Lesson 9
Use Equivalent Expressions
Warmup: 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.

\(\frac{1}{4}+\frac{2}{4}=\frac{3}{4}\)

\(\frac{1}{2}+\frac{1}{4}=\frac{2}{4}\)

\(\frac{3}{4}\frac{1}{2}=\frac{2}{4}\)
Student Response
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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
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).
Supports accessibility for: Organization, Conceptual Processing, Language
Launch
 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

Explain or show why each expression is equivalent to \(\frac{2}{3} + \frac{10}{12}\).
 \(\frac{8}{12} + \frac{10}{12}\)
 \(\frac{4}{6} + \frac{5}{6}\)
 Find the value of the expression \(\frac{2}{3} + \frac{10}{12}\). Explain or show your reasoning.
Student Response
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Advancing Student Thinking
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
Launch
 Groups of 2
Activity
 5 minutes: independent work time
 5 minutes: partner discussion
Student Facing
 Find the value of the expression \(\frac{16}{12}  \frac{3}{6}\). Explain or show your reasoning.
 Compare your strategy with your partner’s strategy. What is the same? What is different?
Student Response
For access, consult one of our IM Certified Partners.
Advancing Student Thinking
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).
Advances: Writing, Representing
Launch
 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.
Student Response
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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.)
Cooldown: Write an Expression (5 minutes)
CoolDown
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