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 draw two diagrams that represent a unit fraction of a unit fraction. Students work with the same unit fractions in both diagrams. The directions were intentionally written to encourage students to partition a unit square in different ways. Students initially partition the square into thirds in the first problem and into fourths in the second problem. Students may complete the diagrams in a way that makes sense to them. During the synthesis, students will connect both of the diagrams to the expressions
\(\frac {1}{4} \times \frac {1}{3}\)
and
\(\frac {1}{3} \times \frac {1}{4}\)
.
Launch
Groups of 2
Activity
2 minutes: Independent work time
10 minutes: partner work time
Monitor for students who:
draw diagrams like those in the student responses
recognize that both diagrams have the same amount shaded
can explain the different ways they partitioned each diagram
Student Facing
Show
\(\frac{1}{3}\)
of the square.
Shade
\(\frac{1}{4}\)
of
\(\frac{1}{3}\)
of the square.
How much of the whole square is shaded?
[IMG]
Show
\(\frac{1}{4}\)
of the square.
Shade
\(\frac{1}{3}\)
of
\(\frac{1}{4}\)
of the square.
How much of the whole square is shaded?
[IMG]
How are the diagrams the same and how are they different?
Student Response
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Advancing Student Thinking
If students do not identify how much of the whole square is shaded, ask: “How can you adapt your diagram to show how many pieces would be in the whole square?”
Activity Synthesis
Ask previously selected students to share their responses in the order given.
“How are the diagrams the same?” (They both have
\(\frac {1}{4}\)
and
\(\frac {1}{3}\)
in them. They both have
\(\frac {1}{12}\)
of the whole square shaded.)
“How are the diagrams different?” (In one diagram, I started by showing
\(\frac {1}{3}\)
of the square and then shaded in
\(\frac {1}{4}\)
of
\(\frac {1}{3}\)
. In the other diagram, I started by showing
\(\frac {1}{4}\)
of the square and then shaded in
\(\frac {1}{3}\)
of
\(\frac {1}{4}\)
.)
Display:
\(\frac{1}{12}\)
,
\(\frac{1}{4}\times\frac{1}{3}\)
,
\(\frac{1}{3}\times\frac{1}{4}\)
“How do the diagrams represent these expressions?”
Adapt diagrams as needed in order to show all three expressions.
“How do we know that both squares have the same amount shaded?” (They both have
\(\frac {1}{12}\)
shaded.)


aligns: no