Lesson 5
Equivalent Multiplication Expressions
Warmup: How Many Do You See? (10 minutes)
Narrative
Launch
 Groups of 2
 “How many thirds do you see? How do you see them?”
Activity
 Display the image.
 1 minute: quiet think time
Student Facing
Student Response
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Activity Synthesis
 For each way that students see thirds, ask: “What expression should we use to represent the groups of thirds that _____ saw?”
 If no students suggest “4 groups of \(\frac{2}{3}\)”, ask them how that might be visible in the diagram. (By combining 2 of the thirds from each strip, we can make 4 groups of \(\frac{2}{3}\).)
 Write \(8 \times \frac{1}{3} =4 \times \frac{2}{3}\), and ask students if they agree or disagree with the statement.
Activity 1: Complete the Equations (15 minutes)
Narrative
The purpose of this activity is for students to think of different ways of using multiplication expressions to represent a nonunit fraction. Students informally use the associative property as they work towards generalizing that \(n \times \frac{a}{b} = \frac{n \times a}{b} = (n \times a) \frac{1}{b}\).
Supports accessibility for: VisualSpatial Processing, Fine Motor Skills
Launch
 Groups of 2
Activity
 “Work with your partner to complete the first problem. Talk about how you know what numbers make the equations true.”
 3 minutes: partner work time
 Monitor for students who use the factors of 12 to complete the equations.
 “Now take a few minutes to complete the rest of the problems independently. Afterwards, share your responses with your partner.”
 7 minutes: independent work time
 3 minutes: partner discussion
 “Did you choose the same numbers as your partner? If not, are both equations correct?”
Student Facing

Find the number that makes each equation true. Draw a diagram if it is helpful.
\(\frac{12}{5} =12 \times\underline{\hspace{.5in}}\)
\(\frac{12}{5} =6 \times\underline{\hspace{.5in}}\)
\(\frac{12}{5} =4 \times\underline{\hspace{.5in}}\)
\(\frac{12}{5} =3 \times\underline{\hspace{.5in}}\)
\(\frac{12}{5} =2 \times\underline{\hspace{.5in}}\)
\(\frac{12}{5} =1 \times\underline{\hspace{.5in}}\)

Here are two sets of numbers:
Set A:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Set B:
\(\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}, \frac{7}{7}\)

Choose a number from set A and a number from set B to complete this equation and make it true:
\(\displaystyle \frac{6}{7} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}\)

Choose a different number from set A and a number from set B to complete the equation to make it true.
\(\displaystyle \frac{6}{7} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}\)


Explain or show how you know that the two equations you wrote are both true.
Student Response
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Activity Synthesis
 “How did you know what fractions to use to complete each equation in the first problem?” (We looked for the numbers to multiply to get \(\frac{12}{5}\). The denominator stays the same, so we find out what is going to be the whole number and numerator by knowing the factor pairs of 12.)
 Select 2–3 partners to share the following equations from the second problem:
 \(\frac{6}{7} = 6 \times \frac{1}{7}\)
 \(\frac{6}{7} = 3 \times \frac{2}{7}\)
 \(\frac{6}{7} = 2 \times \frac{3}{7}\)
 \(\frac{6}{7} = 1 \times \frac{6}{7}\)
 “Why are all these equations true?” (To get \(\frac{6}{7}\), we can multiply the whole number by the numerator and then keep the denominator the same. In all of these, multiplying the whole number and the numerator gives 6. There are different ways to multiply to get 6 as a numerator: \(6\times 1\), \(3\times 2\), \(2 \times 3\) and \(1\times 6\).)
Activity 2: Fractions and Matching Expressions (20 minutes)
Narrative
In this activity, students analyze multiplication expressions, match each to one of a given set of fractions, and explain how they know that certain expressions represent the same fraction (MP7).
Advances: Speaking, Conversing
Launch
 Groups of 2
Activity
 “Match the expressions to a fraction that shows its value. Be prepared to explain how you know which expressions match which fraction.”
 “Each fraction may not have the same number of matching expressions.”
 5–7 minutes: partner work time
 Pause for a discussion before students continue to the second half of the activity.
 Select students to share their matches and their explanations.
 “Are there any expressions without a match? How do you know?” (Yes, expressions G and I. Their values are \(\frac{16}{9}\) and \(\frac{4}{12}\).)
 “You've seen that multiple expressions can represent the same fraction. Some of the expressions have two factors, some have three. All of them show unit fractions.”
 “Now complete each equation in the second problem with two factors that would make the equation true. See if you can use factors that are not unit fractions.”
 5 minutes: independent work time
Student Facing
Here is a set of expressions.
A.
\(6 \times \frac{1}{10}\)
B.
\(2 \times 4 \times \frac{1}{9}\)
C.
\(4 \times \frac{1}{5}\)
D.
\( 3 \times 2 \times \frac{1}{10}\)
E.
\( 5 \times 2 \times \frac{1}{12}\)
F.
\( 2 \times 2 \times \frac{1}{5}\)
G.
\(4 \times 4 \times \frac{1}{9}\)
H.
\(10 \times \frac{1}{12}\)
I.
\(4 \times \frac{1}{12}\)

Match each expression to one of the following fractions, if possible. Record your matches. Be prepared to explain how you know there is or isn't a match.
\(\frac{4}{5}\)
\(\frac{10}{12}\)
\(\frac{6}{10}\)
\(\frac{8}{9}\)

Complete each equation to make it true. Try to do so without using unit fractions.

\( \frac{4}{5} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}\)
\( \frac{4}{5} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}\)

\( \frac{10}{12} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}\)
\( \frac{10}{12} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}}\)

\( \frac{6}{10} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}} \)
\( \frac{6}{10} = \underline{\hspace{.5in}} \times \underline{\hspace{.5in}} \)

\(\frac{8}{9} =\underline{\hspace{.5in}} \times \underline{\hspace{.5in}} \)
\(\frac{8}{9} =\underline{\hspace{.5in}} \times \underline{\hspace{.5in}} \)

Student Response
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Activity Synthesis
 Ask selected students to share expressions for each fraction.

Display the possible expressions for the final equation:
\(8 \times \frac{1}{9}\)
\(2 \times 4 \times \frac{1}{9}\)
\(2 \times \frac{4}{9}\)
\(4 \times 2\times \frac{1}{9}\)
\(4 \times \frac{2}{9}\)  “How can we explain why these expressions have the value of \(\frac{8}{9}\)?”
Lesson Synthesis
Lesson Synthesis
“Today we looked at different expressions to represent the same fraction.”
Display the diagram from the warmup.
Ask students to write as many expressions as they can to describe the value of the shaded parts. Record their responses in a list for all to see. If no students suggest expressions with three factors (\(4 \times 2 \times \frac{1}{3}\) or \(2 \times 4 \times \frac{1}{3}\)), ask them to consider if it's possible to write such expressions.
“Pick two expressions from the list. Talk to your neighbor about how one is related to the other. You can mark up the diagram to support your explanation, if that's helpful.”
Cooldown: Expressions for Fractions (5 minutes)
CoolDown
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