standard_description: Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
problem_activity: Narrative
The purpose of this activity is for students to solve a contextual problem about dividing a fractional amount by a whole number. Students draw a diagram to represent the situation and relate the diagram to a division expression. Because of earlier work in this unit, students may draw one of the familiar square area diagrams showing the product
\(\frac{1}{3} \times \frac{1}{2}\)
. Other students may make a diagram resembling the macaroni and cheese pan and divide it appropriately. The focus in the synthesis is on bringing out how the diagram shows
\(\frac{1}{2} \div 3\)
and how it allows students to answer the question. The relationship between
\(\frac{1}{3} \times \frac{1}{2}\)
and
\(\frac{1}{2} \div 3\)
will be brought out in later lessons. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).
This activity uses
MLR7 Compare and Connect.
Advances: Representing, Conversing.
Representation: Develop Language and Symbols.
Represent the problem in multiple ways to support understanding of the situation. For example, a cut-out or another physical manipulative to represent the pan of macaroni and distribute the “macaroni” to three students or a video that shows the distribution of macaroni and cheese between three people.
Supports accessibility for: Attention, Conceptual Processing
Launch
Groups of 2
Display and read: “Last night, Jada’s aunt baked a pan of macaroni and cheese for dinner. Today, she brought the leftovers to Jada’s home for Jada and her sisters to share.”
“What do you notice? What do you wonder?” (We solved problems about macaroni and cheese before. I wonder how much macaroni and cheese Jada’s aunt brought.)
1–2 minutes: partner discussion
Activity
1–2 minutes: quiet think time
6–8 minutes: partner work time
Monitor for students who:
draw diagrams like the ones in the student responses
explain the situation as
\(\frac {1}{2}\)
divided into 3 equal pieces
recognize each person will get
\(\frac{1}{6}\)
of the whole pan of macaroni and cheese
Student Facing
Jada and her 2 sisters equally share
\(\frac{1}{2}\)
a pan of macaroni and cheese.
Draw a diagram to represent the situation.
Explain how this expression represents the situation:
\(\frac {1}{2} \div 3\)
How much of the whole pan of macaroni and cheese will each person get?
Student Response
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Activity Synthesis
MLR7 Compare and Connect
“Create a visual display that shows your thinking about the problems. You may want to include details such as notes, diagrams, drawings, etc., to help others understand your thinking.”
2–5 minutes: independent or group work
5–7 minutes: gallery walk
“How does each representation show
\(\frac {1}{2}\)
a pan of macaroni and cheese?”
“How does each representation show 3 equal pieces?”
30 seconds: quiet think time
1 minute: partner discussion
“How do we know that each person got the same amount of macaroni and cheese?” (I divided the half into 3 equal shares.)
“How much of the whole pan of macaroni and cheese did each person get?” (
\(\frac {1}{6}\)
)
“How do the diagrams show
\(\frac {1}{6}\)
?” (Each
\(\frac {1}{2}\)
of the pan is divided into 3 equal pieces, so each of those pieces is
\(\frac {1}{6}\)
of the whole pan.)


aligns: no