# Lesson 4

Compare and Order Decimals

## Warm-up: Estimation Exploration (10 minutes)

### Narrative

The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In this case, the given decimal pushes students to think in terms of increments of tenths (0.1) and to relate the fractional measurement to nearby whole numbers.

### 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

The person in the image is 1.7 meters tall.

Estimate the wingspan of the eagle in meters.

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

• “Why might 1.8 meters be too low of an estimate?”
• “Where might a height of 1 meter be on the image of the person? On the image of the eagle?”
• “Is anyone’s estimate less than 2? Is anyone’s estimate greater than 3?”
• “Based on this discussion does anyone want to revise their estimate?”

## Activity 1: All in Order (15 minutes)

### Narrative

This activity prompts students to apply what they know about tenths and hundredths and decimal notation to arrange two sets of numbers in order, first from least to greatest, and then the other way around.

A number line is given here, but students are likely to start seeing its limits as a tool for comparing and ordering decimals. It takes time to plot each value on the number line, the scale of the number line accommodates only a small range of numbers (numbers like 1.25 and 12.05 would go beyond the line), and there are other ways to discern how two decimals compare—by reasoning about the name of the decimals in tenths and hundredths, and by relating to benchmarks such as whole numbers and 5 tenths (0.5, 1.5, 2.5, and so on).

### Launch

• Groups of 2
• Display the six decimals in the first problem.
• “How do we name these decimals in terms of tenths and hundredths? Let’s read each one aloud.”
• Display the six decimals in the second problem.
• “Take turns reading each decimal with your partner. Name them in terms of tenths and hundredths.”
• 1 minute: partner work time

### Activity

• “Take a few quiet minutes to complete the activity. Then, share your responses with your partner.”
• 5 minutes: independent work time
• 2–3 minutes: partner work time
• Monitor for students who order the decimals by:
• plotting the decimals on the number line
• using and comparing the word names of the decimals
• relating each decimal to benchmarks such as 0, 0.5, and 1
• Ask them to share their strategies during the synthesis, in the order as shown.

### Student Facing

1. Order the numbers from least to greatest. Use the number line if it is helpful.

1.08

0.08

0.80

0.9

0.45

0.54

2. Order the numbers from greatest to least. Use the number line if it is helpful.

1.25

0.95

0.4

0.09

12.05

0.25

### Student Response

Students may arrange the numbers by looking only at the digits in the numbers, without attending to the relative sizes of each decimal. (For example, they may say that 0.45 is greater than 0.9 because 45 is greater than 9.) Consider asking them to name the numbers and think about them in terms of tenths and hundredths, or to express them in fraction notation.

### Activity Synthesis

• Select previously identified students to share their responses and reasoning.
• “After seeing these strategies, which one(s) do you prefer to use for ordering decimals? Why?”

## Activity 2: 400-Meter Dash in a Flash (20 minutes)

### Narrative

In this activity, students compare and order decimals in the context of running times. Unlike in preceding activities, in which most decimals they encountered were less than one or were in the low ones, here the numbers all have two-digit whole numbers, prompting students to be more attentive to the place value of the digits. The context of track and field may be unfamiliar, so time is built into the launch for orienting students and for supporting them in making sense of the problem.

When students look carefully at the meaning of each digit in the numbers and interpret them in terms of the running context they are reasoning abstractly and quantitatively and observing place value structure (MP2, MP7).

Representation: Access for Perception. Begin by showing a video of an Olympic Women’s 400-Meter final event to support both engagement and understanding of the context. To emphasize the relative magnitude of decimals in this context, invite students to attend to the running clock, the moment when the athletes cross the finish line, and the table of final results. Ask, “Why are decimals important here?”
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• Groups of 2
• Display a picture of a standard 400-meter running track.
• “How long do you think it would take you to run a lap, about 400 meters? Think about it for a moment, and then share your estimate with your partner.”
• 1 minute: partner discussion
• Explain that in track and field, runners compete to run different distances: 100 meters, 200, 400, 800, and more. The United States, Jamaica, and the Bahamas have produced some of the fastest track runners in the world.
• Display only the opening paragraph, the eight running times, and the table, without revealing the questions.
• “We are going to read this problem 3 times.”
• 1st Read: “The table shows eight of the top runners in the Women’s 400-Meter event. Their best running times, listed here, put the runners in the world’s top 25 in this event.”
• “What is this story about?”
• 1 minute: partner discussion
• Listen for and clarify any questions about the context.
• “Name the quantities. What can we count or measure in this situation?” (times in seconds, years)
• 30 seconds: quiet think time
• Share and record all quantities.
• Reveal the questions.
• “How might we go about matching the times to the right runners?” (Arrange the times in order, from shortest to longest.)

### Activity

• “Work with your partner to complete the activity.”
• 6–8 minutes: partner work time

### Student Facing

The table shows eight of the top runners in the Women’s 400-Meter event. Their best running times, listed here, put the runners in the world’s top 25 for this event.

48.37

49.3

48.7

49.26

49.07

49.28

48.83

49.05

The names in the table are arranged by the runners’ best time. The fastest runner is at the top.

runner best time (seconds) year achieved
Shaunea Miller-Uibo (Bahamas) 2019
Sanya Richards (U.S.A.) 2006
Valerie Brisco-Hooks (U.S.A.) 1984
Chandra Cheesborough (U.S.A.) 1984
Tonique Williams-Darling (Bahamas) 2004
Allyson Felix (U.S.A.) 2015
Pauline Davis (Bahamas) 1996
Lorraine Fenton (Jamaica) 2002
1. Put the times in order, from least to greatest, to match the times with the runners.

2. How many seconds did it take Sanya Richards to run 400 meters?

3. What is Allyson Felix’s best time?

### Activity Synthesis

• Display the table from the activity.
• Invite students to share their ordered list and discuss how they went about arranging the numbers.
• Highlight explanations that are based on place-value reasoning or on understanding of tenths and hundredths.

## Lesson Synthesis

### Lesson Synthesis

“Today we compared decimals and put them in order by their size.”

Display these decimals with missing digits:

$$\boxed{0} \ . \boxed{\phantom{0}}$$

$$\boxed{0} \ . \ \boxed{1} \ \boxed{\phantom{0}}$$

$$\boxed{1} \ \boxed{\phantom{0}} \ . \ \boxed{\phantom{0}} \ \boxed{\phantom{0}}$$

$$\boxed{2} \ . \ \boxed{\phantom{0}}$$

$$\boxed{\phantom{0}} \ . \ \boxed{2}$$

“Are there numbers that we can compare, even though they are all missing digits?” (Yes, we know 1__.__ __ is greater than all the others and 2. __ is greater than 0.__ and 0.1__.)

“Are there numbers that we can’t compare?” (0.__, 0.1__, and __.2)

“What makes it possible for us to compare some decimals but not others?” (Sample responses:

• We know that a number with tens is greater than numbers with only ones.
• We can compare numbers that are greater than 1 and those less than 1.
• We can’t compare numbers when the digit in the place with the largest value is not known.)