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
In this activity, students notice as the divisor increases for a given dividend, the quotient gets smaller. Students may recognize and explain the relationship between multiplication and division. For example, they may notice that dividing one quarter into two equal pieces is the same as finding the product of
\(\frac{1}{2}\times\frac{1}{4}\)
. This relationship is not explicitly brought up in the synthesis, but if students describe this relationship, connect it to the student work that is discussed in the synthesis, as appropriate. When students notice a pattern, they look for and express regularity in repeated reasoning (MP8).
Engagement: Provide Access by Recruiting Interest.
Invite students to share a situation in their own lives that could be used to represent one of the division expressions.
Supports accessibility for: Conceptual Processing, Language, Attention
Launch
Groups of 2
Activity
1–2 minutes: independent think time
3–5 minutes: partner work time
Student Facing
Find the value that makes each equation true. Use a diagram if it is helpful.
\(\frac {1}{4} \div 2 = \underline{\hspace{1 cm}}\)
\(\frac {1}{4} \div 3 = \underline{\hspace{1 cm}}\)
\(\frac {1}{4} \div 4 = \underline{\hspace{1 cm}}\)
What patterns do you notice?
How would you find the value of
\(\frac{1}{4}\)
divided by any whole number? Explain or show your reasoning.
Student Response
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Advancing Student Thinking
If students do not explain how they would find the value of
\(\frac{1}{4}\)
divided by any whole number, prompt them to find the value of
\(\frac{1}{4} \div 5\)
and
\(\frac{1}{4} \div 6\)
and ask, “What is the relationship between the divisor and the quotient?”
Activity Synthesis
Display:
\(\frac {1}{4} \div 2 = \frac {1}{8}\)
\(\frac {1}{4} \div 3 = \frac {1}{12}\)
\(\frac {1}{4} \div 4 = \frac {1}{16}\)
“What patterns do you notice?” (The quotient is getting smaller. The denominator of the quotient is getting bigger. The denominator in the quotient increases by 4. The denominator in the quotient is equal to 4 times the number you are dividing by.)
“Why is the quotient getting smaller?” (Because we are dividing
\(\frac {1}{4}\)
into more pieces each time, so the size of each piece will be smaller.)


aligns: no