Lesson 3
Patterns in Multiplication
Warmup: Choral Count: $\frac{1}{4}$ and $\frac{1}{8}$ (10 minutes)
Narrative
The purpose of this Choral Count is to invite students to practice counting by a unit fraction and notice patterns in the count. These understandings will be helpful later in this lesson when students recognize every fraction can be written as the product of a whole number and unit fraction.
Launch
 “Count by \(\frac{1}{4}\), starting at 0.”
Activity
 Record as students count.
 Stop counting and recording at \(\frac{11}{4}\).
 Repeat with \(\frac{1}{8}\).
 Stop counting and recording at \(\frac{15}{8}\).
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 “What patterns do you notice?” (In both counts, the numerators go up by 1, and denominators stay the same.)
 “How many groups of \(\frac{1}{4}\) do we have?” (11)
 “Where do you see them?” (Each count represents a new group of \(\frac{1}{4}\).)
 “How might we represent 11 groups of \(\frac{1}{4}\) with an expression?” (\(11 \times \frac{1}{4}\))
 “How many groups of \(\frac{1}{8}\) do we have?” (15)
 “How might we represent 15 groups of \(\frac{1}{8}\) with an expression?” (\(15 \times \frac{1}{8}\))
 “How would our count change if we counted by \(\frac{2}{4}\) or \(\frac{2}{8}\)?” (Each numerator would be a multiple of 2 or an even number.)
Activity 1: Describe the Pattern (15 minutes)
Narrative
Students may have previously noticed a connection between the whole number in a given multiplication expression and the numerator of the fraction that is the resulting product. In this activity, they formalize that observation. Students reason repeatedly about the product of a whole number and a unit fraction, observe regularity in the value of the product, and generalize that the numerator in the product is the same as the wholenumber factor (MP8).
Supports accessibility for: Language, Memory
Launch
 Groups of 2
 “Work with your partner to complete the tables. One person should start with Set A and the other with Set B.”
 “Afterwards, analyze your completed tables together and look for patterns.”
Activity
 5–7 minutes: partner work time on the first two problems
 Monitor for the language students use to explain patterns:
 The whole number in each expression is only being multiplied by the numerator of each fraction.
 Language describing patterns in the denominator of the product (The denominator in the product is the same as the unit fraction each time.)
 “Groups of” language to justify or explain patterns (The number of groups of each unit fraction is going up each time because it is one more group.)
 “Pause after you've described the patterns in the second problem.”
 Select 1–2 students to share the patterns they observed.
 “Now let's apply the patterns you noticed to complete the last two problems.”
 3 minutes: independent or partner work time
Student Facing

Here are two tables with expressions. Find the value of each expression. Use a diagram if you find it helpful.
Leave the last two rows of each table blank for now.
Set A
expression value \(1 \times \frac{1}{8}\) \(2 \times \frac{1}{8}\) \(3 \times \frac{1}{8}\) \(4 \times \frac{1}{8}\) \(5 \times \frac{1}{8}\) \(6 \times \frac{1}{8}\) Set B
expression value \(2 \times \frac{1}{3}\) \(2 \times \frac{1}{4}\) \(2 \times \frac{1}{5}\) \(2 \times \frac{1}{6}\) \(2 \times \frac{1}{7}\) \(2 \times \frac{1}{8}\)  Study your completed tables. What patterns do you see in how the expressions and values are related?
 In the last two rows of the table of Set A, write \(\frac{11}{8}\) and \(\frac{13}{8}\) in the “value” column. Write the expressions with that value.
 In the last two rows of the table of Set B, write \(\frac{2}{12}\) and \(\frac{2}{15}\) in the “value” column. Write the expressions with that value.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 “How did you use the patterns to write expressions for \(\frac{11}{8}, \frac{13}{8}, \frac{2}{12}\), and \(\frac{2}{15}\)?” (I knew that each expression had a whole number and a unit fraction. The whole number is the same as the numerator of the product.)
 Select students to share their multiplication expressions for these four fractions.
 “Can you write any fraction as a multiplication expression using its unit fraction?” (Yes, because the numerator is the number of groups and the denominator represents the size of each group.)
 “What would it look like to write \(\frac{3}{10}\) as a multiplication expression using a whole number and a unit fraction?” (\(\frac{3}{10} = 3\times \frac{1}{10}\))
Activity 2: What's Missing? (20 minutes)
Narrative
This activity serves two main purposes. The first is to allow students to apply their understanding that the result of \(a \times \frac{1}{b}\) is \(\frac{a}{b}\). The second is for students to reinforce the idea that any nonunit fraction can be viewed in terms of equal groups of a unit fraction and expressed as a product of a whole number and a unit fraction.
The activity uses a “carousel” structure in which students complete a rotation of steps. Each student writes a nonunit fraction for their group mates to represent in terms of equal groups, using a diagram, and as a multiplication expression. The author of each fraction then verifies that the representations by others indeed show the written fraction. As students discuss and justify their decisions they create viable arguments and critiqe one another’s reasoning (MP3).
Advances: Conversing
Required Materials
Materials to Gather
Launch
 Groups of 3
 “Let's now use the patterns we saw earlier to write some true equations showing multiplication of a whole number and a fraction.”
Activity
 3 minutes: independent work time on the first set of problems
 2 minutes: group discussion
 Select students to explain how they reasoned about the missing numbers in the equations.
 If not mentioned in students' explanations, emphasize that: “We can interpret \(\frac{5}{10}\) as 5 groups of \(\frac{1}{10}\), \(\frac{8}{6}\) as 8 groups of \(\frac{1}{6}\), and so on.”
 “In an earlier activity, we found that we can write any fraction as a multiplication of a whole number and a unit fraction. You'll now show that this is the case using fractions written by your group mates.”
 Demonstrate the 4 steps of the carousel using \(\frac{7}{4}\) for the first step.
 Read each step aloud and complete a practice round as a class.
 “What questions do you have about the task before you begin?”
 5–7 minutes: group work time
Student Facing

Use the patterns you observed earlier to complete each equation so that it’s true.

\(5 \times \frac{1}{10} = \underline{\hspace{0.5in}}\)

\(8 \times \frac{1}{6} = \underline{\hspace{0.5in}}\)

\(4 \times \underline{\hspace{0.5in}} = \frac{4}{5}\)

\(6 \times \underline{\hspace{0.5in}} = \frac{6}{10}\)

\( \underline{\hspace{0.5in}} \times \frac{1}{4}= \frac{3}{4}\)

\( \underline{\hspace{0.5in}} \times \frac{1}{12}= \frac{7}{12}\)


Your teacher will give you a sheet of paper. Work with your group of 3 and complete these steps on the paper. After each step, pass your paper to your right.
 Step 1: Write a fraction with a numerator other than 1 and a denominator no greater than 12.
 Step 2: Write the fraction you received as a product of a whole number and a unit fraction.
 Step 3: Draw a diagram to represent the expression you just received.
 Step 4: Collect your original paper. If you think the work is correct, explain why the expression and the diagram both represent the fraction that you wrote. If not, discuss what revisions are needed.
Student Response
For access, consult one of our IM Certified Partners.
Activity Synthesis
 See lesson synthesis.
Lesson Synthesis
Lesson Synthesis
“Today we looked at two sets of multiplication expressions. In the first set, the number of groups changed while the unit fraction stayed the same. We found a pattern in their values.”
“Then we looked at expressions in which the unit fraction changed and the number of groups stayed the same. We found a pattern there as well.”
Display the two tables that students completed in the first activity.
“In the first table, why does it make sense that the numerator in the product is the same number as the wholenumber factor?” (Because there are as many groups of \(\frac{1}{8}\) as the wholenumber factor)
“In the second table, why does it make sense that the numerator in the product is always 2?” (Because all the expressions represent 2 groups of a unit fraction.)
“We also discussed how we could write any fraction as a product of a whole number and unit fraction. Tell a partner about how we could write \(\frac{8}{3}\) as a product of a whole number and a fraction.” (\(\frac{8}{3} =8 \times \frac{1}{3}\))
Cooldown: Fraction Multiplication (5 minutes)
CoolDown
For access, consult one of our IM Certified Partners.