# Lesson 5

Compare and Order Decimals and Fractions

## Warm-up: Number Talk: Sums of Fractions (10 minutes)

### Narrative

This Number Talk encourages students to rely on what they know about tenths and hundredths and about equivalent fractions to mentally solve problems. The reasoning elicited here will be helpful later in the lesson when students compare and order fractions and decimals.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$\frac{5}{10} + \frac{50}{100}$$
• $$\frac{5}{10} + \frac{55}{100}$$
• $$\frac{6}{10} + \frac{50}{100}$$
• $$\frac{6}{10} + \frac{65}{100}$$

### Activity Synthesis

• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Order Once, Order Twice (25 minutes)

### Narrative

In this activity, students encounter both fraction and decimal notation for tenths and hundredths and are asked to arrange them in order by size. To do so, they need to rely on their knowledge of equivalent fractions and of the relationship between these two ways of expressing values. Students look for and make use of structure (MP7), for instance, by identifying the digits that tell us about the ones, tenths, and hundredths in each number.

 Set A $$1\frac{6}{10}$$ 1.06 2.6 $$\frac{116}{100}$$ 0.96 Set B $$\frac{24}{100}$$ 2.4 2.04 $$1\frac{4}{100}$$ 1.24
MLR8 Discussion Supports. Students should take turns placing a card in order and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____ , so I put . . .” Encourage students to challenge each other when they disagree.
Action and Expression: Develop Expression and Communication. Synthesis: Identify connections between strategies that result in the same outcomes but use differing approaches.
Supports accessibility for: Memory

### Required Materials

Materials to Copy

• Order Once, Order Twice

### Required Preparation

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

### Launch

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

### Activity

• “Work with your group to put the fractions and decimals in order, from least to greatest. Record your ordered set.”
• 4–5 minutes: group work time
• “Next, find a group with a set of cards different than yours. Put all the numbers in order, from least to greatest. Record your ordered set.”
• 8–10 minutes: group work time
• Monitor for the ways students compare fractions and decimals.
• “Complete the last problem independently.”
• 3–4 minutes: independent work time

### Student Facing

Your teacher will give you a set of cards with fractions and decimals.

1. Work with your group to order the numbers from least to greatest. Record your ordered numbers.
2. Find a group whose cards are different than yours. Combine your cards with theirs. Order the combined set from least to greatest. Record your sorted numbers.
3. Use the numbers from your sorted set and <, >, or = symbols to create true comparison statements:

1. $$\underline{\hspace{0.5in}} < \underline{\hspace{0.5in}}$$

2. $$\underline{\hspace{0.5in}} > \underline{\hspace{0.5in}}$$

3. $$\underline{\hspace{0.5in}} < \underline{\hspace{0.5in}}$$

4. $$\underline{\hspace{0.5in}} > \underline{\hspace{0.5in}}$$

### Student Response

Students may choose to order just the fractions and order all the decimals separately. Consider asking: “How might you order both the fractions and decimals?” and “Which decimals would be helpful to think about as fractions (or the other way around)?”

### Activity Synthesis

• Select students to share their ordered collection of 10 cards. Invite the class to agree or disagree with the arrangement.
• “What was the first thing you did or looked at to start ordering? What was the next thing? What came after that?” (We first looked at the digit in the ones place. Next, we decided to write the decimals with the same whole number as fractions and ordered the fractions.)

## Activity 2: Long Jumps (10 minutes)

### Narrative

In this activity, compare and order decimals and fractions to solve problems about distances. As they do so, they practice reasoning about tenths and hundredths expressed in different notations. Some of the distances are written to the tenths of a meter and others are written to the hundredths, prompting students to attend to the size of the decimals.

When students interpret and order the distances, they reason abstractly and quantitatively (MP2).

### Launch

• Groups of 2
• “How far do you think you could jump if you run really fast to gain speed for the jump? Could you jump from one side of the classroom to the other?”
• 1 minute: partner discussion
• Familiarize students with the long jump in track and field. Explain that the best long jumpers in the world, including Carl Lewis, can jump more than 8 meters or more than 26 feet.
• Consider showing a video clip of long jumps.

### Activity

• “Take a few minutes to work on the task. Then, share your responses with your partner.”
• 6–7 minutes: independent work time
• 3–4 minutes: partner discussion
• Monitor for the ways students compare and order decimals and mixed numbers in the last problem.

### Student Facing

American athlete Carl Lewis won 10 Olympic medals and 10 World Championships in track and field—in 100-meter dash, 200-meter dash, and long jump.

Here are some of his long-jump records from his career:
year distance (meters)
1979 8.13
1980 8.35
1982 8.7
1983 8.79
1984 8.24
1987 8.6
1991 8.87
1. On this list, which distance is his shortest jump? Which is his best (longest) jump?
2. Here are the top distances (in meters) of three other American long jumpers:

• Bob Beamon: $$8\frac{9}{10}$$
• Jarrion Lawson: $$8\frac{58}{100}$$
• Mike Powell: $$8\frac{95}{100}$$
Compare their records to Carl Lewis’s best jump. Order the distances from greatest to least.

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we compared tenths and hundredths written as both fractions and decimals.”

“How did you compare Carl Lewis’s best jump with those of the other jumpers and put the numbers in order?” (First, I wrote Carl Lewis’s time as a fraction in hundredths, $$8\frac{87}{100}$$. The one fraction in tenths can be written as $$\frac{90}{100}$$. All the numbers have 8 ones, so we ignored it and compared the hundredths.)

If time permits, invite students to share a general process for comparing any set of tenths and hundredths written in fraction and decimal notation.

## Student Section Summary

### Student Facing

In this section, we learned to express tenths and hundredths as decimals, locate them on a number line, and compare them.

We learned that $$\frac{1}{10}$$ written as a decimal is 0.1, and that this number is also read “1 tenth.” $$\frac{1}{100}$$ written as a decimal is 0.01 and is read “1 hundredth.”

The table shows some more examples of tenths and hundredths in their decimal notation. ​​​​

• Because $$\frac{5}{10}$$ and $$\frac{50}{100}$$ are equivalent, the decimals 0.5 and 0.50 are also equivalent.
• Likewise, $$\frac{17}{10}$$ and $$\frac{170}{100}$$ are equivalent, so 1.7 and 1.70 are also equivalent.
fraction decimal
$$\frac{4}{100}$$ 0.04
$$\frac{23}{100}$$ 0.23
$$\frac{5}{10}$$ 0.5
$$\frac{50}{100}$$ 0.50
$$\frac{17}{10}$$ 1.7
$$\frac{170}{100}$$ 1.70

Just like fractions, decimals can be located on a number line. Doing so can help us compare them.

For instance, 0.24 is equivalent to $$\frac{24}{100}$$, which is between $$\frac{20}{100}$$ and $$\frac{30}{100}$$ (or between $$\frac{2}{10}$$ and $$\frac{3}{10}$$) on the number line. We can see that 0.24 is greater than 0.08 and less than 0.61.