# Lesson 9

Expliquemos la equivalencia

## Warm-up: Conversación numérica: Números conocidos (10 minutes)

### Narrative

This Number Talk encourages students to use the relationship between related numbers (5 and 10, and 6, 12, and 24) and properties of operations to find products. The strategies of doubling and halving elicited here will be helpful later in the lesson when students generate equivalent fractions. In describing strategies, students need to be precise in their word choice and use of language (MP6).

### Launch

• Display one expression.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$10\times6$$
• $$10\times12$$
• $$10\times24$$
• $$5\times24$$

### Activity Synthesis

• “¿Cómo los ayudaron las primeras tres expresiones a encontrar el valor de la última?” // “How did the first three expressions help you find the value of the last one?”

## Activity 1: Discusión puntual (20 minutes)

### Narrative

In this activity, students look closely at the relationships of fractions with denominator 5, 10, and 100. They use their observations and understanding to identify equivalent fractions and to explain why two fractions are or are not equivalent. When students analyze and criticize the reasoning presented in the activity statements and when the discuss their work with classmates, they are critiquing the reasoning of others and improving their arguments (MP3).

Engagement: Develop Effort and Persistence. Check in and provide each group with feedback that encourages collaboration and community.
Supports accessibility for: Social-Emotional Functioning

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Ask students to keep their materials closed.

### Activity

• “Tómense 5 minutos en silencio para responder. Después, discutan con su compañero cómo pensaron” // “Take 5 quiet minutes to answer the problem. Afterwards, discuss your thinking with your partner.”
• 5 minutes: independent work time
• 5 minutes: partner discussion

### Student Facing

Andre, Lin y Clare representan $$\frac{70}{100}$$ en una recta numérica.

• Andre dijo: “¡Oh, no! ¡Debemos partir la recta en 100 partes iguales y contar 70 partes solo para marcar un punto!”.
• Lin dijo: “¿Qué tal si más bien marcamos $$\frac{7}{10}$$? Podemos partir la recta en solo 10 partes y contar 7 partes”.
• Clare dijo: “¿Y si partimos la recta en 5 partes y marcamos $$\frac{3}{5}$$?”.

¿Estás de acuerdo con alguno de ellos? Explica o muestra tu razonamiento.

### Activity Synthesis

MLR1 Stronger and Clearer Each Time

• “Compartan su razonamiento y sus rectas numéricas con su compañero. 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 reasoning and number lines 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.”
• 3–5 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

## Activity 2: ¿Cómo lo sabes? (15 minutes)

### Narrative

This activity gives students opportunities to practice explaining or showing whether two fractions are equivalent. Students may do so using a visual representation, by reasoning about the number and size of the fractional parts in each fraction, or by thinking about multiplicative relationships between the numbers in the given fractions.

Students participate in a gallery walk in which they generate equivalent fractions for the numbers on the posters. Students visit at least two of six posters (or as many as time permits)—at least one poster with two fractions (posters A–C) and one poster with three fractions (posters D–F). Here are the sets shown on the blackline master:

A

​​​​​$$\frac{2}{10}$$, $$\frac{20}{100}$$

B

$$\frac{6}{4}$$, $$\frac{18}{12}$$

C

$$\frac{3}{5}$$, $$\frac{60}{100}$$

D

​​​​​​$$\frac{1}{4}$$, $$\frac{3}{12}$$, $$\frac{30}{100}$$

E

$$\frac{15}{6}$$, $$\frac{7}{4}$$, $$\frac{30}{12}$$

F

$$\frac{7}{3}$$, $$\frac{21}{10}$$, $$\frac{28}{12}$$

Because at the posters with two fractions (A–C) students would need to generate an equivalent fraction that hasn’t already been written by others, generating equivalent fractions becomes more difficult as the activity goes on. Consider using this to differentiate for students who may need an additional challenge: start them at the posters with three fractions (D–F).

MLR8 Discussion Supports. Synthesis. Display sentence frames to support whole-class discussion: “Estoy de acuerdo porque . . .” // “I agree because . . .” and “No estoy de acuerdo porque . . . ” // “I disagree because . . . .”

### Required Materials

Materials to Gather

Materials to Copy

• How Do You Know

### Required Preparation

• Each group needs 4 sticky notes.

### Launch

• Groups of 3–4
• Give each group 4 sticky notes
• Read the task statement as a class. Solicit clarifying questions from students.
• Invite a couple of students to recap the directions in their own words or to demonstrate the process, if helpful.
• Consider assigning each group a starting poster and giving directions for rotation.

### Activity

• 10 minutes: gallery walk
• Tell students who are visiting posters A–C that they could leave feedback about the fraction on a sticky note if they disagree that it is equivalent to the fractions on the poster. They should include their name and be prepared to explain how they know.

### Student Facing

Por todo el salón encontrarás seis pósteres. Cada uno muestra dos o tres fracciones.

Con tu grupo, visita al menos dos pósteres: uno con dos fracciones y uno con tres fracciones.

Para el que tiene 2 fracciones:

• Explica o muestra cómo sabes que las fracciones son equivalentes.
• Escribe una nueva fracción equivalente en una nota adhesiva y agrégala al póster. Piensa en una fracción que nadie más haya escrito.

Visitamos el póster __________, que muestra __________ y __________.

Nueva fracción equivalente: __________

Para el que tiene 3 fracciones:

• Identifica 2 fracciones que sean equivalentes. Explica tu razonamiento.

Visitamos el póster __________, que muestra __________, __________ y __________.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Select a group to share their response and reasoning for each poster.

Highlight visual diagrams or verbal explanations that clearly show how the number and size of the parts of two fractions can differ even though the fractions are the same size.

When students explain their work on posters D–F, ask about the non-equivalent fraction. For instance: “¿Cómo supieron que $$\frac{1}{4}$$ y $$\frac{3}{12}$$ son equivalentes, pero que $$\frac{30}{100}$$ no es equivalente a ellas?” // “How did you know that $$\frac{1}{4}$$ and $$\frac{3}{12}$$ are equivalent but $$\frac{30}{100}$$ is not equivalent to them?”