# Lesson 1

What is One Thousandth?

## Warm-up: Estimation Exploration: One Tiny Piece (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.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

### Activity

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

### Student Facing

What fraction of the whole picture is a single square tile?

Record an estimate that is:

too low about right too high
$$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$ $$\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}$$

### Activity Synthesis

• “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.)
• “We will be investigating very small numbers and how to represent them in the next several lessons.”

## Activity 1: What Do You Know About Thousandths? (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.
• “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. What do you know about 1 tenth?
2. What do you know about 1 hundredth?
3. What do you know about 1 thousandth?

### Activity Synthesis

• Invite students to share their posters.
• “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.)
• “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.)
• “What was challenging about representing 1 thousandth with a drawing or diagram?” (Dividing a rectangle into 1,000 pieces would take forever.)
• “In the next activity we will make and compare drawings of these numbers.”

## Activity 2: Represent Numbers on a Hundredths Grid (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, “What kinds of additional details or language helped you understand the displays?”, “Were there any additional details or language that you have questions about?”, and “Did anyone solve the problem the same way, but would explain it differently?”

### Launch

• Groups of 2
• “Today we are going to investigate some really small numbers.”

### Activity

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

### Student Facing

1. The grid represents 1. What does the shaded region represent?

Be prepared to explain your reasoning.

2. The grid represents 1. What does the shaded region represent?

Be prepared to explain your reasoning.

3. How many of the small rectangular pieces (one of them is shaded) are there in the unit square?

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

How do you think we write the number one thousandth as a decimal? Explain your reasoning.

### Student Response

If students do not explain how many small rectangular pieces are in the whole unit square, ask: “How many small rectangular pieces would be in one row of the unit square?”

### Activity Synthesis

• Display the last image.
• “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.)
• “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.)
• “The number $$\frac{1}{1,\!000}$$ can also be written in decimal form as 0.001. Like the fraction, we call it ‘thousandth.’”
• “How do you think you would write $$\frac{4}{1,\!000}$$ as a decimal?” (0.004)

## Lesson Synthesis

### Lesson Synthesis

“ 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)

“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)