# Lesson 2

Decimales equivalentes

## Warm-up: Verdadero o falso: Fracciones equivalentes (10 minutes)

### Narrative

The purpose of this True or False is to revisit equivalent fractions in tenths and hundredths. The reasoning students do here will be helpful later when students make sense of and identify decimals that are equivalent to given fractions or given decimals.

### Launch

• Display one statement.
• “Hagan una señal cuando sepan si la afirmación es verdadera o no, y puedan explicar cómo lo saben” // “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar tu razonamiento.

• $$\frac{50}{100} = \frac{5}{10}$$
• $$\frac{20}{10} = \frac {20}{100}$$
• $$2 = 1 + \frac{90}{100}$$
• $$3\frac{1}{10} = \frac{31}{10}$$

### Activity Synthesis

• “¿Qué saben de la relación entre décimas y centésimas que les haya ayudado a decidir si cada afirmación es verdadera o falsa?” // “What do you know about the relationship of tenths and hundredths that helped you decide whether each statement is true or false?” (Sample responses:
• One tenth is 10 hundredths.
• One tenth is 10 times 1 hundredth.
• There are 10 tenths in 1 whole.
• There are 100 hundredths in 1 whole.
• If we multiply the numerator and denominator of a fraction in tenths by 10, we get an equivalent fraction in hundredths.)

## Activity 1: Clasificación de tarjetas: Diagramas de fracciones y decimales (15 minutes)

### Narrative

In this activity, students reinforce their understanding of equivalent fractions and decimals by sorting a set of cards by their value. The cards show fractions, decimals, and diagrams. A sorting task gives students opportunities to analyze different representations closely and make connections (MP2, MP7).

Representation: Access for Perception. Synthesis: Display a 10-by-10 grid, as well as a square of the exact same size, but with only the columns shown (therefore representing just tenths). Shade 20 hundredths on the 10-by-10 grid and write 0.20 (twenty hundredths) above it. Shade 2 tenths on the other square and write 0.2 (2 tenths) above it. Invite students to discuss how these diagrams demonstrate equivalence of the two numbers.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Required Materials

Materials to Copy

• Card Sort: Diagrams of Fractions & Decimals

### Required Preparation

• Create a set of cards from the blackline master for each group of 2–4.

### Launch

• Groups of 2–4
• Give one set of cards from the blackline master to each group.

### Activity

• “Con sus compañeros, clasifiquen las tarjetas según su valor” // “Work with your group to sort the set of cards by their value.”
• “Uno de los diagramas no corresponde a ningún grupo. Escriban la fracción y el decimal que ese diagrama representa” // “One diagram has no matching cards. Write the fraction and decimal it represents.”
• 6–7 minutes: group work on the first two problems
• Monitor for the ways students sort the cards and the features of the representations to which they attend.
• “Trabajen en el último problema individualmente” // “Work on the last problem independently.”
• 2–3 minutes: independent work on the last problem

### Student Facing

Tu profesor te va a dar una colección de tarjetas. El cuadrado grande de cada tarjeta representa 1.

1. Clasifica las tarjetas de manera que las representaciones de cada grupo tengan el mismo valor. Anota las decisiones que tomaste para clasificarlos. Prepárate para explicar tu razonamiento.
2. Una de las tarjetas no corresponde a ningún grupo. ¿Qué fracción y qué decimal representa ese diagrama?
3. ¿Son 0.20 y 0.2 equivalentes? Usa fracciones y un diagrama para explicar tu razonamiento.

### Student Response

Students may respond that 0.20 and 0.2 are not “the same.” Consider asking:

• “¿Cómo representarías cada número en una cuadrícula?” // “How would you represent each number on a square grid?”
• “¿En qué se parecen las cantidades y en qué son diferentes?” // “What is the same about the amounts and what is not the same?”

### Activity Synthesis

• Select one group to share each set of sorted cards and explain how they knew the representations belong together.
• “¿Cómo supieron qué fracción y qué decimal escribir para el diagrama que no correspondía a ningún grupo?” // “How did you know what fraction and decimal to write for the diagram without any matches?”
• Select a student to share their response to the last problem. Highlight the equivalence of 0.2 and 0.20 as shown in the Student Responses.

## Activity 2: ¿Verdadero o falso? (20 minutes)

### Narrative

In this activity, students apply their understanding of equivalent fractions and decimals more formally, by analyzing equations and correcting the ones that are false. The last question refers to decimals on a number line and sets the stage for the next lesson where the primary representation is the number line.

As students discuss and justify their decisions about the claim in the last question, they critically analyze student reasoning (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

### Launch

• Groups of 2
• “Antes, vimos ecuaciones que tenían fracciones en ambos lados del signo igual. Ahora exploremos ecuaciones que tienen fracciones y decimales, o solo decimales” // “Earlier, we saw some equations with fractions on both sides of the equal sign. Now let’s look at some equations that include fractions and decimals or just decimals.”

### Activity

• “Tómense unos minutos para terminar la actividad de manera independiente. Luego, compartan con su compañero cómo pensaron” // “Take a few minutes to complete the activity independently. Then, share your thinking with your partner.”
• 6–7 minutes: independent work time
• “En cada pregunta del primer problema, tomen turnos para explicarle a su compañero cómo supieron si la afirmación era verdadera o falsa” // “For each equation in the first problem, take turns explaining to your partner how you know whether it is true or false.”
• 3–4 minutes: partner discussion

### Student Facing

1. En cada caso, decide si la afirmación es verdadera o falsa. Si es falsa, reemplaza uno de los números para que sea verdadera (los números que hay en un lado del signo igual no pueden quedar todos idénticos a los del otro lado). Prepárate para compartir cómo pensaste.

1. $$\frac{50}{100} = 0.50$$

2. $$0.05 = 0.5$$

3. $$0.3 = \frac{3}{10}$$

4. $$0.3 = \frac{30}{100}$$

5. $$0.3 = 0.30$$

6. $$1.1 = 1.10$$

7. $$3.06 = 3.60$$

8. $$2.70 = 0.27$$

2. Jada dice que si ubicamos los números 0.05, 0.5 y 0.50 en la recta numérica, solo quedarán dos puntos marcados. ¿Estás de acuerdo? Explica o muestra tu razonamiento.

### Student Response

Students may be unsure about how to locate 0.05 on a number line. Ask them how they would express the number in words and in fraction notation (in tenths or hundredths). Consider asking them to also name each tick mark on the number line. “Si el espacio entre dos marcas consecutivas representa 10 centésimas, ¿en qué lugar de la recta estará 5 centésimas?” // “If the space between two tick marks represents 10 hundredths, where might 5 hundredths land on the line?”

### Activity Synthesis

MLR1 Stronger and Clearer Each Time

• “Compartan con su pareja su respuesta a la última pregunta. 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 response to the last question 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.”
• 3–4 minutes: structured partner discussion.
• Repeat with 1–2 new 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

## Lesson Synthesis

### Lesson Synthesis

“Hoy exploramos distintas formas de representar decimales que son equivalentes. Usamos cuadrículas, rectas numéricas y fracciones para mostrar que dos decimales pueden representar el mismo valor” // “Today we looked at different ways to represent decimals that are equivalent. We used square grids, number lines, and fractions to show that two decimals can represent the same value.”

“Supongan que un compañero no vino a la clase de hoy. ¿Cómo podrían convencerlo de que 0.3 y 0.30 son equivalentes? Escriban por lo menos dos maneras distintas de hacerlo” // “Suppose a classmate is absent today. How would you convince them that 0.3 and 0.30 are equivalent? Write down at least two different ways.”

Select students to share their thinking. Display the representations they used, or display the following:

“0.3 es 3 décimas y 0.30 es 30 centésimas. La misma parte sombreada representa 3 décimas y también 30 centésimas” // “0.3 is 3 tenths and 0.30 is 30 hundredths. The same shaded part represents 3 tenths and 30 hundredths.”

“3 décimas y 30 centésimas se marcan en la recta numérica con el mismo punto” // “Both 3 tenths and 30 hundredths share the same point on the number line.”

“0.3 es $$\frac{3}{10}$$ y 0.30 es $$\frac{30}{100}$$. Las dos fracciones son equivalentes” // “0.3 is $$\frac{3}{10}$$ and 0.30 is $$\frac{30}{100}$$. The two fractions are equivalent.”

$$\frac {3 \ \times \ 10}{10 \ \times \ 10}=\frac{30}{100}$$

$$\frac{3}{10} = \frac{30}{100}$$