# Lesson 14

¿Cómo comparas fracciones?

## Warm-up: Conversación numérica: ¿Cuáles números enteros? (10 minutes)

### Narrative

This Number Talk encourages students to use what they know about the meaning of fractions and about properties of operations to mentally relate fractions that are equivalent to whole numbers.

### Launch

• Display one fraction.
• “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 fractions and work displayed.
• Repeat with each fraction.

### Student Facing

En cada caso, encuentra a qué número entero es equivalente la fracción.

• $$\frac{16}{1}$$
• $$\frac{16}{2}$$
• $$\frac{16}{4}$$
• $$\frac{20}{4}$$

### Activity Synthesis

• “¿Cómo les ayudaron las primeras fracciones a encontrar el número entero en el caso de la última fracción?” // “How did the earlier fractions help you find the whole number for the last fraction?”
• “¿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 el problema de otra forma?” // “Did anyone approach the problem in a different way?”

## Activity 1: ¿Son equivalentes o no? (25 minutes)

### Narrative

The purpose of this activity is for students to analyze pairs of fractions to determine if they are equivalent. Students may use any representation that makes sense to them. Students will create a visual display and have a gallery walk to consider the different ways of looking for equivalence. Highlight representations such as diagrams and number lines, which will support students as they learn to compare fractions with the same numerator or denominator in this section.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

Engagement: Develop Effort and Persistence. Invite students to generate a list of shared expectations for group work. Record responses on a display and keep visible during the activity.
Supports accessibility for: Social-Emotional Functioning

### Launch

• Groups of 2
• “Decidan si las fracciones de cada pareja son equivalentes y muestren cómo pensaron en cada caso. Pueden usar cualquier representación que tenga sentido para ustedes” // “Decide if these pairs of fractions are equivalent and show your thinking for each one. You can use any representation that makes sense to you.”

### Activity

• 3–5 minutes: independent work time
• “Compartan sus ideas con su compañero” // “Share your ideas with your partner.”
• 2–3 minutes: partner discussion
MLR7 Compare and Connect
• “Con su compañero, creen una presentación visual que muestre cómo pensaron sobre $$\frac{4}{6}$$ y $$\frac{5}{6}$$. Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender sus ideas” // “Work with your partner to create a visual display that shows your thinking about $$\frac{4}{6}$$ and $$\frac{5}{6}$$. You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
• Give students materials for creating a visual display.
• 2–5 minutes: partner work time
• 5–7 minutes: gallery walk

### Student Facing

¿Estas fracciones son equivalentes? Muestra cómo pensaste. Usa diagramas, símbolos u otras representaciones.

1. $$\frac{1}{2}$$ y $$\frac{1}{3}$$
2. $$\frac{4}{6}$$ y $$\frac{5}{6}$$
3. $$\frac{3}{4}$$ y $$\frac{6}{8}$$

### Student Response

If students don’t determine whether or not fractions are equivalent, consider asking:

• “¿Qué sabes sobre estas fracciones?” // “What do you know about these fractions?”
• “¿Cómo puedes usar diagramas o tus tiras de fracciones para decidir si son equivalentes?” // “How could you use your fraction strips or diagrams to decide if they are equivalent?”

### Activity Synthesis

• “¿Qué representaciones diferentes podemos usar para decidir si $$\frac{4}{6}$$ y $$\frac{5}{6}$$ son equivalentes o no?” // “What are the different representations we can use to decide if $$\frac{4}{6}$$ and $$\frac{5}{6}$$ are equivalent?” (diagrams, number lines)
• “¿Cómo mostró cada representación que $$\frac{4}{6}$$ y $$\frac{5}{6}$$ no son equivalentes?” // “How did each representation show that $$\frac{4}{6}$$ and $$\frac{5}{6}$$ are not equivalent?” (In the diagram, you can see that $$\frac{5}{6}$$ has more space shaded. On the number line, they are not at the same location.)

## Activity 2: Mismas fracciones, ¿diferente resultado? (10 minutes)

### Narrative

The purpose of this activity is for students to recognize that fraction comparisons are only valid when they refer to the same whole. Previously, students analyzed pairs of fractions to determine if they were equivalent. In doing so, they were likely to have used comparison language, such as “larger or smaller than” or “greater or less than.” In this activity, students encounter a pair of fractions they saw earlier ($$\frac{4}{6}$$ and $$\frac{5}{6}$$) and compare them more explicitly. The student work in this activity uses the number line, but this might also come up with student-drawn diagrams.

In order to interpret Lin’s argument that $$\frac{4}{6}$$ is greater than $$\frac{5}{6}$$ students will need to articulate the meaning of fractions and highlight the fact that the two wholes Lin is comparing are not equal (MP6).

MLR8 Discussion Supports. Synthesis: Revoice student ideas to demonstrate and amplify mathematical language use. For example, revoice the student statement “the number lines are different” as “the size of the whole from 0 to 1 is different.”

### Launch

• Groups of 2
• “Al igual que ustedes, Han y Lin comparan $$\frac{4}{6}$$ y $$\frac{5}{6}$$. Tómense un minuto para examinar su trabajo” // “Han and Lin are comparing $$\frac{4}{6}$$ and $$\frac{5}{6}$$ like you did. Take a minute to look at their work.”
• 1 minute: quiet think time

### Activity

• “Hablen con su compañero sobre cómo pudieron Han y Lin obtener resultados diferentes” // “Talk with your partner about how Han and Lin could get different results.”
• 2–3 minutes: partner discussion
• Monitor for students who notice that the whole is different in Lin’s number lines, which makes her think that $$\frac{4}{6}$$ is greater.

### Student Facing

Han dice que $$\frac{4}{6}$$ es menor que $$\frac{5}{6}$$. Este es su trabajo.

Lin dice que $$\frac{4}{6}$$ es mayor que $$\frac{5}{6}$$. Este es su trabajo.

¿Por qué Han y Lin hacen afirmaciones de comparación diferentes sobre las mismas fracciones?

### Activity Synthesis

• Select previously identified students to share how Han and Lin could make different comparison statements for the same fractions.
• “Cuando comparamos fracciones, es importante recordar que esas fracciones deben hacer referencia a la misma unidad” // “It is important to remember when we are comparing fractions that those fractions need to refer to the same whole.”
• “Cuando la unidad de 0 a 1 es del mismo tamaño, podemos ver que $$\frac{4}{6}$$ es menor que $$\frac{5}{6}$$” // “When the whole from 0 to 1 is the same size, we can see that $$\frac{4}{6}$$ is less than $$\frac{5}{6}$$.”
• “Esto es cierto independientemente de que dibujemos una recta numérica, usemos tiras de fracciones o dibujemos un diagrama” // “This is true whether we are drawing a number line, using fraction strips, or drawing a diagram.”
• Consider asking, “¿Qué entendieron Lin y Han sobre representar fracciones en una recta numérica?” // “What did both Lin and Han understand about representing fractions on a number line?” (The number lines need to be partitioned into equal parts. The denominator tells us then number of parts. The numerator tells us how many parts to count to locate a point. Points farther to the right are greater than those to the left.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy estudiamos parejas de fracciones para saber si eran equivalentes o no. ¿Cómo decidieron si dos fracciones eran equivalentes?” // “Today, we studied pairs of fractions to see if they were equivalent or not. How did you decide if two fractions were equivalent?” (Drew diagrams to see if the parts that represent the fractions were the same size. Represent the fractions on number lines and see if they were at the same location.)

“Si las fracciones no eran equivalentes, esto significa que una de las fracciones era mayor que la otra y una de las fracciones era menor que la otra. Vamos a aprender más sobre esto en otras lecciones” //  “If the fractions were not equivalent, it means that one of the fractions was greater than the other, and one of the fractions was less than the other. We'll learn more about this in future lessons.”

Draw two number lines (or diagrams) with different lengths (or areas) representing 1 whole. Partition each into 3 parts.

“¿Cuál podría ser un problema si comparamos $$\frac{2}{3}$$ y $$\frac{3}{3}$$ usando estas rectas numéricas (o diagramas)?” // “What might be a problem with comparing $$\frac{2}{3}$$ and $$\frac{3}{3}$$ using these number lines (or diagrams)?” (The 1 whole is not the same size, so we can’t use the number lines (or diagrams) to compare the fractions.)