (15 minutes)

### Narrative

This activity is optional because it provides an opportunity for students to apply concepts from previous activities that not all classes may need. It allows students to practice using numerical strategies to find equivalent fractions by sorting a set of 36 cards. Students are not expected to find all equivalent fractions in the set. When students look for equivalent fractions they use their understanding of multiples and the meaning of fractions (MP7).

### Required Materials

Materials to Copy

• Fractions Galore

### Required Preparation

• Create a set of Fraction Galore cards from the blackline for each group of 3.

### Launch

• Groups of 3–4
• Give each group one set of cards created from the blackline master.

### Activity

• “Hagan grupos de tarjetas que tengan fracciones equivalentes. Encuentren todos los grupos de fracciones equivalentes que puedan. Algunas fracciones no tienen fracciones equivalentes” // “Work with your group to sort the cards by equivalence. Find as many sets of equivalent fractions as you can. Some fractions have no equivalent fractions.”
• 8–10 minutes: small group work time

### Student Facing

Tu profesor te dará un grupo de tarjetas. Encuentra todos los grupos de fracciones equivalentes que puedas. Prepárate para explicar o mostrar tu razonamiento.

Anota aquí los grupos de fracciones equivalentes.

Anota aquí las fracciones que no tienen una fracción equivalente.

### Activity Synthesis

• “¿Qué estrategia usó su grupo para encontrar fracciones equivalentes? ¿Qué tan bien funcionó la estrategia? ¿Qué tan eficiente fue?” // “What strategy did your group use to find equivalent fractions? How well did the strategy work? How efficient was it?” (We looked at the numerators and denominators to see if they were multiples or factors we recognized.)
• “¿Observaron patrones nuevos en las fracciones que son equivalentes?” // “Did you notice any new patterns in the fractions that are equivalent?”

## Lesson Synthesis

### Lesson Synthesis

“Hoy vimos otra forma de encontrar fracciones equivalentes. Dividimos el numerador y el denominador de una fracción entre un factor que tenían en común” // “Today we looked at another way to find equivalent fractions. We divided the numerator and denominator of a fraction by a factor they have in common.”

“¿Cómo decidieron entre usar la multiplicación o usar la división para escribir una fracción equivalente?” // “How did you decide whether to use multiplication or division to write an equivalent fraction?” (Sample response: It depends on the numbers in the fraction. When the numbers are large to start with and both have a factor in common, we’d divide by that factor. When the numbers are small and have no shared factors, we’d multiply.)

## Student Section Summary

### Student Facing

En esta sección, aprendimos a identificar y escribir fracciones equivalentes. Ubicamos fracciones en rectas numéricas y vimos que dos fracciones que ocupan el mismo lugar en una recta numérica son equivalentes.

También vimos estrategias para encontrar fracciones equivalentes y aprendimos que al multiplicar el numerador y el denominador por el mismo número o al dividirlos entre el mismo número se obtiene una fracción equivalente. Estos son algunos ejemplos:

$$\frac{1 \ \times \ 2}{5 \ \times \ 2} = \frac{2}{10}$$

$$\frac{1 \ \times \ 4}{5 \ \times \ 4} = \frac{4}{20}$$

$$\frac{1}{5}$$ es equivalente a $$\frac{2}{10}$$ y a $$\frac{4}{20}$$.

$$\frac{8 \ \div \ 2}{12 \ \div \ 2} = \frac{4}{6}$$

$$\frac{8 \ \div \ 4}{12 \ \div \ 4} = \frac{2}{3}$$

$$\frac{8}{12}$$ es equivalente a $$\frac{4}{6}$$ y a $$\frac{2}{3}$$.