# Lesson 12

Fracciones equivalentes en una recta numérica

## Warm-up: Observa y pregúntate: Correr por un sendero (10 minutes)

### Narrative

The purpose of this warm-up is to elicit the idea that fractions can be used to describe lengths. While students may notice and wonder many things about this statement, the idea that Han and Tyler could have run the same distance or different distances are the important discussion points.

### Launch

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

### Activity

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

### Student Facing

¿Qué observas? ¿Qué te preguntas?

Tyler corrió parte de la longitud de un sendero.
Han corrió parte de la longitud del mismo sendero.

### Activity Synthesis

• “¿Cómo pueden las fracciones darnos más información sobre qué distancia corrieron Tyler y Han?” // “How could fractions give us more information about how far Tyler and Han ran?” (Tyler and Han ran $$\frac{1}{2}$$ of the field. Tyler ran $$\frac{3}{4}$$ of the field and Han ran $$\frac{7}{8}$$ of the field.)
• “¿Qué preguntas podemos hacer sobre la situación?” // “What questions could we ask about the situation?” (Did they run the same distance? Who ran farthest? How much farther did one student run than the other?)

## Activity 1: Correr parte de un sendero (10 minutes)

### Narrative

The purpose of this activity is for students to explain equivalence using a number line. Students are given situations in a measurement context and have to determine whether the distance is the same. Students are encouraged to use a number line to provide an opportunity to explain fraction equivalence as fractions that are at the same location. They may choose to use two number lines for each question (one for each fraction). Choosing to use one number line or two will be discussed in the synthesis of the next activity.

When they identify whether or not two fractions of the same trail represent the same distance, students reason abstractly and quantitatively (MP2).

MLR8 Discussion Supports. Display sentence frames to support whole group discussion. “Primero, yo _____ porque . . .” // “First, I _____ because . . .”, “Observé ____, entonces yo . . .” // “I noticed _____ so I . . . .”

• Groups of 2

### Activity

• “Decidan con su compañero si los estudiantes de cada pareja corrieron la misma distancia o no. Si les ayuda, pueden usar rectas numéricas para explicar su razonamiento” // “Work with your partner to decide whether each pair of students ran the same distance or not. You can use number lines to explain your reasoning if they’re helpful to you.”
• 5–7 minutes: partner work time
• Monitor for students who use the number lines to explain that the students ran the same distance if the fractions are at the same location on the number line.

### Student Facing

Algunos estudiantes corren por un sendero en un parque. Decide si los estudiantes de cada pareja corrieron la misma distancia.

Puedes usar rectas numéricas si te ayudan.

1. Elena corrió $$\frac{3}{6}$$ del sendero.

Han corrió $$\frac{1}{2}$$ del sendero.

2. Jada corrió $$\frac{1}{4}$$ del sendero.

Kiran corrió $$\frac{2}{8}$$ del sendero.

3. Lin corrió $$\frac{2}{3}$$ del sendero.

Mai corrió $$\frac{5}{6}$$ del sendero.

### Activity Synthesis

• Display a student-created number line that shows $$\frac{1}{4}$$ and $$\frac{2}{8}$$ at the same location.
• “¿Cómo muestra esto que Jada y Kiran corrieron la misma distancia?” // “How does this show that Jada and Kiran ran the same distance?” (The points that represent them are at the same location between 0 and 1 on the number line.)
• “Aprendimos que dos fracciones son equivalentes si tienen el mismo tamaño. Ahora también sabemos que dos números son equivalentes si están en la misma ubicación en una recta numérica. Como $$\frac{1}{4}$$ y $$\frac{2}{8}$$ están en la misma ubicación, podemos decir que son equivalentes” // “We’ve learned that two fractions are equivalent if they are the same size. Now we also know that two numbers are equivalent if they are at the same location on a number line. Because $$\frac{1}{4}$$ and $$\frac{2}{8}$$ are at the same location, we can say they are equivalent.”
• “¿Cómo podemos usar el signo igual para escribir fracciones que son equivalentes?” // “How could we use the equal sign to record fractions that are equivalent?” ($$\frac{3}{6} = \frac{1}{2}$$, $$\frac{1}{4} = \frac{2}{8}$$)
• Share and record responses.

## Activity 2: Ubiquemos y emparejemos (10 minutes)

### Narrative

The purpose of this activity is for students to locate fractions on the number line, and find pairs of fractions that are equivalent. Students can use a separate number line for each denominator, but they can also place fractions with different denominators on the same number line to show equivalence. Focus explanations about why fractions are equivalent on the fact that they share the same location. In the synthesis, discuss how one number line or two can be used to compare fractions.

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk.
Supports accessibility for: Attention, Organization

• Groups of 2

### Activity

• “Individualmente, ubiquen estos números en la recta numérica. Después, encuentren 4 parejas de fracciones que sean equivalentes. Prepárense para explicar su razonamiento” // “Work independently to locate these numbers on the number line. Then, find 4 pairs of fractions that are equivalent. Be prepared to explain your reasoning.”
• 3–5 minutes: independent work time
• “Ahora compartan con su compañero las parejas de fracciones que escribieron y expliquen cómo saben que son equivalentes” // “Now, share the pairs of fractions you wrote with your partner and explain how you know they are equivalent.”
• 2–3 minutes: partner discussion
• Monitor for students who compare fractions on a single number line and those who compare fractions on separate number lines.

### Student Facing

1. Ubica y marca los siguientes números en una recta numérica. Puedes usar más de una recta numérica si quieres.

$$\frac{1}{2}$$, $$\frac{1}{3}$$, $$\frac{1}{4}$$, $$\frac{2}{3}$$, $$\frac{2}{6}$$, $$\frac{3}{8}$$, $$\frac{3}{4}$$, $$\frac{4}{6}$$, $$\frac{4}{8}$$, $$\frac{6}{8}$$, $$\frac{7}{8}$$

2. Encuentra 4 parejas de fracciones que sean equivalentes. Escribe ecuaciones para representarlas.

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

$$\underline{\hspace{1 cm}} = \underline{\hspace{1 cm}}$$

Si te queda tiempo: Usa las rectas numéricas para generar todas las fracciones equivalentes que puedas.

### Activity Synthesis

• Select previously identified students to display how a single number line or separate number lines can be used to show equivalent fractions.
• “¿Cuándo tiene sentido usar una sola recta numérica y cuándo ayuda usar dos rectas numéricas?” // “When might it make sense to use a single number line and when might it be helpful to use two number lines?“ (One number line might make sense if one fraction can be partitioned to get to another, like halves to fourths. If a number line is too crowded or has fractions that could be hard to partition together, like halves and thirds, it might be helpful to use two number lines.)

## Activity 3: Lánzate a hacer fracciones equivalentes (15 minutes)

### Narrative

The purpose of this activity is for students to practice generating equivalent fractions. The goal of each round is to use the numbers on the number cubes to complete a statement that shows that two fractions are equivalent. Students roll 6 number cubes and try to use 4 of the numbers to create a statement that shows two equivalent fractions. If students roll a 5 (or a blank), they may choose any number to use. Students may choose to re-roll any of their number cubes up to 2 times. Students get a point for every true statement they make. Students may choose to use fraction strips, diagrams, or number lines to prove that their fractions are equivalent. If students choose to use diagrams, monitor to make sure they are drawing equal-sized wholes.

### Required Materials

Materials to Gather

### Required Preparation

• Each group of 2 needs 6 number cubes.

### Launch

• Groups of 2
• Give each group 6 number cubes.
• “Vamos a jugar un juego que se llama ‘Lánzate a hacer fracciones equivalentes’. Leamos las instrucciones y juguemos 1 ronda juntos” // “We’re going to play a game called Rolling for Equivalent Fractions. Let’s read through the directions and play 1 round together.”
• Read through the directions with the class and play a round against the class, displaying the fractions from the cards, drawing tape diagrams, and thinking through decisions aloud.

### Activity

• “Ahora jueguen con su compañero. Traten de obtener 5 puntos” // “Now, play the game with your partner. See if you can get 5 points.”
• 8–10 minutes: partner work time
• Monitor for students to highlight during the synthesis that:
• create a diagram of their fraction and generate an equivalent fraction with larger parts, such as picturing $$\frac{2}{8}$$ as $$\frac{1}{4}$$
• create a diagram of the fraction they draw and further partition their fraction to make smaller pieces, such as further partitioning $$\frac{1}{2}$$ to make $$\frac{2}{4}$$
• use a pattern to generate equivalent fractions, such as knowing that there are two sixths in each third, so $$\frac{2}{3}$$ is equivalent to $$\frac{4}{6}$$

### Student Facing

1. Lanza 6 dados numéricos. Si sacas cincos, úsalos como comodines. Cada cinco puede ser el número que quieras.
2. ¿Puedes poner en los cuadros los números que sacaste y hacer una afirmación que muestre fracciones equivalentes? Decídelo con tu compañero.
3. Si no puedes, lanza otra vez los dados que quieras. Puedes volver a lanzar tus dados dos veces.
4. Si puedes hacer fracciones equivalentes, anota tu afirmación y muestra o explica cómo sabes que las fracciones son equivalentes. Obtienes 1 punto por cada pareja de fracciones equivalentes que escribas.

Ronda 1:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Ronda 2:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Ronda 3:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Ronda 4:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Round 5:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Ronda 6:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Ronda 7:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

Ronda 8:

$$\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}=\frac{\boxed{\phantom{\frac{333}{333}}}}{\boxed{\phantom{\frac{333}{333}}}}$$

Muestra o explica cómo sabes que tus fracciones son equivalentes.

### Student Response

If students say they aren’t sure what fractions they can make that would be equivalent, consider asking:

• “¿Qué fracciones puedes hacer con lo que sacaste?” // “What fractions could you make with what you rolled?”
• “¿Cómo puedes usar tus tiras de fracciones para decidir si algunas de las fracciones son equivalentes?” // “How could you use your fraction strips to decide if any of the fractions are equivalent?”

### Activity Synthesis

• Display number cubes showing 1, 1, 4, 2, 2, 5
• “Si sacaran estos números en su último lanzamiento, ¿qué fracciones equivalentes podrían hacer?” // “If you got these numbers on your last roll, what equivalent fractions could you make?” ($$\frac{2}{4} = \frac{1}{2}$$ or $$\frac{1}{4} = \frac{2}{8}$$)
• If needed, ask, “¿Qué número debo usar para mi comodín?” // “What number should I use for my wild card?”

## Lesson Synthesis

### Lesson Synthesis

Display a number line that shows two fractions that are at the same location, such as $$\frac {3}{2}$$ and $$\frac{6}{4}$$.

“Al principio de la unidad, usamos tiras de fracciones para ver y encontrar fracciones equivalentes. Aquí, usamos rectas numéricas para encontrar fracciones equivalentes” // “Earlier in the unit, we used fraction strips to see and find equivalent fractions. Here, we use number lines to find equivalent fractions.”

“¿En qué se parecen las dos formas de mostrar fracciones equivalentes?” // “How are the two ways of showing equivalent fractions alike?” (They both involve partitioning a whole and identifying two or more fractions.)

“¿En qué son diferentes?” // “How are they different?” (Instead of looking for parts that are the same size, we are looking for the same point or location on the number line.)

“Hoy vimos que puede ser útil usar una o dos rectas numéricas para mostrar que las fracciones son equivalentes. Tengan eso en mente durante el cierre” // “Today, we saw that it can be helpful to use one or two number lines to show that fractions are equivalent. Keep that in mind during the cool-down.”