# Lesson 3

## Warm-up: Cuál es diferente: Distintas maneras de expresar un número decimal (10 minutes)

### Narrative

This warm-up prompts students to compare four expressions. The particular expressions chosen give students a chance to focus on several important features, including:

• the operations
• the values of the expressions
• the types of numbers in the expressions (whole numbers versus decimals)

Students work in this lesson to express decimals in many different forms, and this warm-up gives students some familiarity thinking about some of those different forms.

### Launch

• Groups of 2
• Display the image.
• “Escojan una que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

¿Cuál es diferente?

1. $$26\div100$$
2. $$0.26$$
3. $$26 \times 0.001$$
4. $$(2 \times 0.1) + (6 \times 0.01)$$

### Activity Synthesis

• Display the expression: $$2 \times 0.1 + 6 \times 0.01$$
• “¿En qué se diferencia esta expresión de las demás?” // “How is this expression different from the others?” (It is written as a sum. The different place values are written separately.)
• “Hoy vamos a representar números decimales de esta manera” // “Today we are going to represent decimal numbers in this way.”

## Activity 1: Forma desarrollada (20 minutes)

### Narrative

In a previous course, students multiplied a decimal fraction by a whole number. In previous lessons, students wrote decimal fractions in decimal form. The purpose of this activity is for students to use expanded form of a decimal number to the thousandth. Students relate expanded form to both diagrams and decimal numbers. The expanded form of a decimal number highlights the value of each digit. For the number 0.835, for example, the 8 represents 8 tenths. This is shown in expanded form by writing the $$0.8$$ as $$8 \times 0.1$$. Students practice relating decimals, diagrams, and expanded form and then are formally introduced to the term expanded form, as it applies to decimals, in the activity synthesis (MP2). The notation of expanded form is a generalization of what students saw in a previous grade with whole numbers.

• Groups of 2

### Activity

• 15 minutes: partner work time

### Student Facing

1. Explica o muestra por qué la región sombreada representa $$(4 \times 0.1) + (1 \times 0.01) + (9 \times 0.001)$$.
1. Sombrea la cuadrícula para representar $$(8 \times 0.1) + (3 \times 0.01) + (5 \times 0.001)$$.

2. Escribe el número $$(8 \times 0.1) + (3 \times 0.01) + (5 \times 0.001)$$ en forma decimal.
1. Mai dice que el número decimal 0.105 representa $$(1 \times 0.1) + (5 \times 0.01)$$. ¿Estás de acuerdo con Mai? Explica o muestra tu razonamiento.

### Activity Synthesis

• Display the diagram from the student workbook that shows $$0.835$$.
• Consider circling each part of the diagram as students respond to the questions.
• “¿Qué parte del diagrama muestra $$8 \times 0.1$$? ¿Cómo lo saben?” // “What part of the diagram shows $$8 \times 0.1$$? How do you know?” (The 8 shaded horizontal strips because they are each tenths or 0.1.)
• “¿Qué parte del diagrama muestra $$3 \times 0.01$$? ¿Cómo lo saben?” // “What part of the diagram shows $$3 \times 0.01$$? How do you know?” (The 3 single shaded squares because they are hundredths or 0.01.)
• “¿Qué parte del diagrama muestra $$5 \times 0.001$$? ¿Cómo lo saben?” // “What part of the diagram shows $$5 \times 0.001$$? How do you know?” (The 5 tiny shaded pieces because they are each thousandths or 0.001.)
• “¿Qué representa el 5 en 0.105?” // “What does the 5 in 0.105 represent?” (5 thousandths)
• “Puedo escribir 0.105 como $$1 \times 0.1 + 5 \times 0.001$$ para mostrar que es 1 décima y 5 milésimas. Esto se llama la ‘forma desarrollada’ de un número decimal” // “I can write 0.105 as $$1 \times 0.1 + 5 \times 0.001$$ to show that it is 1 tenth and 5 thousandths. This is called the expanded form of a decimal.”

## Activity 2: Números decimales de numerosas maneras (15 minutes)

### Narrative

The purpose of this activity is for students to practice different ways of expressing decimal numbers to the thousandth. In addition to standard decimal digit form, these ways include:

• diagrams
• expanded form
• fractions
• words
The goal of the activity synthesis is to show how the different ways to represent a decimal are interrelated. This gives students an opportunity to make sense of each form and how it relates to the others (MP2).

### Launch

• Groups of 2
• Display the first image that represents the number 0.742 in the student workbook.
• “¿De qué maneras podemos representar el número que se muestra en el diagrama?” // “What are some different ways we can represent the number shown in the diagram?”
• 0.742
• seven hundred forty-two thousandths
• $$0.7 + 0.04 + 0.002$$
• $$(7\times 0.1)+(4 \times 0.01)+(2\times 0.001)$$
• $$\frac {742}{1,000}$$
• 1 minute: quiet think time
• Share and record responses.

### Activity

• “Ahora representen cada número de tantas maneras como puedan” // “Now find as many ways as you can to represent each number.”
• 2 minutes: independent work time
• 6 minutes: partner work time

### Student Facing

Representa cada número de tantas maneras como puedas.

1.

2. $$\frac{477}{1,000}$$

3. ciento treinta y seis milésimas

4. $$(3 \times 0.1) + (6 \times 0.01) + (8 \times 0.001)$$

### Student Response

If students are not able to find multiple ways to represent a given number, refer to the list from the launch of the activity.

### Activity Synthesis

• Invite students to share their representations of $$\left(3 \times 0.1\right) + \left(6 \times 0.01\right) + \left(8 \times 0.001\right)$$.
• Display: 0.368
• “¿Cómo se relaciona cada dígito del número decimal con la forma desarrollada?” // “How does each digit in the decimal relate to the expanded form?” (The 3 is 3 groups of 1 tenth or 0.3, the 6 is six groups of 1 hundredths or 0.06, and the 8 is 8 groups of 1 thousandth or 0.008.)
• Display: $$\frac{368}{1,\!000}$$
• “¿Cómo se relaciona la forma desarrollada con la fracción?” // “How does the expanded form relate to the fraction?” (The 300 is the 3 tenths, the 60 is the 6 hundredths, and the 8 is the 8 thousandths.)
• Display student work that shows 0.368 represented on the hundredths grid.
• “¿Cómo se relaciona la forma desarrollada con el diagrama?” // “How does the expanded form relate to the diagram?” (The 3 tenths are the top 3 rows. The 6 hundredths are the 6 squares in the next row. The 8 thousandths are the small rectangles.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy representamos números decimales de muchas maneras” // “Today we represented decimal numbers in many ways.”

Display: 0.315

“Mencionen varias maneras en las que pueden representar este número. ¿Cuál es su manera favorita?” // “What are some different ways you can represent this number? What is your favorite way?” (Three hundred fifteen thousandths, $$\frac {315}{1,000}$$ , $$(3 \times 0.1) + (1 \times 0.01) + (5 \times 0.001)$$, or I could draw a diagram. My favorite way is the decimal because it's the shortest.)

Display responses for all to see.