# Lesson 13

Put It All Together: Add and Subtract Fractions

## Warm-up: Number Talk: Sums with $\frac{1}{8}$ (10 minutes)

### Narrative

The purpose of this Number Talk is for students to use different strategies to add fractions. Each pair of fractions has \(\frac{1}{8}\) and the difference between the expressions is the denominator of the second fraction which is chosen to suggest different strategies for finding a common denominator. Students will explore these strategies in depth in this lesson.

### Launch

- Groups of 2
- Display one problem.
- “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 problems and work displayed.
- Repeat with each problem.
- “Discuss your thinking with your partner.”

### Student Facing

Find the value of each expression mentally.

- \(\frac{1}{8} + \frac{5}{8}\)
- \(\frac{1}{8} + \frac{6}{16}\)
- \(\frac{1}{8} + \frac{1}{3}\)
- \(\frac{1}{8} + \frac{5}{12}\)

### Student Response

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### Activity Synthesis

- “How did you decide which denominator to use when you found the sums of the fractions?” (The first two were the easiest because I could use eighths. For the others I used the product of the denominators for 8 and 3 and then I knew 24 would work for 8 and 12.)

## Activity 1: Common Denominators (20 minutes)

### Narrative

The purpose of this activity is for students to recognize that different strategies can be used to find common denominators. The two strategies highlighted in this activity are the ones students have used throughout the last several lessons.

- Use the product of the two denominators.
- Use a smaller, recognizable common multiple of the two denominators.

Both strategies are valuable and students can consider using these strategies in the next activity when they practice finding a variety of sums and differences.

When students evaluate the student claims about common denominators they critique the reasoning of others (MP3).

*MLR8 Discussion Supports.*During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.

*Advances: Listening, Speaking*

### Launch

- Groups of 2

### Activity

- 2 minutes: independent work time
- 8 minutes: small-group work time
- Monitor for students who:
- can explain why 18 is not a common denominator
- can explain why 48 can be used as a common denominator
- can explain why 24 can be used as a common denominator

### Student Facing

\(\frac{4}{6}+\frac{5}{8}\)

Tyler says: “To find the sum, I can use 18 as a common denominator.”

Han says: “To find the sum, I can use 24 as a common denominator.”

Clare says: “To find the sum, I can use 48 as a common denominator.”

- Whom do you agree with? Explain or show your reasoning.
- What is the value of \(\frac{4}{6}+\frac{5}{8}\)?
- Are there other common denominators you could use to find the sum? Explain or show your reasoning.

### Student Response

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### Activity Synthesis

- Invite previously selected students to share.
- “How do you know that 18 will not work as a common denominator?” (It is not a multiple of 8.)
- “How do you know that 24 will work as a common denominator?” (It is a multiple of 6 and 8.)
- “How do you know that 48 will work as a common denominator?” (\(6\times8=48\))
- “Which denominator do you prefer to use and why?” (I like 24 because it’s smaller so the arithmetic is easier. I like 48 because I know right away which multiples to take to get the common denominator.)
- Invite students to share other common denominators that they found.
- “Would you use any of these denominators to find the sum? Why or why not?” (No, it makes the numbers bigger and I need to figure out which multiple to use to get these common denominators.)

## Activity 2: Unlike Denominators (15 minutes)

### Narrative

The purpose of this activity is for students to consider which common denominators will be most helpful to add and subtract fractions. The numbers for each problem are chosen to highlight different strategies. The first problem has denominators where one is a multiple of the other. Students will likely recognize they can use one of them as a common denominator, reducing the number of computations needed. The second problem does not have one denominator that is a multiple of the other. The numbers are small and the product is a good choice for a common denominator. The other problems have larger denominators that share common factors. For these problems, some students may prefer to find a smaller common denominator as it can make the arithmetic simpler. Other students may prefer taking the product of the denominators because that way they don’t need to work to find a common multiple of the two denominators.

*Engagement: Develop Effort and Persistence.*Check in and provide each group with feedback that encourages collaboration and community. For example, ask students to identify the similarities and differences between their choices of denominators.

*Supports accessibility for: Conceptual Processing; Social-Emotional Functioning*

### Launch

- Groups of 2

### Activity

- 5 minutes: independent work time
- 5 minutes: small group work time
- Monitor for students who use different common denominators including the product of the denominators.

### Student Facing

Find the value of each expression. Explain or show your reasoning.

- \(\frac{2}{5}+\frac{13}{15}\)
- \(\frac{6}{5}-\frac{1}{3}\)
- \(\frac{11}{12}+3\frac{5}{9}\)
- \(\frac{6}{10}-\frac{9}{25}\)

### Student Response

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### Activity Synthesis

- Display expression: \(\frac{11}{12} + 3\frac{5}{9}\)
- Invite students to share the denominators they used to find the value of the expression.
- “Why are there different choices for a common denominator?” (Any number that is a multiple of both 9 and 12 works.)
- Display expression: \(\frac{6}{10} - \frac{9}{25}\)
- “Which common denominator did you use for these fractions?” (50 or 100 because I could see that they are common multiples and know what factors to multiply 10 and 25 by to get those numbers.)
- “Which strategy do you prefer to use to find a common denominator?” (Using the product always works, but the multiplication can be challenging sometimes. Finding a smaller common denominator can be helpful because I might not have to multiply large numbers, but it can be time consuming.)

## Lesson Synthesis

### Lesson Synthesis

“Today we added and subtracted fractions with unlike denominators. We found common denominators including the product of the denominators.”

Display: \(\frac{6}{10}+\frac{9}{25}\)

“Why might someone use 250 as a common denominator to add these fractions?” (They can just multiply the two denominators.)

“Why might someone use 50 as a common denominator?” (They want to use a smaller common denominator to simplify the arithmetic or visualize the answer more easily.)

Highlight the idea that when we add and subtract fractions with unlike denominators, we replace the given fractions with equivalent fractions that have the same denominator, whether that common denominator is found by multiplying the original denominators or finding a smaller common multiple of the two.

## Cool-down: Fraction Addition and Subtraction (5 minutes)

### Cool-Down

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