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 practice using an algorithm that uses partial quotients to divide multi-digit numbers by two-digit divisors. Before finding the quotient, students estimate the value of the quotient which both helps students decide which partial quotients to use and helps them evaluate the reasonableness of their solution (MP8).  For example, if students estimate that the value of
\(529 \div 23\)
is a little more than 20, that means that a good choice for first partial quotient is 20. Whenever possible, ask students to explain the steps they are taking.
To add movement to the activity, after students have solved each problem, they can partner with other students who used different partial quotients or got a different solution. This gives students the opportunity to explain their reasoning to one another and make adjustments to their work, as needed.
MLR7 Compare and Connect.
Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What did the approaches have in common?”, “How were they different?” and “Why did the different approaches lead to the same outcome?”.
Advances: Representing, Conversing
Engagement: Develop Effort and Persistence.
Chunk this task into more manageable parts. Check in with students to provide feedback and encouragement after each chunk. Provide feedback on their use of algorithms using partial quotients.
Supports accessibility for: Social-Emotional Functioning
Launch
Groups of 2
Activity
5 minutes: independent work time
5 minutes: partner work time
Monitor for students who use different sets of partial quotients for the same problem.
Student Facing
Estimate the value of each quotient. Then, use an algorithm using partial quotients to find the value.
A reasonable estimate for
\(612 \div 34\)
is:
[IMG]
A reasonable estimate for
\(529 \div 23\)
is:
[IMG]
A reasonable estimate for
\(1,\!044 \div 29\)
is:
[IMG]
Student Response
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Activity Synthesis
Ask 2–3 students to share their work for the same problem that shows different partial quotients.
“How can you make sure that the whole number quotient you got at the end is reasonable?” (It should be close to my estimate. I can multiply the quotient and divisor and that should give me the dividend.)
If students pair and share with other partners, ask, “How did explaining your work to others help you today?” or “What did someone say today that helped you in your understanding of division?” (I learned that it’s ok to take more steps because I was comfortable with the multiples I used.)


aligns: no