The purpose of this Choral Count is to invite students to practice counting by a unit fraction and to notice patterns in the count. These understandings will be helpful later in this lesson when students recognize that every fraction can be written as the product of a whole number and a unit fraction.
Understand a fraction as a multiple of . For example, use a visual fraction model to represent as the product , recording the conclusion by the equation
Students previously may have 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 whole-number factor (MP8).
Representation: Develop Language and Symbols. Provide students with access to a chart that shows definitions and examples of the terms that will help them articulate the patterns they see, including “whole number,” “fraction,” “numerator,” “denominator,” “unit fraction,” and “product.” 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.”
“Afterward, 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, such as:
The whole number in each expression is only multiplied by the numerator of each fraction.
The denominator in the product is the same as in the unit fraction each time.
The number of groups of each unit fraction is going up each time because it is 1 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
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
Set B
expression
value
Look at your completed tables. What patterns do you see in how the expressions and the values are related?
In the last two rows of the table of Set A, write in one row and in the other, in the “value” column. Write the expressions with those values.
In the last two rows of the table of Set B, write in one row and in the other, in the “value” column. Write the expressions with those values.
Student Response
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Advancing Student Thinking
Activity Synthesis
“How did you use the patterns to write expressions for , and ?” (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 as a multiplication expression, using a whole number and a unit fraction?” ()
Activity 2
Standards Alignment
Building On
Addressing
4.NF.B.4.a
Understand a fraction as a multiple of . For example, use a visual fraction model to represent as the product , recording the conclusion by the equation
This activity serves two main purposes. The first is to allow students to apply their understanding that the result of is . The second is for students to reinforce the idea that any non-unit 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 non-unit fraction for their group 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 critique one another’s reasoning (MP3).
MLR8 Discussion Supports. Display sentence frames to support small-group discussion after checking students’ fraction diagrams and equations. Examples: “I agree because . . . ,” and “I disagree because . . . .”
Advances: Conversing
Launch
Groups of 3
“Let's now use the patterns we saw earlier to write equations that show 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 as 5 groups of , as 8 groups of , 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.”
Demonstrate the 4 steps of the carousel, using 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
Activity Synthesis
See the 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 whole-number factor?” (Because there are as many groups of as the whole-number 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 a unit fraction. Tell a partner how we could write as a product of a whole number and a fraction.” ()
Standards Alignment
Building On
Addressing
4.NF.B.4
Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
Use the patterns you observed earlier to complete each equation so that it’s true.
Your teacher will give you a sheet of paper. Work with your group to 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 equation 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.
