# Lesson 16

Compare and Order Fractions

## Warm-up: Number Talk: Multiples of 6 and 12 (10 minutes)

### Narrative

This Number Talk encourages students to think about multiples of 5, 6, and 12—numbers that students will see as denominators later in the lesson. It also prompts students to rely on doubling and on properties of operations to mentally solve multiplication problems. The reasoning elicited here will be helpful later in the lesson when students compare fractions by finding equivalent fractions with a common denominator.

To find products by doubling or by using properties of operations, students need to look for and make use of structure (MP7).

### 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.

• $$5 \times 6$$
• $$5 \times 12$$
• $$6 \times 12$$
• $$11 \times 12$$

### Activity Synthesis

• “How do the first three expressions help you find the value of the last one?”

## Activity 1: Compare Fractions Game (20 minutes)

### Narrative

This activity allows students to practice comparing fractions and apply the comparison strategies they learned through a game. Students use fraction cards from an earlier lesson to play a game in groups of 2, 3, or 4. To win the game is to have the greater (or greatest) fraction of the cards played as many times as possible. This is stage 5 of the Compare center.

Consider arranging students in groups of 2 for the first game or two (so that students would need to compare only 2 fractions at a time), and arranging groups of 3 or 4 for subsequent games. Before students begin playing, ask them to keep track of and record pairs of fractions that they find challenging to compare.

MLR8 Discussion Supports. Students should take turns explaining their reasoning to their partner. Display the following sentence frames for all to see: “_____ is greater than _____ because  . . .”, and “_____ and _____ are equivalent because . . . .” Encourage students to challenge each other when they disagree.

### Required Materials

Materials to Copy

• Compare Stage 3-8 Directions

### Required Preparation

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

### Launch

• Groups of 2–4
• Give each group a set of fraction cards.
• Tell students that they will play one or more games of Compare Fractions.
• Demonstrate how to play the game. Invite a student to be your opponent in the demonstration game.
• Read the rules as a class and clarify any questions students might have.
• Groups of 2 for the first game or two, then groups of 3–4 for subsequent games, if time permits

### Activity

• “Play one game with your partner.”
• “As you play, you may come across one or more sets of fractions that are tricky to compare. Record those fractions. Be prepared to explain how you eventually figure out which fraction is greater.”
• “If you finish before time is up, play another game with the same partner, or play a game with the players from another group.”
• 15 minutes: group work time

### Student Facing

Play Compare Fractions with 2 players:

• Split the deck between the players.
• Each player turns over a card.
• Compare the fractions. The player with the greater fraction keeps both cards.
• If the fractions are equivalent, each player turns over one more card. The player with the greater fraction keeps all four cards.
• Play until you run out of cards. The player with the most cards at the end of the game wins.

Play Compare Fractions with 3 or 4 players:

• The player with the greatest fraction wins the round.
• If 2 or more players have the greatest fraction, those players turn one more card over. The player with the greatest fraction keeps all the cards.

Record any sets of fractions that are challenging to compare here.

_________ and _________

_________ and _________

_________ and _________

_________ and _________

### Activity Synthesis

• Invite groups to share some of the challenging sets of fractions they recorded and how they eventually determined the greater one in each pair.
• As one group shares, ask others if they have other ideas about how the fractions could be compared.

## Activity 2: Fractions in Order (15 minutes)

### Narrative

This activity prompts students to compare multiple fractions and put them in order by size. The work gives students opportunities to look for and make use of structure (MP7) in each set of fractions and make comparisons strategically. For instance, rather than comparing two fractions at a time and in the order they are listed, students could first classify the given fractions as greater or less than $$\frac{1}{2}$$ or 1, look for fractions with a common numerator or denominator, and so on.

If time is limited, consider asking students to choose two sets of fractions to compare and order.

Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most useful when putting fractions in order. Display the sentence frame, “The next time I put fractions in order, I will pay attention to . . . .”
Supports accessibility for: Memory

• Groups of 2

### Activity

• “Work independently on two sets. Then, discuss your work with your partner and complete the rest together.”
• 10 minutes: independent work time
• Monitor for students who look for and make use of structure. Ask them to share during lesson synthesis.
• 3–4 minutes: partner discussion

### Student Facing

Put each set of fractions in order, from least to greatest. Be prepared to explain your reasoning.

1. $$\frac{3}{12} \qquad \frac{2}{4} \qquad \frac{2}{3} \qquad \frac{1}{8}$$
2. $$\frac{8}{5} \qquad \frac{5}{6} \qquad \frac{11}{12} \qquad \frac{11}{10}$$
3. $$\frac{21}{20} \qquad \frac{9}{10} \qquad \frac{6}{5} \qquad \frac{101}{100}$$
4. $$\frac{5}{8} \qquad \frac{2}{5} \qquad \frac{3}{7} \qquad \frac{3}{6}$$

### Student Response

Some students may try to write equivalent fractions with a common denominator for all four fractions in each set before comparing them but may be unable to do so. Encourage them to try reasoning about two fractions at a time, and to use what they know about the fractions to determine how they compare (to one another or to familiar benchmarks).

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

Invite students to share their strategies for comparing and ordering the fractions in the last activity. Record their responses.

Ask students to reflect on their understanding of fractions in this unit.

“What are some things about writing, representing, or comparing fractions that you didn’t know at the beginning of the unit but you know quite well now? Think of at least two specific things.”

## Student Section Summary

### Student Facing

In this section, we compared fractions using what we know about the size of fractions, benchmarks such as $$\frac12$$ and 1, and equivalent fractions. For example, to compare $$\frac38$$ and $$\frac{6}{10}$$, we can reason that:

• $$\frac{4}{8}$$ is equivalent to $$\frac12$$, so $$\frac38$$ is less than $$\frac12$$.
• $$\frac{5}{10}$$ is equivalent to $$\frac12$$, so $$\frac{6}{10}$$ is more than $$\frac12$$.

This means that $$\frac{6}{10}$$ is greater than $$\frac38$$ (or $$\frac38$$ is less than $$\frac{6}{10}$$).

We can also compare by writing equivalent fractions with the same denominator. For example, to compare $$\frac{3}{4}$$ and $$\frac{4}{6}$$, we can use 12 as the denominator:

$$\frac{3}{4} = \frac{9}{12} \hspace{2cm} \frac{4}{6} = \frac{8}{12}$$

Because $$\frac{9}{12}$$ is greater than $$\frac{8}{12}$$, we know that $$\frac{3}{4}$$ is greater than $$\frac{4}{6}$$.