The purpose of this Warm-up is to elicit the idea of partial quotients, which will be useful when students learn a new way to record partial quotients in a later activity. While students may notice and wonder many things about these equations, the idea of dividing a number by decomposing the dividend into smaller and more-familiar parts are the important discussion points.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.
Student Lesson in Spanish
Launch
Arrange students in groups of 2. Display the problem stem and four equations for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder about. Give students another minute to discuss their observations and questions.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory; Organization
Activity
None
Here are Kiran’s calculations for finding :
What do you notice? What do you wonder?
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the equations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If no students mentioned that Kiran did not decompose the 657 strictly by place value, ask students to discuss this idea and whether it would be just as productive to decompose 657 into 600, 50, and 7.
10.2
Activity
Standards Alignment
Building On
5.NBT.B.6
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
In this activity, students analyze another method of division in which one portion of the dividend is divided at a time. They see that there is more than one way to record this process and that different partial quotients can be used for any given division.
The first division problem, , involves the same whole-number dividend and divisor as seen in the Warm-up. The calculations also show the same partial quotients of 200, 10, and 9 being added to give the quotient of 219. Unlike the strategy shown in the Warm-up, however:
All the calculations are arranged vertically in a single stack rather than written as equations.
The process involves repeatedly subtracting some amounts from the original dividend until there are no remainders.
Each partial quotient is recorded above the line with the dividend and divisor. Partial dividends that are being subtracted and remainders are recorded below that line.
In analyzing Andre’s calculations, students need to think about what calculations are performed, what each number means, why it is placed where it is or aligned as it is, and how the numbers are related. The reasoning here gives students opportunities to persevere in making sense of a problem (MP1) and to attend to precision (MP6).
Monitor for students who use different partial quotients to calculate . Select them to share later.
Launch
Arrange students in groups of 3–4. Display Andre’s calculations for all to see. Read aloud (or invite a student to read) the description of Andre’s steps. Give students 5–6 minutes to make sense of Andre’s method and to answer the first set of questions with their group. Pause for a brief whole-class discussion afterward.
Invite students to share their responses. As students explain, annotate Andre’s calculations to illustrate the connections that students are making.
Tell students that, just like Kiran’s method, Andre’s method uses partial quotients. In both methods, one part of the dividend is divided at a time, which gives a part of the quotient. Both Kiran and Andre started by thinking about “how many groups of 3 are in 600?” or “what is 600 divided by 3?” but one person represented the question and the answer with a division equation () while the other recorded the relationship between the numbers in a vertical arrangement. Consider asking:
“Why might it make sense to divide 30 and 27 by 3 instead of dividing 57 by 3?” (30 and 27 are familiar multiples of 3.)
“Could Kiran have gotten the same quotient if he wrote instead?” (Yes) “What if he wrote and instead?” (All the partial quotients would still add up to 219.)
Give students 3–4 minutes to calculate the quotient in the last question, using Andre’s partial quotients method. Provide access to graph paper. Tell students that they may find a grid helpful for aligning the digits when arranging their calculations vertically as Andre has done.
MLR5 Co-Craft Questions. Keep books or devices closed. Display the images and descriptions of Andre’s steps without revealing the questions. Ask students to record possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the task. Ask, “What do these questions have in common? How are they different?” Reveal the intended questions for this task and invite additional connections. Advances: Reading, Writing
Activity Synthesis
Ask previously selected students to share their calculations for , starting from the most involved and ending with the most streamlined.
Highlight that there are no rules about how many partial quotients we could use or what partial dividends to use. That said, it is often helpful to take out the amount in each place value and distribute it into groups. Once the entirety of the dividend is accounted for, the partial quotients can be added and the division is complete.
10.3
Activity
Standards Alignment
Building On
5.NBT.B.6
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
In this activity, students choose how to find quotients of two whole numbers. As students work with larger dividends and divisors, they begin to see the merits and potential drawbacks of each method.
Here are some strategies students are likely to use, listed in the order of efficiency, from less straightforward to more straightforward:
Using base-ten blocks or diagrams.
Writing a series of equations to find partial quotients (similar to Kiran’s equations in the Warm-up).
Using vertical calculations to find partial quotients (similar to Andre’s method).
Monitor for students who use these strategies, as well as for those who use other ways to divide.
Launch
Invite students to share some of the strategies for dividing two whole numbers that they have seen so far. Then, tell students that in this activity they choose a method to divide but should use partial quotients at least once. Encourage them to refer to Elena's, Kiran's, and Andre’s methods of reasoning from earlier activities, if needed.
Keep students in groups of 3–4. Give students 2–3 minutes of quiet time to work on the first division task, followed by a few minutes to discuss their response and reasoning with their group. Then give students 8–10 minutes of quiet work time on the remaining questions. Provide access to base-ten blocks or paper cutouts of base-ten representations, as well as to graph paper.
Select students who used each strategy described in the Activity Narrative, and ask them to share later. Aim to elicit both key mathematical ideas and a variety of student contributions, especially from students who haven’t shared recently.
Action and Expression: Develop Expression and Communication. Provide continued access to base-ten blocks or paper cut-outs for students who found them helpful earlier in the lesson. Supports accessibility for: Conceptual Processing, Organization
Activity
None
Find the value of each quotient. Show your reasoning. Use vertical calculations at least once.
Activity Synthesis
The purpose of this discussion is to compare the different strategies students used to divide to and draw attention to advantages and disadvantages.
Ask previously selected students to share their method for finding quotients. Sequence the discussion of the methods in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
Connect the different responses to the learning goals by asking questions such as:
“What are some benefits of writing equations as Kiran has done?” (It’s a clear way to organize the calculations and is quicker than drawing diagrams. We’re dealing with one division at a time. We can choose any number to divide.)
“What might be some drawbacks of this method?” (We need to mentally keep track of how much is left of the dividend that is yet to be divided. We’re writing “divide by 9” or “" over and over.)
“What are some benefits of using vertical calculations as Andre has done?” (It’s quicker than drawing diagrams. The partial dividends and quotients are all recorded in one place. The amount that still needs to be divided can be found by subtracting and is shown as a part of the process.)
“What might be some challenges with this method?” (It’s not always clear what number to use each time. It involves multiplying and subtracting repeatedly, so if one calculation is wrong, the rest may also be wrong.)
Lesson Synthesis
Two key ideas from this lesson are:
Division can be done in parts and recorded in more than one way.
We can divide efficiently by making use of place value and the structure of the base-ten system.
To highlight these ideas, ask questions such as:
“How is using partial quotients to divide similar to other division methods you have used before?” (It still involves thinking about place value and equal-size groups.)
“How is it different?” (There is flexibility in how to divide and how much to divide each time. The vertical arrangement of numbers and repeated subtraction are new.)
“Can we find partial quotients in any order? For example, to find , can we subtract 16 for 4 groups of 4, then 80 for 4 groups of 20, and then 800 for 4 groups of 300? (Yes.)
“Many of you started with the highest place value in the dividend. For example, when dividing 1,115 by 5, you started with . Why might that be?” (It helps to subtract the largest multiple of the divisor first, which means looking at the highest place value.)
Student Lesson Summary
Another way to find the quotient of is by using partial quotients, in which we keep subtracting 3 groups of some amount from 345. We can organize the steps and record the partial quotients in a vertical calculation.
Here are two calculations for finding :
2 partial quotients methods of 345 divided by 3. First calculation, 11 rows. First row: 115. Second row: 5. Third row: 10. Fourth row: 100. Fifth row: 3, long division symbol with 345 inside. Sixth row: minus 300, 3 groups of 100. Horizontal line. Seventh row: 45. Eighth row: minus 30, 3 groups of 10. Horizontal line. Ninth row: 15. Tenth row: minus 15, 3 groups of 5. Horizontal line. Eleventh row: 0. Second calculation, 11 rows. First row: 115. Second row: 50. Third row: 50. Fourth row: 15. Fifth row: 3, long division symbol with 345 inside. Sixth row: minus 45, 3 groups of 15. Horizontal line. Seventh row: 300. Eighth row: minus 150, 3 groups of 50. Horizontal line. Ninth row: 150. Tenth row: minus 150, 3 groups of 50. Horizontal line. Eleventh row: 0
In the calculation on the left, first we subtract 3 groups of 100, then 3 groups of 10, and then 3 groups of 5. Adding up the partial quotients () gives us 115.
The calculation on the right shows a different amount per group subtracted each time (3 groups of 15, 3 groups of 50, and 3 more groups of 50), but the total amount in each of the 3 groups is still 115.
There are other ways of calculating using partial quotients. We can calculate with fewer steps by removing groups of larger sizes.
Standards Alignment
Building On
4.NBT.B.6
Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Andre calculated using a method that was different from Kiran's.
Andre subtracted 600 from 657. What does the 600 represent?
Andre wrote 10 above the 200, and then subtracted 30 from 57. How is the 30 related to the 10?
What do the numbers 200, 10, and 9 represent?
What is the meaning of the 0 at the bottom of Andre’s work?
How might Andre calculate ? Explain or show your reasoning.
Student Response
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Building on Student Thinking
Students who know that 57 is might wonder why Andre subtracted and separately rather than in one step. Encourage these students to write out the calculation that they think would be more efficient, compare it with Andre’s calculation, and point out how the process is streamlined. Clarify that there are many ways to divide in portions, and that some may be quicker than others.
