# Lesson 10

Usemos múltiplos para encontrar fracciones equivalentes

## Warm-up: Observa y pregúntate: Cuatro ecuaciones (10 minutes)

### Narrative

The purpose of this warm-up is to draw students’ attention to the multiplicative relationships between the numerators and denominators of two equivalent fractions. These observations will be helpful later as students use the idea of multiples to generate equivalent fractions.

While students may notice and wonder many things about these equations, highlight observations about a factor relating the numbers in the two sides of each equation.

### 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 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?

• $$\frac{1}{3} = \frac{2}{6}$$

• $$\frac{2}{3} = \frac{4}{6}$$

• $$\frac{3}{3} = \frac{6}{6}$$

• $$\frac{4}{3} = \frac{8}{6}$$

### Activity Synthesis

• “¿Cómo se relacionan los números al lado derecho de cada signo igual con los números al lado izquierdo?” // “How are the numbers on the right side of each equal sign related to the numbers on the left?” (Each number on the right is twice the number on the left.)
• “¿El tamaño de las fracciones a la derecha es el doble del tamaño de las fracciones a la izquierda?” // “Are the fractions on the right twice the size of the fractions on the left?” (No, they are the same size.)

## Activity 1: La forma de Elena (20 minutes)

### Narrative

In an earlier lesson, students used visual representations to generate equivalent fractions. They did so by partitioning each increment on a number line into smaller equal-size parts. In this activity, they connect that action to a numerical process—one that involves multiplying both the numerator and denominator by the same factor. When students notice that they can multiply the numerator and denominator of a fraction by any whole number to get an equivalent fraction they observe regularity in repeated reasoning (MP8).

MLR2 Collect and Display. Collect the language students use to reason about how to find equivalent fractions. Display words and phrases such as: equivalent fraction, equation, number line, numerator, denominator, multiply, multiples, etc. During the activity, invite students to suggest ways to update the display: “¿Qué otras palabras o frases debemos incluir?” //  “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.

### Launch

• Groups of 2
• “Miren las rectas numéricas de Andre con las que trabajaron en una lección anterior” // “Take a look at Andre’s number lines you worked with in a previous lesson.”

### Activity

• “Piensen en silencio durante un par de minutos sobre lo que hizo Elena y cómo se relaciona con las rectas numéricas de Andre” // “Think quietly for a couple of minutes about what Elena did and how it relates to Andre’s number lines.”
• 1–2 minutes: quiet think time for the first problem
• 3–4 minutes: partner discussion on the first problem
• Pause for a brief whole-class discussion. Invite students to share their ideas about Elena’s work and how it is related to Andre’s number lines.
• 4–5 minutes: independent work time for the last problem
• Monitor for students who find equivalent fractions for $$\frac{1}{8}$$ by multiplying by a factor other than 2, 3 or 4.

### Student Facing

A Elena se le ocurrió otra forma de encontrar fracciones equivalentes. Ella escribió:

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

$$\frac{1 \ \times \ 3}{5 \ \times \ 3} = \frac{3}{15}$$

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

$$\frac{1 \ \times \ 5}{5 \ \times \ 5} = \frac{5}{25}$$

$$\frac{1 \ \times \ 10}{5 \ \times \ 10} = \frac{10}{50}$$

1. Analiza el trabajo de Elena. Después, discute con un compañero:

1. ¿Cómo se relacionan las ecuaciones de Elena con las rectas numéricas de Andre?
2. ¿Cómo podría Elena encontrar otras fracciones que sean equivalentes a $$\frac{1}{5}$$? Muestra un par de ejemplos.
2. Usa la estrategia de Elena para encontrar cinco fracciones que sean equivalentes a $$\frac{1}{8}$$. Si te ayuda, usa rectas numéricas para comprobar lo que pensaste.

### Activity Synthesis

• Select 1–2 students to share their equivalent fractions for $$\frac{1}{8}$$ and their reasoning. Display their equivalent fractions as equations (for example, $$\frac{1}{8} = \frac{3}{24}$$, $$\frac{1}{8} = \frac{4}{32}$$, $$\frac{1}{8} = \frac{7}{56}$$, and $$\frac{1}{8} = \frac{8}{64}$$).
• “¿Estas ecuaciones muestran los mismos patrones que las ecuaciones del calentamiento? ¿En qué se parecen o en qué son diferentes?” // “Do these equations show the same patterns as the equations in the warm-up? How are they alike or different?” (Alike: In all of the equations the numerators and denominators are multiplied by the same amount. Different: In the warm up each numerator and denominator was multiplied by 2. In these problems each the numerator and denominator of the fraction is multiplied by different numbers each time.)

## Activity 2: En busca de equivalencias (15 minutes)

### Narrative

In this activity, students identify equivalent fractions. In the first problem, they use the numerical strategy they learned earlier to determine if two fractions are equivalent. In the second problem, they can use any strategy in their toolkit—which now includes a numerical method—to identify equivalent fractions.

Students encounter some fractions with unfamiliar denominators such as 9, 16, 32, 40, and 80, but they will not be assessed on such fractions. These denominators are multiples of familiar denominators such as 2, 3, 4, 5, 8, or 10, and are included to give students opportunities to generalize their reasoning about equivalence.

Action and Expression: Internalize Executive Functions. Synthesis: Check for understanding by inviting students to rephrase directions in their own words. Keep a display of directions visible throughout the activity.
Supports accessibility for: Memory, Organization

### Launch

• Groups of 2
• “Ahora vamos a ver si se puede usar el método de Elena para comprobar si las fracciones son equivalentes” // “Now you are going to see whether you can use Elena’s method to see if fractions are equivalent.”

### Activity

• 3–4 minutes: independent time to work on the first problem
• Pause for a brief whole-class discussion.
• “¿Cómo supieron por cuál número multiplicar el numerador y el denominador para comprobar la equivalencia?” // “How did you know what number to multiply to the numerator and denominator to check equivalence?” (Sample responses:
• See if there’s a whole number that can be multiplied by 5 to get 10, multiplied by 2 to get 8, and so on.
• Divide 10 by 5, or 8 by 2, and so on, and see if what the result is and whether it’s a whole number.)
• “Identifiquen con su compañero todas las fracciones en la lista que sean equivalentes a $$\frac{3}{4}$$. Prepárense para mostrar cómo lo saben” //  “Work with your partner to identify all fractions on the list that are equivalent to $$\frac{3}{4}$$. Be prepared to show how you know.”
• 6–7 minutes: group work time for the second problem

### Student Facing

Examina la estrategia de Elena de la actividad anterior.

1. ¿Su estrategia podría ayudarnos a saber si dos fracciones son equivalentes? Intenta usarla para comprobar la equivalencia de estas fracciones:

1. $$\frac{5}{2}$$ y $$\frac{10}{8}$$
2. $$\frac{2}{6}$$ y $$\frac{4}{12}$$

Escribe una ecuación si las fracciones son equivalentes.

2. Encuentra todas las fracciones en la lista que son equivalentes a $$\frac{3}{4}$$. Prepárate para explicar o mostrar cómo lo sabes.

$$\frac{2}{10}$$

$$\frac{6}{8}$$

$$\frac{12}{15}$$

$$\frac{30}{40}$$

$$\frac{8}{9}$$

$$\frac{12}{20}$$

$$\frac{12}{16}$$

$$\frac{15}{20}$$

$$\frac{8}{10}$$

$$\frac{24}{32}$$

$$\frac{75}{100}$$

$$\frac{60}{80}$$

### Activity Synthesis

• “Revisen su lista de fracciones equivalentes con otra pareja de compañeros” // “Check your list of equivalent fractions with another group.”
• “Discutan cualquier desacuerdo sobre una fracción hasta que todos estén de acuerdo en si la fracción es o no equivalente a $$\frac{3}{4}$$” // “Discuss any disagreement about a fraction until both groups agree whether or not it is equivalent to $$\frac{3}{4}$$.”
• 3 minutes: Check list with another group.

## Lesson Synthesis

### Lesson Synthesis

“Hoy usamos una estrategia numérica para encontrar fracciones equivalentes y comprobar si las fracciones son equivalentes” // “Today we used a numerical strategy for finding equivalent fractions and for checking if fractions are equivalent.”

“Supongamos que un compañero no estuvo hoy en la clase. Después, vio algunos ejemplos de cómo encontrar fracciones equivalentes a $$\frac{1}{3}$$ usando esta estrategia, pero no entiende del todo los ejemplos” //  “Suppose a classmate was absent today. They later saw some examples of how to find equivalent fractions for $$\frac{1}{3}$$ using this strategy, but they don’t fully follow the examples.”

Display: $$\frac{1\times4}{3\times4} = \frac{4}{12}$$ and $$\frac{1\times6}{3\times6} = \frac{6}{18}$$

“¿Qué le dirían a su compañero para ayudarlo a entender lo que está ocurriendo en las ecuaciones? ¿Cómo explicarían la multiplicación por 4 o por 6?” // “What would you say to help your classmate understand what is happening in the equations? How would you explain the multiplication by 4 or by 6?”