# Lesson 2

Thousandths on Grids and in Words

## Warm-up: Estimation Exploration: What Part of the Square is Shaded? (10 minutes)

### 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.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Record responses.

### Student Facing

How much of the square is shaded?

Record an estimate that is:

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

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “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: Represent Thousandths on a Grid (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: “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
• “Today we are going to represent decimal numbers with diagrams.”
• “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

1. Shade each grid to represent the given number.

2. For each diagram, write a decimal number to represent how much is shaded. Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

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

• “How does the diagram represent the number you wrote?“
• “How does the diagram represent each of the digits in the number you wrote?“

### Activity Synthesis

• Display the diagram that shows 0.625.
• “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.
• “Where do you see 0.6 in the diagram?” (There are 6 rows shaded and each row is 0.1 or a tenth.)
• “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.)
• “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: Say What? (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

Several students look at the diagram and describe the shaded region in different ways. Who do you agree with? Why?

1. Jada says it’s “15 hundredths.”
2. Priya says it’s “150 thousandths.”
3. Tyler says it’s “15 thousandths.”
4. Diego says it’s “1 tenth and 5 hundredths.”
5. Mai says it’s “1 tenth and half of a tenth.”

### Student Response

For access, consult one of our IM Certified Partners.

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, “How does the description represent the hundredths grid?” Then, refer to the incorrect description and ask, “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

“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.

“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.

## Cool-down: Shading Thousandths (5 minutes)

### Cool-Down

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