Lesson 5
Compare and Order Decimals and Fractions
Warmup: 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
 Record answers and strategy.
 Keep expressions and work displayed.
 Repeat with each expression.
Student Facing
Find the value of each expression mentally.
 \(\frac{5}{10} + \frac{50}{100}\)
 \(\frac{5}{10} + \frac{55}{100}\)
 \(\frac{6}{10} + \frac{50}{100}\)
 \(\frac{6}{10} + \frac{65}{100}\)
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 Consider asking:
 “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  \(\frac{24}{100}\)  2.40  2.04  \(1\frac{4}{100}\)  1.24 
Advances: Conversing, Representing
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.
 Work with your group to order the numbers from least to greatest. Record your ordered numbers.
 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.

Use the numbers from your sorted set and <, >, or = symbols to create true comparison statements:

\(\underline{\hspace{0.5in}} < \underline{\hspace{0.5in}}\)

\(\underline{\hspace{0.5in}} > \underline{\hspace{0.5in}}\)

\(\underline{\hspace{0.5in}} < \underline{\hspace{0.5in}}\)

\(\underline{\hspace{0.5in}} > \underline{\hspace{0.5in}}\)

Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
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?”
 “Think about it for a moment, and then share your estimate with your partner.”
 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 100meter dash, 200meter dash, and long jump.
Here are some of his longjump 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 
 On this list, which distance is his shortest jump? Which is his best (longest) jump?

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: \(8\frac{95}{100}\)
Student Response
For access, consult one of our IM Certified Partners.
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.
Cooldown: Order Up (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.
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.