# Lesson 9

Use Equivalent Expressions

## Warm-up: True or False: Fraction Addition and Subtraction (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate strategies they have for using equivalent fractions to add and subtract fractions with different denominators. These mental calculations prepare students for working with more complex common denominators during this 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.

• $$\frac{1}{4}+\frac{2}{4}=\frac{3}{4}$$
• $$\frac{1}{2}+\frac{1}{4}=\frac{2}{4}$$
• $$\frac{3}{4}-\frac{1}{2}=\frac{2}{4}$$

### Activity Synthesis

• “How can we find the correct value of $$\frac{3}{4}-\frac{1}{2}$$?” ($$\frac{1}{2}=\frac{2}{4}$$ so $$\frac{3}{4}-\frac{2}{4}=\frac{1}{4}$$.)

## Activity 1: Equal Sums (15 minutes)

### Narrative

The purpose of this activity is for students to identify equivalent sums of fractions and use them to find the value of sums of fractions with different denominators. In a previous course, students learned to use factors and multiples to generate and identify equivalent fractions. They recall that technique here and then use those equivalent fractions to find sums. This helps reinforce the idea that it is helpful, when adding two fractions, if the fractions have the same denominator while also recalling how to find an equivalent fraction with a different denominator.
When students identify that equivalent fractions with the same denominator help to find the value of a sum they notice and take advantage of the meaning and structure of fractions (MP7).
Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for explaining each equivalent expression before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language

• Groups of 2

### Activity

• 5–8 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
• use multiplication to explain why the expressions are equivalent. For example, multiply  $$\frac{2 \times 4}{3 \times 4}$$ to show why $$\frac{2}{3}=\frac{8}{12}$$
• use division to explain why the expressions are equivalent. For example, divide $$\frac{10 \div 2}{12 \div 2}$$ to show why $$\frac{10}{12}=\frac{5}{6}$$

### Student Facing

1. Explain or show why each expression is equivalent to $$\frac{2}{3} + \frac{10}{12}$$.

• $$\frac{8}{12} + \frac{10}{12}$$
• $$\frac{4}{6} + \frac{5}{6}$$
2. Find the value of the expression $$\frac{2}{3} + \frac{10}{12}$$. Explain or show your reasoning.

### Student Response

If a student needs help getting started, suggest they draw 2 different number lines to represent $$\frac{2}{3}$$. Then, for each number line, ask, “How can you adapt the diagram to show $$\frac{8}{12}$$? $$\frac{4}{6}$$?”

### Activity Synthesis

• Invite previously selected students to share how they know the expressions $$\frac{8}{12} + \frac{10}{12}$$ and $$\frac{4}{6} + \frac{5}{6}$$ are equivalent to $$\frac{2}{3} + \frac{10}{12}$$.
• “How do you know that $$\frac{8}{12} + \frac{10}{12} = \frac{2}{3} + \frac{10}{12}$$?” (I can divide each $$\frac{1}{3}$$ into 4 equal parts. Those parts are $$\frac{1}{12}$$s and there are 8 of them.)
• “Why is the expression $$\frac{8}{12} + \frac{10}{12}$$ helpful for finding the sum?” (It’s all twelfths. I have 8 of them and 10 more so that's $$\frac{18}{12}$$.)
• “Which expression did you choose to find the sum?” (Sample response: I used $$\frac{4}{6} + \frac{5}{6}$$ because the numbers were smaller.)

## Activity 2: Find the Value of the Difference (15 minutes)

### Narrative

This activity builds on the previous activity where students saw how equivalent expressions can be a valuable tool to add or subtract fractions. The purpose of this activity is for students to generate an equivalent expression in order to find the value of a difference of fractions. Monitor for students who:

• find equivalent fractions with smaller numerators and denominators than the given fractions
• find equivalent fractions with larger numerators and denominators than the given fractions

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: partner discussion

### Student Facing

1. Find the value of the expression $$\frac{16}{12} - \frac{3}{6}$$. Explain or show your reasoning.
2. Compare your strategy with your partner’s strategy. What is the same? What is different?

### Student Response

If students try to use $$1\frac{2}{6}-\frac{3}{6}$$ to find the value of $$\frac{16}{12}-\frac{3}{6}$$ and do not get the correct value, ask, “How can you use a number line to represent the expression $$1\frac{2}{6}-\frac{3}{6}$$?”

### Activity Synthesis

• Invite previously selected students to share how they found the value of $$\frac{16}{12} - \frac{3}{6}$$.
• “How are the strategies for finding the value of the expression the same?” (They both change one of the fractions to an equivalent fraction so the fractions have the same denominator.)
• “How are the strategies for finding the value of the expression different?” (To make the denominator bigger I multiply by a whole number. To make the denominator smaller I divide by a whole number.)
• “Why is it important to have the same denominator?” (Then I can add or subtract the number of parts because they are the same size.)

## Activity 3: Grow Plants [OPTIONAL] (10 minutes)

### Narrative

The purpose of this activity is for students to solve a problem that involves finding the difference of fractions. Students may use addition or subtraction to solve the problem. Either way they will need to find a common denominator for the fractions. One of the numbers is a mixed number so students may:

• convert the mixed number to a fraction
• find the difference in steps, adding on or subtracting

When students recognize mathematical features of objects in the real world, they model with mathematics (MP4).

MLR8 Discussion Supports. Students who are working toward verbal output may benefit from access to mini-whiteboards, sticky notes, or spare paper to write down and show their responses to their partner.

• Groups of 2

### Activity

• 5 minutes: independent work time
• 1–2 minutes: partner discussion

### Student Facing

Jada and Andre compare the growth of their plants. Jada’s plant grew $$1\frac{3}{4}$$ inches since last week. Andre’s plant grew $$\frac{7}{8}$$ inches. How much more did Jada’s plant grow? Explain or show your reasoning.

### Activity Synthesis

• Continue to lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Today we used equivalent expressions to add and subtract fractions with unlike denominators.”

Display: $$\frac{15}{12}-\frac{3}{4}$$

“Describe to your partner how you would find the value of this expression.” (I need to find a common denominator so I would figure out how many twelfths are equal to $$\frac{3}{4}$$. $$\frac{3}{4}=\frac{9}{12}$$. Then, I would find the difference between $$\frac{15}{12}$$ and $$\frac{9}{12}$$. I can use fourths as a common denominator because $$\frac{15}{12}=\frac{5}{4}$$ so the difference is $$\frac{2}{4}$$.)

“How do you decide which common denominator to use when you are adding or subtracting fractions with unlike denominators?” (Here one denominator is 3 times the other. So I can use that as my common denominator by splitting the fourths into 3 equal pieces or combining the twelfths to make fourths.)