# Lesson 4

## Warm-up: Observa y pregúntate: Una tira de fracciones y una recta numérica (10 minutes)

### Narrative

The purpose of this warm-up is to revisit the idea from grade 3 that tape diagrams and number lines are related, which will be useful later in the lesson, when students transition from using fraction strips to using the number line to represent fractions and reason about their size.

While students may notice and wonder many things about these representations, the connections between the tape diagram and number line (the number and size of the parts in relation to 1) are important to note.

### 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 pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

### Student Facing

¿Qué observas? ¿Qué te preguntas?

### Activity Synthesis

• “¿En qué se parecen estas representaciones? ¿En qué son diferentes?” // “How are these representations alike? How are they different?”
• “Algunas marcas de la recta numérica no tienen números. ¿Qué números creen que serían apropiados?” // “Some tick marks on the number line are not labeled. What labels do you think would be appropriate for them?” ($$\frac{1}{4}$$, $$\frac{2}{4}$$, $$\frac{3}{4}$$, or $$\frac{1}{4}$$, $$\frac{1}{2}$$, $$\frac{3}{4}$$, or $$\frac{3}{12}$$, $$\frac{6}{12}$$, $$\frac{9}{12}$$)

## Activity 1: Mismo tamaño, números diferentes (20 minutes)

### Narrative

This activity serves two main goals: to revisit the idea of equivalence from grade 3, and to represent non-unit fractions with denominator 10 and 12. Students use diagrams of fraction strips, which allow them to see and reason about fractions that are the same size. In the next activity, students will apply a similar process of partitioning to represent these fractional parts on number lines.

Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. Provide access to colored pencils students can use to label each rectangle.
Supports accessibility for: Attention, Organization

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Give each student a straightedge.
• “Este diagrama de tiras de fracciones es como el de antes, pero con dos filas nuevas” // “Here is a diagram of fraction strips you saw before, with two new rows added.”
• “¿Cómo podemos mostrar décimos y doceavos en las dos filas? Piensen en silencio por un minuto” // “How can we show tenths and twelfths in the two rows? Think quietly for a minute.”
• 1 minute: quiet think time

### Activity

• “Trabajen individualmente en las dos primeras preguntas. Después, discutan sus respuestas con su pareja” // “Work on the first two questions on your own. Afterward, discuss your responses with your partner.”
• “Usen una regla cuando dibujen su diagrama” // “Use a straightedge when drawing your diagram.”
• 5–6 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who found the size of tenths and twelfths as noted in student responses.
• Pause for a brief discussion. Select students who used different strategies to find tenths and twelfths to share.
• After each person shares, ask if others in the class did it the same way or if they had anything to add to the explanation.
• “Miren el diagrama que completaron. ¿Qué pueden decir sobre la relación entre $$\frac{1}{5}$$ y $$\frac{1}{10}$$?” // “Look at your completed diagram. What can you say about the relationship between $$\frac{1}{5}$$ and $$\frac{1}{10}$$?” (There are two $$\frac{1}{10}$$s in every $$\frac{1}{5}$$. One fifth is twice one tenth. One fifth is the same size as 2 tenths.)
• “¿Qué pueden decir sobre la relación entre $$\frac{1}{6}$$ y $$\frac{1}{12}$$?” // “What can you say about the relationship between $$\frac{1}{6}$$ and $$\frac{1}{12}$$?” (There are two $$\frac{1}{12}$$s in every $$\frac{1}{6}$$. One sixth is twice one twelfth. One sixth is the same size as 2 twelfths.)
• “Respondan la última pregunta en 2 minutos” // “Take 2 minutes to answer the last question.”
• 2 minutes: independent or group work time

### Student Facing

Este es un diagrama de tiras de fracciones, con dos filas más para décimos y doceavos.

1. Usa una de las tiras en blanco para mostrar décimos. Marca las partes. ¿Cómo partiste la tira?

2. Usa una de las tiras en blanco para mostrar doceavos. Marca las partes. ¿Cómo partiste la tira?

3. Jada dice: “Observé que una parte de $$\frac{1}{2}$$ tiene el mismo tamaño que dos partes de $$\frac{1}{4}$$ y que tres partes de $$\frac{1}{6}$$. Entonces $$\frac{1}{2}$$, $$\frac{2}{4}$$ y $$\frac{3}{6}$$ deben ser equivalentes”.

En cada caso, encuentra una fracción que sea equivalente a la fracción dada. Prepárate para explicar tu razonamiento.

1. $$\frac{1}{6}$$
2. $$\frac{2}{10}$$
3. $$\frac{3}{3}$$

### Activity Synthesis

• Invite students to share their response to the last question and how they found equivalent fractions.
• Highlight the idea that two fractions that are the same size are equivalent, even if they have different numbers for the numerators and denominators.
• If needed, “¿Cuántas fracciones equivalentes a $$\frac{3}{3}$$ ven en el diagrama?” // “How many fractions that are equivalent to $$\frac{3}{3}$$ do you see on the diagram?” (Every strip on the diagram shows a fraction equivalent to $$\frac{3}{3}$$.)

## Activity 2: Fracciones en rectas numéricas (15 minutes)

### Narrative

The purpose of this activity is to remind students of their work in grade 3 using number lines as a way to reason about fractions. Students see that they can partition number lines in a similar way as they partitioned fraction strips and diagrams.

The activity gives students another opportunity to notice the relationship between two fractions whose denominator is a multiple or a factor of each other, and then use this relationship to locate fractions on a number line. In doing so, students practice looking for and making use of structure (MP7).

The work prepares students to use number lines to think about equivalent fractions in the next lesson.

### Launch

• Groups of 2

MLR5 Co-craft Questions

• Display only the four number lines without revealing the question(s).
• “Escriban una lista de preguntas matemáticas que se podrían hacer sobre esta situación” // “Write a list of mathematical questions that could be asked about this situation.”
• 2 minutes: independent work time
• 2–3 minutes: partner discussion
• Invite several students to share one question with the class. Record responses.
• “¿Qué tienen en común estas preguntas? ¿En qué son diferentes?” // “What do these questions have in common? How are they different?”

### Activity

• “Completen las primeras dos preguntas en 5 minutos” // “Take 5 minutes to complete the first two questions.”
• 5 minutes: independent work time
• “En parejas, discutan su trabajo y pónganse de acuerdo en los valores que escribieron en las rectas numéricas. Asegúrense de poder explicar su razonamiento antes de pasar a la última pregunta” // “Discuss your work with a partner. Make sure you and your partner agree on the labels for the number lines and can explain how you know before moving on to the last question.”
• 2 minutes: partner discussion
• Monitor for students who use the tick marks for $$\frac{1}{3}$$, $$\frac{1}{4}$$, and $$\frac{1}{5}$$ on the given number lines to locate $$\frac{1}{6}$$, $$\frac{1}{8}$$, and $$\frac{1}{10}$$.

### Student Facing

1. Estas son algunas rectas numéricas. El punto que está sobre esta recta numérica muestra la fracción $$\frac{1}{2}$$.

Escribe la fracción que corresponde debajo de las marcas de cada recta numérica.

2. Supongamos que vas a ubicar $$\frac{1}{6}$$, $$\frac{1}{8}$$ y $$\frac{1}{10}$$ en una de las rectas numéricas.

1. ¿Cuál recta numérica usarías para cada fracción? Prepárate para explicar tu razonamiento.

2. Ubica y marca cada fracción ($$\frac{1}{6}$$, $$\frac{1}{8}$$ y $$\frac{1}{10}$$) en una recta numérica diferente.

3. Ubica y marca cada una de estas fracciones en una de las rectas numéricas.

$$\frac{2}{3}$$

$$\frac{2}{8}$$

$$\frac{2}{5}$$

$$\frac{3}{5}$$

$$\frac{4}{6}$$

$$\frac{4}{8}$$

$$\frac{4}{10}$$

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

$$\frac{6}{10}$$

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

### Student Response

Students may place the labels between the tick marks on the number line, rather than at or below the tick marks. Clarify that the numbers on a number line represent distance from 0. Consider asking: “Si empezamos en 0, nos movemos hacia 1 y paramos a medio camino, ¿dónde debemos poner ‘$$\frac{1}{2}$$’ para marcar el punto del medio? ¿Por qué?” // “If we start from 0 and move halfway toward 1, where should we put the ‘$$\frac{1}{2}$$’ to mark the halfway point? Why?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Select 1–2 students to share their completed number lines from the last activity, with points marked on the lines to represent the given fractions.

• “¿Cómo decidieron cuál recta numérica escoger para $$\frac{1}{6}$$$$\frac{1}{8}$$ y $$\frac{1}{10}$$?” // “How did you know which number line to choose for $$\frac{1}{6}$$$$\frac{1}{8}$$, and $$\frac{1}{10}$$?” (We could locate $$\frac16$$, for example, on any of the number lines. But since we know that $$3 \times 2 = 6$$, we can split each part in the number line that shows thirds into 2 to make 6 parts, which makes it easiest to locate $$\frac16$$.)
• “¿Cómo supieron dónde poner un punto para, por ejemplo, $$\frac{4}{10}$$?” // “How did you know where to put a point for, say, $$\frac{4}{10}$$?” (Starting from 0, count as many tick marks as the number in the numerator. For $$\frac{4}{10}$$, count 4 tick marks on the number line that show tenths.)

Display a completed diagram of fraction strips from an earlier activity.

“¿En qué se parecen representar una fracción como $$\frac{6}{10}$$ en una recta numérica y representarla en una tira de fracciones? ¿En qué son diferentes?” // “How is representing a fraction like $$\frac{6}{10}$$ on a number line like representing it on a fraction strip? How is it different?” (Sample responses:
• Alike: They both involve identifying the right fractional parts—by looking at the denominator—and then counting as many parts as the numerator of the fraction.
• Different: One involves the size of parts that are folded and the other involves a specific place on the number line.)