# Lesson 2

Equivalent Decimals

## Warm-up: True or False: Equivalent Fractions (10 minutes)

### Narrative

The purpose of this True or False is to revisit equivalent fractions in tenths and hundredths. The reasoning students do here will be helpful later when students make sense of and identify decimals that are equivalent to given fractions or given decimals.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each statement.

### Student Facing

Decide whether each statement is true or false. Be prepared to explain your reasoning.

• $$\frac{50}{100} = \frac{5}{10}$$
• $$\frac{20}{10} = \frac {20}{100}$$
• $$2 = 1 + \frac{90}{100}$$
• $$3\frac{1}{10} = \frac{31}{10}$$

### Activity Synthesis

• “What do you know about the relationship of tenths and hundredths that helped you decide whether each statement is true or false?” (Sample responses:
• One tenth is 10 hundredths.
• One tenth is 10 times 1 hundredth.
• There are 10 tenths in 1 whole.
• There are 100 hundredths in 1 whole.
• If we multiply the numerator and denominator of a fraction in tenths by 10, we get an equivalent fraction in hundredths.)

## Activity 1: Card Sort: Diagrams of Fractions and Decimals (15 minutes)

### Narrative

In this activity, students reinforce their understanding of equivalent fractions and decimals by sorting a set of cards by their value. The cards show fractions, decimals, and diagrams. A sorting task gives students opportunities to analyze different representations closely and make connections (MP2, MP7).

Representation: Access for Perception. Synthesis: Display a 10-by-10 grid, as well as a square of the exact same size, but with only the columns shown (therefore representing just tenths). Shade 20 hundredths on the 10-by-10 grid and write 0.20 (twenty hundredths) above it. Shade 2 tenths on the other square and write 0.2 (2 tenths) above it. Invite students to discuss how these diagrams demonstrate equivalence of the two numbers.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Required Materials

Materials to Copy

• Card Sort: Diagrams of Fractions & Decimals

### Required Preparation

• Create a set of cards from the blackline master for each group of 2–4.

### Launch

• Groups of 2–4
• Give one set of cards from the blackline master to each group.

### Activity

• “Work with your group to sort the set of cards by their value.”
• “One diagram has no matching cards. Write the fraction and decimal it represents.”
• 6–7 minutes: group work on the first two problems
• Monitor for the ways students sort the cards and the features of the representations to which they attend.
• “Work on the last problem independently.”
• 2–3 minutes: independent work on the last problem

### Student Facing

Your teacher will give you a set of cards. Each large square on the cards represents 1.

1. Sort the cards into groups so that the representations in each group have the same value. Record your sorting decisions. Be prepared to explain your reasoning.
2. One of the diagrams has no matching fraction or decimal. What fraction and decimal does it represent?
3. Are 0.20 and 0.2 equivalent? Use fractions and a diagram to explain your reasoning.

### Student Response

Students may respond that 0.20 and 0.2 are not “the same.” Consider asking:

• “How would you represent each number on a square grid?”
• “What is the same about the amounts and what is not the same?”

### Activity Synthesis

• Select one group to share each set of sorted cards and explain how they knew the representations belong together.
• “How did you know what fraction and decimal to write for the diagram without any matches?”
• Select a student to share their response to the last problem. Highlight the equivalence of 0.2 and 0.20 as shown in the Student Responses.

## Activity 2: True or Not True? (20 minutes)

### Narrative

In this activity, students apply their understanding of equivalent fractions and decimals more formally, by analyzing equations and correcting the ones that are false. The last question refers to decimals on a number line and sets the stage for the next lesson where the primary representation is the number line.

As students discuss and justify their decisions about the claim in the last question, they critically analyze student reasoning (MP3).

This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing

### Launch

• Groups of 2
• “Earlier, we saw some equations with fractions on both sides of the equal sign. Now let’s look at some equations that include fractions and decimals or just decimals.”

### Activity

• “Take a few minutes to complete the activity independently. Then, share your thinking with your partner.”
• 6–7 minutes: independent work time
• “For each equation in the first problem, take turns explaining to your partner how you know whether it is true or false.”
• 3–4 minutes: partner discussion

### Student Facing

1. Decide whether each statement is true or false. For each statement that is false, replace one of the numbers to make it true. (The numbers on the two sides of the equal sign should not be identical.) Be prepared to share your thinking.

1. $$\frac{50}{100} = 0.50$$

2. $$0.05 = 0.5$$

3. $$0.3 = \frac{3}{10}$$

4. $$0.3 = \frac{30}{100}$$

5. $$0.3 = 0.30$$

6. $$1.1 = 1.10$$

7. $$3.06 = 3.60$$

8. $$2.70 = 0.27$$

2. Jada says that if we locate the numbers 0.05, 0.5, and 0.50 on the number line, we would end up with only two points. Do you agree? Explain or show your reasoning.

### Student Response

Students may be unsure about how to locate 0.05 on a number line. Ask them how they would express the number in words and in fraction notation (in tenths or hundredths). Consider asking them to also name each tick mark on the number line. “If the space between two tick marks represents 10 hundredths, where might 5 hundredths land on the line?”

### Activity Synthesis

MLR1 Stronger and Clearer Each Time
• “Share your response to the last question with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–4 minutes: structured partner discussion.
• Repeat with 1–2 new partners.
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

## Lesson Synthesis

### Lesson Synthesis

“Today we looked at different ways to represent decimals that are equivalent. We used square grids, number lines, and fractions to show that two decimals can represent the same value.”

“Suppose a classmate is absent today. How would you convince them that 0.3 and 0.30 are equivalent? Write down at least two different ways.”

Select students to share their thinking. Display the representations they used, or display the following:

“0.3 is 3 tenths and 0.30 is 30 hundredths. The same shaded part represents 3 tenths and 30 hundredths.”

“Both 3 tenths and 30 hundredths share the same point on the number line.”

“0.3 is $$\frac{3}{10}$$ and 0.30 is $$\frac{30}{100}$$. The two fractions are equivalent.”

$$\frac {3 \ \times \ 10}{10 \ \times \ 10}=\frac{30}{100}$$

$$\frac{3}{10} = \frac{30}{100}$$