# Lesson 2

Milésimas en cuadrículas y en palabras

### Narrative

The purpose of this Estimation Exploration is for students to recognize the structure of the hundredths grid. Students have used this grid in this course and earlier courses. Without the hundredths grid, it is difficult to estimate the shaded region. This grid helps students to see tenths, hundredths, and with some extra work, even thousandths.

When students reflect about how the hundredths grid could help refine their estimate, they observe the value and power if its structure (MP7).

### Launch

• Groups of 2
• Display the image.
• “¿Qué estimación sería muy alta?, ¿muy baja?, ¿razonable?” // “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

### Activity

• “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

### Student Facing

Escribe una estimación que sea:

muy baja razonable muy alta
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### Activity Synthesis

• “¿Por qué es más difícil estimar la región sombreada sin las rectas que dividen a una cuadrícula de centésimas?” // “Why is estimating the shaded region more difficult without the gridlines of a hundredths grid?” (The gridlines show me the tenths and hundredths. Without that, I can only guess or estimate.)

## Activity 1: Representemos milésimas en una cuadrícula (20 minutes)

### Narrative

The purpose of this activity is for students to shade diagrams to represent fractions and decimals to the thousandths place. The first problem reviews grade 4 work in which students filled in the same diagrams to show decimal fractions and decimals to hundredths. After this review, the problems all involve thousandths. First students interpret how much of a square is shaded and then they shade a part of a square to represent a three-digit decimal. Because the thousandths are so small students may struggle to count the shaded thousandths and may disagree about how many thousandths are shaded in the diagrams.

Monitor for students who interpret and draw diagrams of decimals by thinking about each individual digit in a number. For example, in order to show 0.327 in a diagram, students can think of this as:

• 3 tenths
• 2 hundredths
• 7 thousandths

When shading the thousandths and naming them, students must be precise and pay close attention to what they decide to shade (MP6).

MLR2 Collect and Display. Circulate, listen for, and collect the language students use as they shade and interpret diagrams. On a visible display, record words and phrases such as: “fracción”, “parte de”, “decimal”, “décimas”, “fila”, “centésimas”, “milésimas”, “representa”, “muestra” // “fraction,” “part of,” “decimal,” “tenths,” “row,” “hundredths,” “thousandths,” “represents,” “shows.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from starting with representing smaller values on the grid. For example, represent one thousandth or two thousandths on the grid.
Supports accessibility for: Conceptual Processing, Attention

### Launch

• Groups of 2
• “Hoy vamos a representar números decimales con diagramas” // “Today we are going to represent decimal numbers with diagrams.”
• “¿Qué significa el número decimal 0.001?” // “What does the decimal 0.001 mean?” (1 thousandth)

### Activity

• 8–10 minutes: independent work time
• Monitor for students who relate the diagrams to the decimal numbers by thinking about the tenths, hundredths, and thousandths shaded in the diagrams.

### Student Facing

2. En cada caso, escribe un número decimal que represente cuánto está sombreado en el diagrama. Explica o muestra tu razonamiento.

3. Sombrea 0.328 en el diagrama. Explica o muestra tu razonamiento.

### Student Response

If students do not write the correct number to represent the shaded hundredths grids, ask:

• “¿Cómo está representado en el diagrama el número que escribiste?” // “How does the diagram represent the number you wrote?“
• “¿Cómo está representado en el diagrama cada dígito del número que escribiste?” // “How does the diagram represent each of the digits in the number you wrote?“

### Activity Synthesis

• Display the diagram that shows 0.625.
• “¿Qué número está representado por este diagrama?” // “What number does this diagram represent?” (six hundred twenty-five thousandths)
• As students respond to each of the following questions, highlight on the diagram the tenths, hundredths, and thousandths.
• “¿Cómo ven 0.6 en el diagrama?” // “Where do you see 0.6 in the diagram?” (There are 6 rows shaded and each row is 0.1 or a tenth.)
• “¿Cómo ven 0.62 en el diagrama?” // “Where do you see 0.62 in the diagram?” (There are 62 small squares shaded and each one is 0.01 or a hundredth so that's 0.62 or 62 hundredths.)
• “¿Cómo ven 0.625 en el diagrama?” // “Where do you see 0.625 in the diagram?” (If we divide each small square into ten tiny rectangles there will be 625 of them and they are each 0.001.)

## Activity 2: ¿Qué dicen? (15 minutes)

### Narrative

In this activity, students consider different ways to name a decimal shown on a hundredths grid. The meaning of a decimal such as 0.150 is 1 tenth, 5 hundredths, and 0 thousandths. In words, however, it is usually read as one hundred fifty thousandths. Students see, using a diagram, that 1 tenth and 5 hundredths is equivalent to 150 thousandths. When students interpret the different descriptions of the shaded region they construct viable arguments and critically analyze claims (MP3).

• Groups of 2

### Activity

• 2 minutes: quiet think time
• 6 minutes: partner work time
• Monitor for students who use fractions or decimals to represent the language each student uses.

### Student Facing

Varios estudiantes ven el diagrama y describen la región sombreada de distintas maneras. ¿Con quién o con quiénes estás de acuerdo? ¿Por qué?

1. Jada dice que es “15 centésimas”.
2. Priya dice que es “150 milésimas”.
3. Tyler dice que es “15 milésimas”.
4. Diego dice que es “1 décima y 5 centésimas”.
5. Mai dice que es “1 décima y la mitad de una décima”.

### Student Response

If students need support when identifying the correct ways to represent 0.15 with words, refer to each of the correct student descriptions and ask, “¿Cómo está representada la cuadrícula de centésimas en la descripción?” // “How does the description represent the hundredths grid?” Then, refer to the incorrect description and ask, “¿Por qué esta descripción no tiene sentido?” // “Why doesn’t this description make sense?”

### Activity Synthesis

• Invite students to share their answer and reasoning for each student’s statement.
• As students share, represent their language with fractions and decimals and show their reasoning on the hundredths grid.

## Lesson Synthesis

### Lesson Synthesis

“Hoy representamos números decimales de distintas maneras” // “Today we represented decimal numbers in different ways.” Display a shaded grid, such as a student response to represent the decimal 0.34 in the first activity.

“Mencionen varias maneras en las que podemos decir este número” // “What are some different ways we can say this number?” (34 hundredths, 340 thousandths, 3 tenths and 4 hundredths)

Show or ask students to show how the diagram shows each way of saying the number.