# Lesson 9

Differences of Fractions

## Warm-up: True or False: Sums of Tenths (10 minutes)

### Narrative

This warm-up prompts students to reason about sums of fractions with the same denominator and to apply their understanding of equivalence, especially of whole numbers and fractions. The reasoning here will be helpful as students explore subtraction of fractions later in the lesson.

### 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 if each statement is true or false. Be prepared to explain your reasoning.

• $$\frac{1}{10}+\frac{2}{10}+\frac{3}{10}=1$$
• $$1+\frac{7}{10}=\frac{3}{10} + \frac{4}{10} + \frac{10}{10}$$
• $$\frac{5}{10}+1=\frac{6}{10}$$
• $$\frac{2}{10}+\frac{10}{10}= 1 + \frac {1}{5}$$

### Activity Synthesis

• “How can you explain your answer without finding the value of both sides?”
• “Who can restate ___ ’s reasoning in a different way?”
• “Does anyone want to add on to _____ ’s reasoning?”

### Narrative

In this activity, students reason about differences of fractions on a number line and write equations for number line diagrams that represent subtraction. They subtract a fraction from another fraction, as well as a whole number from a fraction, applying what they know about equivalence of whole numbers and fractions to facilitate their reasoning. When students decide whether or not they agree with Noah and explain their reasoning, they critique the reasoning of others (MP3).

In earlier grades, students used number lines to reason about subtraction of whole numbers. To find the value of $$42 - 15$$, for example, they could start at 42 and jump 15 spaces to the left (or jump 2 spaces to 40, and then 10 spaces to 30, and 3 more to 27). They could also think in terms of addition—“What number added to 15 gives 42?”—and start at 15 and see how many spaces it takes to get to 42.

Students may reason about subtraction of fractions the same way here. For instance, to find $$\frac{8}{3} - \frac{2}{3}$$, they may:

• Start at $$\frac{8}{3}$$ and jump 2 thirds to the left. The difference is the point where the arrow lands.
• Start at $$\frac{2}{3}$$ and see how many thirds are needed to reach $$\frac{8}{3}$$. The difference is the distance between the two points.​​​​​
MLR8 Discussion Supports. Synthesis: Display sentence frames to support small-group discussion: “_____ and _____ are the same because . . .”, and “I wonder if . . . .”
Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from reviewing subtraction on the number line with whole numbers. For example, display a number line with two jumps that represents $$11-7$$. Invite students to write and discuss the equation represented in the diagram.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• Groups of 2
• Display the first three number line diagrams in the activity.
• “How are these diagrams the same as the diagrams we saw in an earlier lesson? How are they different?” (Same: They use jumps to show a change. Each space between tick marks represents a unit fraction. Different: There is only one jump. The arrows point to the left.)
• 1 minute: quiet think time
• Share responses.
• “How do we know that the point represents $$\frac{11}{6}$$?” (Each space represents 1 sixth. The point is 11 sixths from 0.)

### Activity

• “Each of Noah's diagrams represents subtraction from $$\frac{11}{6}$$. Think about what number is being subtracted and what the result of the subtraction might be.”
• “Work with your partner on the first two problems.”
• 5–7 minutes: partner work time
• Invite students to share their responses to the first problem.
• “Where do you see the numbers being subtracted?” (The number of spaces jumped)
• “Where do you see the result of the subtraction?” (The point where the arrow lands)
• For the second problem, poll the class on which equations they thought the diagram could represent (for example: only the first, only the second, only the third, the first two, all three, and so on). Invite students from each camp to share their reasoning.
• Make sure students recognize why the diagram can represent all three equations. (See Student Responses.)
• 2 minutes: independent work on the last problem

### Student Facing

1. To subtract different fractions from $$\frac{11}{6}$$, Noah draws “jumps” on number lines.
1. The first diagram shows how he finds $$\frac{11}{6} - \frac{7}{6}$$. What is the value of $$\frac{11}{6} - \frac{7}{6}$$ ?
2. Write an equation to show the difference represented by each of Noah’s diagrams.
2. Here is another diagram Noah draws:

Which equations could the diagram represent? Explain your reasoning.

$$\frac{11}{6} - \frac{6}{6} = \frac{5}{6}$$

$$\frac{11}{6} - 1 = \frac{5}{6}$$

$$1\frac{5}{6} - 1= \frac{5}{6}$$

3. Use a number line to represent each difference and to find its value.

1. $$\frac{8}{3} - \frac{2}{3}$$
2. $$\frac{8}{3} - \frac{4}{3}$$
3. $$\frac{8}{3} - 1$$

### Student Response

If students find a value for $$\frac{8}{3} – 1$$ other than $$\frac{5}{3}$$ or $$1\frac{2}{3}$$, consider asking: “How else can we represent 1 whole using thirds?” and “How might we use this to help show the difference on this number line?”

### Activity Synthesis

• Focus the discussion on the last expression $$\frac{8}{3} – 1$$.
• “How did you subtract 1, a whole number, from $$\frac{8}{3}$$, a fraction?” (Start at $$\frac{8}{3}$$ and jump to the left 3 thirds, to land at $$\frac{5}{3}$$. Start at 1 and find out how far to jump to the right to reach $$\frac{8}{3}$$.)
• “How could you subtract 1 from $$\frac{8}{3}$$ if you didn’t have a number line?” (I could:
• Think of 1 as $$\frac{3}{3}$$ and subtract $$\frac{3}{3}$$ from $$\frac{8}{3}$$, which gives $$\frac{5}{3}$$.
• Think about how many thirds to add to $$\frac{3}{3}$$ to get $$\frac{8}{3}$$.
• Think of $$\frac{8}{3}$$ as $$2\frac{2}{3}$$ and subtract 1 from it, which gives $$1\frac{2}{3}$$.)

## Activity 2: What’s the Difference? (15 minutes)

### Narrative

In this activity, students use number lines to represent subtraction of a fraction by another fraction with the same denominator—including a mixed number—and by a whole number. Locating a fraction greater than 1 on the number line prompts students to decompose the fraction mentally into a whole number and a fractional part, rather than to rely on counting tick marks. Representing subtraction of a whole number on the number line encourages students to use their knowledge of whole-number equivalents of fractions and to look for and make use of structure (MP7). For example, when subtracting 1 from $$\frac{13}{8}$$, it helps to think of 1 as $$\frac{8}{8}$$, and when subtracting 1 from a mixed number such as $$1\frac{5}{8}$$, it helps to notice that $$1\frac{5}{8}$$ is $$1 + \frac{5}{8}$$.

As before, students may reason about subtraction in terms of removing an amount or finding an unknown addend, resulting in different number line diagrams. While students may rely on number lines to find each difference, the reasoning they do here prompts them to notice patterns and to think flexibly, preparing them to reason numerically in upcoming lessons.

• Groups of 2

### Activity

• “Take a few quiet minutes to work on the task. Then, share your responses with your partner.”
• 5–7 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for the different strategies students use to locate $$\frac{13}{8}$$ on the number line and to represent subtraction by 1 and $$1\frac{4}{8}$$.

### Student Facing

Use a number line to represent each difference and to find its value.

1. $$\frac{13}{8} - \frac{2}{8}$$

2. $$\frac{13}{8} - \frac{6}{8}$$

3. $$\frac{13}{8} - 1$$

4. $$1\frac{5}{8} - \frac{7}{8}$$

5. $$1\frac{5}{8} - 1$$

6. $$1\frac{5}{8} - 1\frac{4}{8}$$

### Activity Synthesis

• Select students to share their responses and completed diagrams.
• Focus the discussion on expressions that involve subtraction by a whole number or by a mixed number.
• Display the third and fifth expressions side by side: $$\frac{13}{8} - 1$$ and $$1\frac{5}{8} - 1$$.
• “Did you use the same strategy to represent these expressions on the number line and to find the value? If so, what was your strategy? If not, what was different?”
• Display the fourth and last expressions side by side: $$1\frac{5}{8} - \frac{4}{8}$$ and $$1\frac{5}{8} - 1\frac{4}{8}$$. Ask students to compare how they found the value of these expressions.
• See lesson synthesis.

## Activity 3: Make a Jump, Subtraction Edition [OPTIONAL] (15 minutes)

### Narrative

This optional activity gives students an additional opportunity to practice using “jumps” on number lines to subtract fractions, decomposing any whole numbers as needed along the way. (It is similar in structure to an optional activity in an earlier lesson on addition.)

Students are given four number lines with a point marked on each. They then draw a card with a fraction on it. All fractions on the cards, shown here, are greater than the values of the points. Students make one or more jumps to find the difference of the two points and then represent it with a subtraction equation.

$$\frac{8}{5} \qquad \frac{9}{5} \qquad \frac{10}{5} \qquad \frac{11}{5} \qquad \frac{12}{5} \qquad \frac{13}{5} \qquad \frac{14}{5} \qquad \frac{15}{5}$$

### Required Materials

Materials to Copy

• Make a Jump, Subtraction Edition

### Required Preparation

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

### Launch

• Groups of 2
• Give each group a set of fraction cards from the blackline master.
• If students did not complete the optional activity in the previous lesson (which had a similar structure):
• Display the four number lines.
• “What do you notice? What do you wonder?”
• 30 seconds: quiet think time
• 30 seconds: partner discussion

### Activity

• “Label each point on the number line with a fraction it represents. This is your target.”
• 1 minute: independent work time
• “You will now make one or more jumps from another fraction to your target and write an equation to represent the difference between the two.”
• Explain how to use the cards and how to complete the task.
• Monitor for students who facilitate subtraction by:
• rewriting a fraction on their card as a whole number or a mixed number
• labeling their number lines with whole numbers beyond 1

### Student Facing

Here are four number lines, each with a point on it. Label each point with a fraction it represents.

The point you labeled is your target.

• Pick a card from the set given to you. Locate and label the fraction on the number line.
• From that point, draw one or more jumps to reach the target. What do you need to subtract? Label each jump you draw.
• Write an equation to represent the difference of your two fractions.

### Activity Synthesis

• Select students to share their responses to the first couple of diagrams (or more if time permits).

## Lesson Synthesis

### Lesson Synthesis

“Today we learned to subtract a fraction from a fraction and a whole number from a fraction. We used number lines to help us.”

“How could we find the value of $$\frac{11}{8} - \frac{7}{8}$$?” (We could:

• Start at $$\frac{11}{8}$$ on a number line that is partitioned into eighths, and move to the left 7 spaces (7 eighths) to land at $$\frac{4}{8}$$.
• Subtract the numerators: removing 7 eighths from 11 eighths gives 4 eighths.
• Start at $$\frac{7}{8}$$ and think about how many eighths to add to reach $$\frac{11}{8}$$.)

“How could we find the value of $$\frac{11}{8} - 1$$?” (We could:

• Start at $$\frac{11}{8}$$ and move to the left 8 spaces—because 1 is 8 eighths—to land at $$\frac{3}{8}$$.
• Think about how many eighths to add to 1 or $$\frac{8}{8}$$ to get to $$\frac{11}{8}$$.
• Find $$\frac{11}{8} - \frac{8}{8}$$, which is $$\frac{3}{8}$$.)

“How could we find the value of $$\frac{11}{8} -1\frac{1}{8}$$?” (We could:

• Think of $$1\frac{1}{8}$$ as $$\frac{9}{8}$$ and move left 9 spaces from $$\frac{11}{8}$$ on the number line.
• Jump to the left 8 spaces to represent subtraction by 1 and then another 1 space for the $$\frac{1}{8}$$.
• Write $$\frac{11}{8}$$ as $$1\frac{3}{8}$$ and think about how far away it is from $$1\frac{1}{8}$$.
• Subtract the whole number and the fractional part separately. $$1 - 1 = 0$$ and $$\frac{3}{8} - \frac{1}{8}$$ = $$\frac{2}{8}$$.)