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.
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
Student Task Statement
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.
18.2
Activity
15 mins
Using Partial Quotients to Calculate Quotients
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.
Teacher Notes for IM 6–8 Math Accelerated v.360
Adjust this activity to 10 minutes. Consider discussing only one set of partial quotients before continuing on to the next activity.
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.
18.3
Activity
25 mins
Lin Uses Long Division
Standards Alignment
Building On
Addressing
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
This activity introduces the use of long division to calculate a quotient of whole numbers. Students make sense of the process of long division by studying an annotated example and relating it to the use of partial quotients and base-ten diagrams. They begin to see that long division is a variant of the method that uses partial quotients, but it is calculated and recorded differently.
When using partial quotients, the division is done in installments. The size of each installment can vary, but it is always a multiple of the divisor. Each partial quotient is written above the dividend and stacked. The sum of all partial quotients is the quotient.
In long division, the division is performed digit by digit, from the largest place to the smallest, so the resulting quotient is also recorded one digit at a time. In each step, one more digit of the quotient is calculated. Students notice that although only one digit of the quotient is written down at a time, the value that it represents is communicated through its placement.
As students examine the numbers, their placements, and the operations in the algorithm and interpret them in terms of place value and the process of dividing, they engage in abstract and quantitative reasoning (MP2).
To become proficient in long division requires time, encounters with a variety of division problems, and considerable practice. Students will have opportunities to study the algorithm more closely and to use it to divide increasingly more challenging numbers over several upcoming lessons.
Teacher Notes for IM 6–8 Math Accelerated v.360
Adjust this activity to 20 minutes. During the Launch, consider displaying Lin’s calculations without any explanations and asking students to share what they notice and wonder.
Here are Lin’s calculations for finding .
Students may notice:
Lin arranged her calculations vertically.
There are no partial quotients being added, but the quotient is still 219, just as in other calculations.
The quotient is shown one digit at a time, starting with the hundreds, then the tens, then the ones.
Lin subtracted 6, then 3, and then 27.
There are arrows pointing down in the tens place and ones place.
Students may wonder:
What do the arrows mean?
Why did Lin divide one number at a time?
Why did Lin subtract one-digit numbers at first, 6 and 3, but then subtract a two-digit number, 27, at the end?
Launch
Tell students that they will now consider a third method—called long division—for solving the same division problem that Kiran and Andre had calculated using partial quotients.
Display both Andre’s and Lin’s calculations for all to see.
Arrange students in groups of 2. Give students 2–3 minutes of quiet time to read and make sense of the annotated example of long division, and then 3–4 minutes to discuss the first set of questions with a partner. Pause for a whole-class discussion before students use long division to answer the second set of questions.
Invite 1–2 students to explain, in their own words, what Lin had done. Then discuss students’ responses to the first set of questions. Follow with questions such as:
“After writing down the 2, Lin subtracted 6. Why was the result of the subtraction not 651 given that ?” (Though she wrote a subtraction of 6, she actually subtracted 600 from 657.)
“Could Lin have written the full amounts being subtracted instead of just the non-zero digit—for example, writing 600, 50, and 7 instead of aligning 6, 5, and 7 to certain places)?” (Yes, it would involve more writing, but it works just as well.)
Consider demonstrating the long-division process with another example such as before asking students to complete the rest of the task. Provide access to graph paper. Tell students that the grid could help them line up the digits.
MLR8 Discussion Supports. Invite students to repeat their explanations for Lin’s steps using mathematical language: "Can you say that again, using the terms ‘place value,’ ‘ones place,’ and ‘tenths place’?” Advances: Speaking, Representing
Representation: Access for Perception. Read Lin’s method for calculating the quotient of aloud. Students who both listen to and read the information will benefit from extra processing time. Supports accessibility for: Language, Attention
Activity Synthesis
Display the worked-out long divisions for all to see. Select a student to explain the steps for at least one of the division problems.
Draw students’ attention to the division , in which the first digit of the dividend is smaller than the divisor. Select 1–2 students to share how they approached this situation. If not brought up in students’ explanation, discuss how we could reason about these.
“If we were using base-ten diagrams to represent , we would have 1 piece representing a thousand. How would we divide that piece into 4 groups?” (Decompose it into 10 hundreds, add them to the 8 pieces representing 8 hundreds, and then distribute the 18 hundreds into 4 groups.)
“How can we apply the same idea to long division when there are not enough thousands to divide into 4 groups?” (We can think of the 1 thousand and 8 hundreds as 18 hundreds and divide that value instead.)
“How many hundreds would go into each group if we divide 18 hundreds into 4 groups?” (4 hundreds, with a remainder of 2 hundreds.)
“Where should we write the 4? Why?” (In the hundreds place, because it represents 4 hundreds.)
“How do we deal with the remaining 2 hundreds?” (Decompose them into 20 tens, combine those with the 1 ten, and divide 21 tens by 4.)
18.4
Activity
Optional
15 mins
Dividing Whole Numbers
Standards Alignment
Building On
Addressing
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
In this activity, students continue to practice using long division to find quotients. Here, the presence of 0’s in the dividend and the quotient presents an added layer of complexity, prompting students to reason abstractly and quantitatively (MP2) as they make sense of the meaning of each digit in the numbers they are dealing with.
Launch
Keep students in groups of 2. Give students 6–7 minutes to work independently or with their partner. Follow with a whole-class discussion. Provide access to graph paper.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation, such as “First, I because . . .,” “Why did you . . .?,” “How did you get . . .?” Supports accessibility for: Language, Social-Emotional Functioning
Activity
None
Student Task Statement
Use long division to calculate each quotient.
Here is Priya’s calculation of .
Long division of 906 divided by 3. 6 rows. First row: 320. Second row: 3, long division symbol with 906 inside. Third row: minus 9. Horizontal line. Fourth row: 0 6. Fifth row: minus 6. Horizontal line. Sixth row: 0.
Priya wrote 320 for the value of . Check her answer by multiplying it by 3. What product do you get?
What does the product tell you about Priya’s answer? Explain your reasoning. If you think her answer is incorrect, describe the error and show the correct calculation and answer.
Student Response
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Building on Student Thinking
Activity Synthesis
Display the solutions to the first two division problems and give students time to check their work. Then, focus the discussion on ways to tell if a long division is done correctly. Discuss questions such as:
“How can we check if 143 is the quotient of ?” (We can multiply 143 by 7. If the division was done correctly, then the product would be 1,001, which is the case.)
“Why does multiplying Priya’s answer, 320, by the divisor 3, give 960 instead of 906? Where did the error happen?” (Priya placed the result of dividing the 6 in 906 by 3 in the tens place, even though the 6 represents ones. It should’ve been 2 ones.)
Make sure students notice that although checking an answer can tell us that we have made a mistake, it will not necessarily show where the mistake is. This way of checking also works only if we perform the multiplication correctly.
MLR8 Discussion Supports. Display sentence frames to support students in explaining their analysis of Priya’s answer. Examples:
“The product shows that Priya’s answer is correct (or incorrect) because . . . .”
“She did correctly and (but) she . . . .”
“The correct answer should be because . . . .”
Advances: Writing, Conversing
Lesson Synthesis
Invite students to compare and contrast long division with other methods of division. Ask questions such as:
“How is long division like other methods of dividing that we have used before?” (It involves putting base-ten units into equal-size groups, decomposing larger units into smaller ones, and paying attention to place value.)
“How is long division similar to the vertical calculations that use partial quotients? (We divide in “installments” and subtract parts of the dividend below it until there is nothing left. The quotient is above the dividend. The steps are recorded in a vertical format.)
“How is it different?” (There is a very specific order to follow. We divide digit by digit—from left to right, and subtract as large a group as possible at any step. Not all the digits of the amounts being divided, multiplied, or subtracted are written out. The place of the digit tells us its value.)
Then use an example from the lesson to highlight some ideas about long division:
We start by dividing the digit with the highest place value and work downward.
Any remainder in one place is worth 10 times as much in the place to its right and is combined with the value that is in that place before being divided. This keeps going until there is nothing left of the dividend.
The placement of each digit of the quotient matters because it conveys the value of the digit
Student Lesson Summary
One way to find the quotient of is by using partial quotients, in which we keep subtracting 3 groups of some amount from 948. 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 948 divided by 3. First calculation, 11 rows. First row: 316. Second row: 1. Third row: 15. Fourth row: 300. Fifth row: 3, long division symbol with 948 inside. Sixth row: minus 900, 3 groups of 300. Horizontal line. Seventh row: 48. Eighth row: minus 45, 3 groups of 15. Horizontal line. Ninth row: 3. Tenth row: minus 3, 3 groups of 1. Horizontal line. Eleventh row: 0. Second calculation, 11 rows. First row: 316. Second row: 150. Third row: 150. Fourth row: 16. Fifth row: 3, long division symbol with 948 inside. Sixth row: minus 48, 3 groups of 16. Horizontal line. Seventh row: 900. Eighth row: minus 450, 3 groups of 150. Horizontal line. Ninth row: 450. Tenth row: minus 450, 3 groups of 150. Horizontal line. Eleventh row: 0
In the calculation on the left, first we subtract 3 groups of 300, then 3 groups of 15, and then 3 groups of 1. Adding up the partial quotients () gives us 316.
The calculation on the right shows a different amount per group subtracted each time (3 groups of 16, 3 groups of 150, and 3 more groups of 150), but the total amount in each of the 3 groups is still 316.
There are other ways of calculating using partial quotients. We can calculate with fewer steps by removing groups of larger sizes.
Long division is another method for calculating quotients. It relies on place value to perform and record the division.
When we use long division, we work from left to right and with one digit at a time, starting with the leftmost digit of the dividend. We remove the largest group possible each time, using the placement of the digit to indicate the size of each group.
Here is an example of how to find using long division.
Long division of 948 divided by 3. 8 rows. First row: 316. Second row: 3, long division symbol with 948 inside. Third row: minus 9, 3 groups of 3 hundreds. Horizontal line. Fourth row: 4. Fifth row: minus 3, 3 groups of 1 ten. Horizontal line. Sixth row: 18. Seventh row: minus 18, 3 groups of 6 ones. Horizontal line. Eight row: 0.
We start by dividing 9 hundreds into 3 groups, which means 3 hundreds in each group. Instead of writing 300, we simply write 3 in the hundreds place, knowing that it means 3 hundreds.
There are no remaining hundreds, so we work with the tens. We can make 3 groups of 1 ten out of 4 tens, so we write 1 in the tens place above the 4 of 948. Subtracting 3 tens from 4 tens, we have a remainder of 1 ten.
We know that 1 ten is 10 ones. Combining these with the 8 ones from 948, we have 18 ones. We can make 3 groups of 6 ones, so we write 6 in the ones place.
In total, there are 3 groups of 3 hundreds, 1 ten, and 6 ones in 948, so .
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.
Activity
None
Student Task Statement
Here is how Lin found the quotient of .
Study Lin’s steps. Then discuss with your partner:
In the first step, Lin divided 6 by 3 to get 2. Why do you think she put the 2 over the 6?
Lin subtracted then , and lastly . Earlier, Andre subtracted then , and lastly . Why did they have the same quotient?
In the third step, why do you think Lin wrote the 7 next to the remainder of 2 rather than adding 7 and 2 to get 9?
Lin’s method is called long division. Use this method to find the following quotients. Check your answer by multiplying it by the divisor.
Student Response
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Building on Student Thinking
Students may recognize that they need to subtract multiples of the divisor in each place but not always subtract the greatest multiple. For example, when calculating they might subtract 3 instead of 6 from the 8 in the hundreds place, and end up with 54 for the next round of division, which would lead to a two-digit quotient for that round. Urge students to check if they have found the greatest whole-number multiple of the divisor in each place before moving on to the next place. One way to check is to see if the result of the subtraction (in this case, 5, the result of ), is greater than the divisor, and if so, to adjust the calculation accordingly.
