# Lesson 1

Números decimales

### Narrative

The purpose of this warm-up is to elicit the use of a square grid to represent fractions in hundredths. Both the representation and the fractions will be useful later in the lesson, when students write fractions as decimals. While students may notice and wonder many things about the diagram, focus on expressing the fractions of the large square that are shaded and unshaded.

### 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?

### Activity Synthesis

• “Si el cuadrado grande representa 1, ¿qué fracción del cuadrado está sombreada? ¿Qué fracción no está sombreada? ¿Cómo lo saben?” // “If the large square represents 1, what fraction of it is shaded? What fraction is not shaded? How do you know?” ($$\frac{6}{100}$$ are shaded. There are 100 small squares, and 6 of them shaded and 94 are not.)

## Activity 1: Fracciones sombreadas (20 minutes)

### Narrative

In this activity, students use a square grid of 100 to revisit the meaning of tenths and hundredths and to make sense of the decimal notation for these fractions. They begin to make connections between the familiar representations of a fraction—using a diagram, fraction notation, and words—and the newly introduced decimal notation, and to notice similarities in their structure (MP2, MP7). It is important for students to consistently hear numbers read as decimals, for example 1.7 as one and 7 tenths so that they can connect decimal notation to visual representations and fraction notation. Later, in the lesson synthesis, the connections between the decimal notation and numbers in base-ten will begin to be made explicit.

Students may begin to notice that there are different ways to write decimals that represent the same fraction. It is not essential to discuss this in depth, as students will look at equivalent decimals more closely in upcoming activities.

Representation: Develop Language and Symbols. Some students may need additional examples before writing fractions in decimal notation independently. To emphasize connections between representations, create a two-column table with examples of tenths and hundredths in both decimal and word form on each side. This might include 0.04 (4 hundredths), 0.25 (25 hundredths), and 0.50 (50 hundredths) on one side and 0.4 (4 tenths), 0.2 (2 tenths), and 0.5 (5 tenths) on the other. Invite students to notice patterns. Check for understanding by asking students how they would write 7 hundredths, 70 hundredths, and 7 tenths in decimal notation.
Supports accessibility for: Visual-Spatial Processing, Organization, Attention

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• “Trabajen en el primer problema de manera independiente” // “Work on the first problems independently.”
• 3–4 minutes: independent work time
• Pause for a brief whole-class discussion. Display the diagrams in the first problem.
• “¿Qué notación podemos usar para escribir cada fracción? ¿Cómo decimos cada fracción en palabras?” // “What notation can we write to show each fraction? How do we say the fraction in words?”
• Record students’ responses in both notation and words.
• Display the diagrams of one hundredth and one tenth (in the next problem). Explain that the first fraction, $$\frac{1}{100}$$, can also be expressed as 0.01 and is still read “1 centésima” // “1 hundredth”. This form of writing is called decimal notation.
• “¿Y $$\frac{9}{100}$$?” // “What about $$\frac{9}{100}$$?” (nine hundredths, 0.09) “¿Y $$\frac{10}{100}$$?” // $$\frac{10}{100}$$?” (ten hundredths, 0.10)
• “Sabemos que $$\frac{10}{100}$$ también se puede expresar como $$\frac{1}{10}$$. ¿Cómo lo decimos en palabras y cómo lo escribimos en notación decimal?” // “We know that $$\frac{10}{100}$$ can also be expressed as $$\frac{1}{10}$$. How do we say it in words and write it in decimal notation?” (one tenth, 0.1)
• “En los números que se escriben como 0.1 y 0.01 hay puntos decimales. El dígito que está a la izquierda del punto decimal está en la posición de las unidades. El 0 significa que no hay unidades” // “In the numbers written like 0.1 and 0.01, there are decimal points. The digit to the left of the decimal point is in the ones place. The 0 means that there are no ones.”
• “Ahora intenten escribir las fracciones del primer problema como decimales. Luego, completen el resto de la actividad” // “Now try writing the fractions from the first problem as decimals. Then, complete the rest of the activity.”
• 5–6 minutes: independent or partner work time
• Monitor for the ways students write the fractions in the last problem, which may inform how they write corresponding decimals. For instance, they may write:
• mixed numbers ($$1\frac{20}{100}$$)
• sums ($$\frac{100}{100} + \frac{20}{100}$$ or $$1 + \frac{20}{100}$$)
• fractions with no whole numbers ($$\frac{120}{100}$$)

### Student Facing

2. La parte sombreada de este diagrama representa 0.01, que llamamos “1 centésima”.

Las partes sombreadas de este diagrama representan 0.10, que llamamos “10 centésimas”.
También representan 0.1 o “1 décima”.

Los números como 0.01, 0.10 y 0.1 están escritos como decimales.

Observa las partes sombreadas de cada diagrama del primer problema. Escribe como decimales los números que ellas representan.

​​​​​

### Student Response

Students may write only fractions for the last problem as $$\frac{120}{100}$$ and $$\frac{133}{100}$$. Ask them to try expressing them as mixed numbers, and see how doing so might help with the decimal notation.

### Activity Synthesis

• Invite students to share the decimals for the diagrams in the first problem. Record their responses for all to see.
• “¿En qué se diferencian los diagramas del último problema de los diagramas del primer problema?” // “How are the diagrams in the last problem different from those in the first problem?” (They represent numbers greater than 1.)
• “¿Cómo supieron de qué manera escribir cada fracción como un decimal?” // “How did you figure out how to write each fraction as a decimal?”
• If not mentioned in students’ responses, point out that we can think of $$1\frac{20}{100}$$ as $$1 + \frac{20}{100}$$. The 1 whole goes in the ones place, to the left of the decimal point, and the 20 hundredths goes on the right of the decimal point.
• “El decimal 1.20 se puede leer ‘uno y 20 centésimas’. El decimal 1.33 se puede leer ‘uno y 33 centésimas’” // “The decimal 1.20 can be read ‘one and 20 hundredths’. The decimal 1.33 can be read ‘one and 33 hundredths’.”

## Activity 2: Formas de expresar un número (15 minutes)

### Narrative

In this activity, students practice representing and writing decimals given another representation (fraction notation or a diagram). The idea that two decimals can be equivalent, just like two fractions can be equivalent, is made explicit here. When students make connections between quantities in word form, decimal form, and fraction form, they reason abstractly and quantitatively (MP2).

MLR2 Collect and Display. Synthesis: Direct attention to words collected and displayed from the previous activity. Invite students to borrow language from the display as needed, and update it throughout the lesson.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “Practiquemos cómo representar cantidades usando diagramas, fracciones y decimales” // “Let’s practice representing amounts using diagrams, fractions, and decimals.”

### Activity

• “En silencio, trabajen unos minutos en la actividad. Luego, compartan sus respuestas con su compañero” // “Take a few quiet minutes to work on the activity. Then, share your responses with your partner.”
• 7–8 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for students who use the idea of equivalent fractions to explain why 0.6 and 0.60 refer to the same amount.

### Student Facing

Fracción: ________

Decimal: 0.78

Fracción: $$\frac{8}{10}$$

Decimal: ________

Fracción: $$\frac{55}{100}$$

Decimal: ________

Fracción: $$\frac{107}{100}$$

Decimal: ________

Fracción: ________

Decimal: 1.6

3. Han y Elena están no están de acuerdo sobre qué número está representado por la porción sombreada.

Han dice que la porción representa 0.60 y Elena dice que representa 0.6.

Explica por qué tanto Han como Elena tienen razón.

### Activity Synthesis

• Ask students to use words and decimal notation to express each amount in the first problem.
• Clarify that 0.02 can be named “dos centésimas” // “two hundredths” and 0.22 can be named “veintidós centésimas” // “twenty-two hundredths” and so on.
• Select students to share their representations for $$\frac{107}{100}$$ and 1.6 and how they reasoned about them.
• Display 1.07 and 1.6. “¿Cómo dirían estos decimales en palabras?” // “How would you say these decimals in words?” (One and seven hundredths or one point zero seven for 1.07, and one and sixth tenths or one point six for 1.6.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy aprendimos que una fracción se puede escribir como un decimal, sin importar si es menor o mayor que 1” //  “Today we learned that a fraction can be written as a decimal, regardless of whether it is less or greater than 1.”

Display the number 0.78. Ask students to identify what each digit represents. Annotate the numbers as shown. (Note that students are not expected to do this independently at this time.)

“¿Por qué puede tener sentido llamar a este decimal ‘setenta y ocho centésimas’?” // “Why might it make sense to name this decimal seventy-eight hundredths?” (Seven tenths and 8 hundredths is equivalent to 78 hundredths.)

Display the numbers 0.6 and 0.60.

“¿Cómo decimos estos números en palabras?” // “How do we say these numbers in words?” (Six tenths for 0.6, and sixty hundredths for 0.60)

“En ambos números, ¿qué representa el 0 que está a la izquierda del punto decimal?” // “In both numbers, what does the 0 to the left of the decimal point represent?” (Zero ones)

“En 0.6, ¿qué representa el 6?” // “In 0.6, what does the 6 represent?” (Six tenths)

“En 0.60, ¿qué representa el 60?” // “In 0.60, what does the 60 represent?” (Sixty hundredths)

“¿Por qué podemos usar el mismo diagrama para representar 0.6 y 0.60?” // “Why can we use the same diagram to represent 0.6 and 0.60?” (They represent the same amount. Six tenths is equivalent to sixty hundredths, so 0.6 and 0.60 are equivalent.)

Reiterate that the decimal point separates the whole number and the fractional amount.