# Lesson 1

¿Qué es una milésima?

## Warm-up: Exploración de estimación: Una pequeña pieza (10 minutes)

### Narrative

The purpose of this Estimation Exploration is to invite students to think about small fractions of a quantity in context. The mosaic pictured here is made up of many small square tiles. They are arranged in a complex pattern and are not identical in size but students can relate the denominator of a fraction giving the size of each tile, relative to the whole mosaic, to the total number of tiles making the mosaic. This helps them think of a fraction with a large denominator which prepares them to think about the fraction $$\frac{1}{1,000}$$ in this lesson.

### 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.”
• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Escribe una estimación que sea:

muy baja razonable muy alta
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Activity Synthesis

• “¿Cómo hicieron su estimación?” // “How did you make your estimate?” (I tried to estimate out how many tiles made the whole picture and then used that as my denominator.)
• “En varias de las siguientes lecciones vamos a examinar números muy pequeños y a descubrir cómo representarlos” // “We will be investigating very small numbers and how to represent them in the next several lessons.”

## Activity 1: ¿Qué sabes sobre los milésimos? (20 minutes)

### Narrative

The purpose of this activity is for students to share what they know about one tenth and one hundredth, and consider what they might know about one thousandth. Students make a poster showing what they know about these numbers and then discuss different representations they made. If students show tenths, hundredths or thousandths on a number line or with base ten diagrams, highlight these representations in the synthesis, as they are familiar from grade 4.

This activity is meant to be an invitational opportunity for students to bring their lived experience into the math classroom. Consider taking a walk through the community where your students live and noticing places that decimals are seen and used. Take pictures or notes that capture the details of your observations. Be prepared to share these artifacts with students during the synthesis.

Representation: Develop Language and Symbols. Synthesis: Maintain a visible display to record new vocabulary. Invite students to suggest details (words or pictures) that will help them remember the meaning of thousandths and the thousandths' connection to tenths and hundredths.
Supports accessibility for: Conceptual Processing, Memory

### Required Materials

Materials to Gather

Materials to Copy

• Small Grids

### Launch

• Groups of 2
• Give students access to chart paper and colored pencils, crayons, or markers.
• “Con su compañero, hagan un póster que muestre lo que saben sobre estos números: 1 décimo, 1 centésimo y 1 milésimo” // “Work with your partner to make a poster showing what you know about the numbers 1 tenth, 1 hundredth, and 1 thousandth.”

### Activity

• 6-8 minutes: partner work time
• Monitor for students who:
• represent the numbers with fractions
• represent the numbers with decimals
• represent the numbers with diagrams

### Student Facing

1. ¿Qué sabes sobre 1 décimo?
2. ¿Qué sabes sobre 1 centésimo?
3. ¿Qué sabes sobre 1 milésimo?

### Activity Synthesis

• Invite students to share their posters.
• “¿Cuáles son algunas maneras de representar 1 décimo?” // “What are some ways to represent 1 tenth?” (I can write it as a fraction or a decimal. I can divide a rectangle into 10 equal pieces. I can put it on the number line.)
• “¿Qué fue retador al representar 1 centésimo con un dibujo o con un diagrama?” // “What was challenging about representing 1 hundredth with a drawing or diagram?” (Answers vary. It was hard to divide something into 100 equal pieces because that’s a lot.)
• “¿Qué fue retador al representar 1 milésimo con un dibujo o con un diagrama?” // “What was challenging about representing 1 thousandth with a drawing or diagram?” (Dividing a rectangle into 1,000 pieces would take forever.)
• “En la siguiente actividad vamos a hacer y a comparar dibujos de estos números” // “In the next activity we will make and compare drawings of these numbers.”

## Activity 2: Representemos números en una cuadrícula de centésimas (15 minutes)

### Narrative

The purpose of this activity is for students to represent tenths, hundredths, and thousandths with diagrams and decimals. The diagrams highlight the relationships between these quantities:

• there are 10 tenths in a whole
• there are 10 hundredths in a tenth
• there are 10 thousandths in a hundredth

This relationship can be seen when numbers are written as decimals or fractions. When students see this common relationship between the decimal place values they look for and make use of structure (MP7).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “¿Qué tipos de lenguaje o de detalles adicionales les ayudaron a entender las presentaciones?” // “What kinds of additional details or language helped you understand the displays?”, “¿Tienen preguntas sobre el lenguaje o sobre algún otro detalle?” // “Were there any additional details or language that you have questions about?”, and “¿Alguien resolvió el problema de la misma manera, pero lo explicaría de otra forma?” // “Did anyone solve the problem the same way, but would explain it differently?”

### Launch

• Groups of 2
• “Hoy vamos a examinar algunos números realmente pequeños” // “Today we are going to investigate some really small numbers.”

### Activity

• 2 minutes: independent work time
• 8 minutes: partner work time

### Student Facing

Prepárate para explicar tu razonamiento.

Prepárate para explicar tu razonamiento.

Explica o muestra cómo pensaste.

4. Fracción Decimal
$$\frac{1}{10}$$ 0.1
$$\frac{1}{100}$$ 0.01
$$\frac{1}{1,000}$$ ?

¿Cómo crees que escribimos el número ‘un milésimo’ como un número decimal? Explica tu razonamiento.

### Student Response

If students do not explain how many small rectangular pieces are in the whole unit square, ask: “¿Cuántas partes pequeñas rectangulares habría en una fila del cuadrado unitario?” // “How many small rectangular pieces would be in one row of the unit square?”

### Activity Synthesis

• Display the last image.
• “¿Cuántos rectángulos pequeños sombreados hay en todo el cuadrado unitario? ¿Cómo lo saben?” // “How many of the tiny shaded rectangles are there in the whole unit square? How do you know?" (1,000 because there are 10 in the small square and 100 of those squares in the whole.)
• “De todo el cuadrado, ¿cuánto está sombreado?” // “How much of the whole square is shaded?” (There is one tiny rectangle shaded and there are 1,000 of those in the whole so $$\frac{1}{1,\!000}$$ is shaded. There is $$\frac{1}{10}$$ of $$\frac{1}{100}$$ shaded. That’s $$\frac{1}{10} \times \frac{1}{100}$$. There is $$\frac{1}{100}\div10$$ because the hundredth is divided into 10 equal pieces.)
• “El número $$\frac{1}{1,\!000}$$ también se puede escribir en forma decimal como 0.001. De forma similar a la fracción, al número decimal lo llamamos ‘milésima’” // “The number $$\frac{1}{1,\!000}$$ can also be written in decimal form as 0.001. Like the fraction, we call it ‘thousandth.’”
• “¿Cómo piensan que se podría escribir $$\frac{4}{1,\!000}$$ como un número decimal?” // “How do you think you would write $$\frac{4}{1,\!000}$$ as a decimal?” (0.004)

## Lesson Synthesis

### Lesson Synthesis

“Hoy representamos 1 décimo, 1 centésimo y 1 milésimo de distintas formas. ¿De qué formas pueden representar 1 centésimo?” // “Today we represented 1 tenth, 1 hundredth, and 1 thousandth in different ways. What are some different ways that you can represent 1 hundredth?” (as a fraction $$\frac{1}{100}$$, as a decimal 0.01, or with a drawing)

“¿De qué formas pueden representar 1 milésimo?” // “What are some different ways that you can represent 1 thousandth?” ($$\frac{1}{1,\!000}$$, 0.001, or with a drawing, but it’s so small and there are so many of them in the whole that the drawing is not that helpful)