Lesson 10

This number talk prompts students to make reasonable estimates of quotients using their knowledge of numbers, division, and structures. Only two problems are given to allow more time for students to share their estimation strategies.

Making reasonable estimates helps to develop arithmetic fluency. Here, it relies on understanding the relationship between multiplication and division, and on the different properties of operations (commutative, associative, and distributive). For example, to find \(500 \div 7\), students might think of the multiplication equation \(7 \boldcdot {?} = 500\). Since they know that \(7 \boldcdot 100 = 700\) and \(7 \boldcdot 30 = 210\), and that \(500 = 700-200\), they could reason that 500 would be approximately \(7 \boldcdot (100-30)\) or \(7 \boldcdot 70\). 

Display each problem one at a time for all to see. Give students 1 minute of quiet think time and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Students might try to calculate exact answers and take longer to produce an answer. Encourage them to approximate the actual answer by rounding the dividend or divisor or starting with friendlier numbers.

Invite students to share their strategies. Record and display student explanations for all to see. Ask students to explain if or how the dividend or divisor impacted their choice of strategy. To involve more students in the conversation, consider asking:

• “Did anyone reason about the problem the same way but would explain it differently?”
• “Did anyone estimate in a different way?”
• “Does anyone want to add on to _____’s reasoning?”
• “Do you agree or disagree? Why?”

At the end of the discussion, if time permits, ask a few students to share a story problem or context that \(1,394 \div 9\) could represent.

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 partial quotients method, but it is calculated and recorded differently.

In the partial quotients method, the division is done in installments, resulting in a series of partial quotients. 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. 

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.  

Tell students that they will now consider a third method—called long division—for solving the same division problem that Elena and Andre had calculated using base-ten diagrams and the partial quotients method. Encourage students to refer to their work on those activities, or display Elena and Andre’s methods for all to see. Ask a couple of students to briefly explain how Elena and Andre calculated \(657 \div 3\). 

Arrange students in groups of 2. Give students 2–3 minutes of quiet think time to make sense of the annotated example of long division, and then 3–4 minutes to discuss the first set of questions with a partner. Follow with a whole-class discussion before students use long division to answer the second set of questions.  

Display Lin’s method for all to see and ask a student to explain what Lin had done in his or her own words. Then, discuss students’ responses to the first set of questions and these questions:

• In the first step, Lin divided 6 by 3 to get 2. Did it matter where Lin wrote the 2? Why did she put it over the 6? (Yes, it mattered. Because the 6 represents 600, she was really dividing 600 by 3, which is 200. The 2 needs to be written in the hundreds place to tell us its actual value.)

• After writing down the 2, Lin subtracted 6. Why? And why was the result of the subtraction not 651 (since \(657-6=651\))? (Though she wrote a subtraction of 6, she actually subtracted 600. Because she had just divided 600 by 3, that portion of 657 is already accounted for.)

• Could Lin have written the full amounts being subtracted instead of just the non-zero digit (e.g., subtracting by 600, 50 and 7, instead of subtracting by 6, 5, and 7 after aligning them to certain places)? (Yes, it would involve more writing, but it works just as well.)

• How is this process similar to and different than the partial quotients method? (It is a similar idea of taking out a certain multiple of 3 at a time, but in long division we do it digit by digit and in the order of place value—from the largest unit to the smallest.) 

Consider demonstrating the long-division process with another example such as \(912 \div 4\) 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.

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. Highlight two ideas about long division: 1) we start by dividing the largest base-ten units and work toward smaller units, and 2) the placement of each digit of the quotient matters because it conveys the value of the digit.

Draw students’ attention to the second problem (\(1,\!816 \div 4\)) or third problem (\(768 \div 12\)), in which the first digit of the dividend is smaller than the divisor. Select 1–2 students to share how they approached these situations. If not brought up in students’ explanation, discuss how we could reason about these.

• “Let’s take \(1,\!816 \div 4\) as an example. If we were using base-ten diagrams, we would have 1 piece representing a thousand. How would we divide that piece into 4 groups?” (We would unbundle 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? If there is not enough thousands to divide into 4 groups, what can we do?” (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 2 hundreds?” (Since there is not enough to distribute into 4 groups, we can unbundle them into 20 tens, combine them with the 1 ten, and divide 21 tens by 4.)

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 really make sense of the the meaning of each digit in numbers they are dealing with (MP7).

Give students 6–7 minutes of quiet work time. Follow with a whole-class discussion. Provide access to graph paper. 

Focus class discussion on attending to the meaning of each digit in performing division. Discuss:

• How did you deal with the 0’s in 1,001? Would they cause any difficulty when doing long division? (I brought down the first 0 and then performed division like I would have done with any other digit. After subtracting one 7 from 10, I was left with 3. Putting a 0 after the 3 changes the value to 30. So even though the 0 alone has no value, it changes the value of the numbers in front of it).

• How can you check your answer to a division problem such as \(1,\!001 \div 147\)? (We can check by multiplying the quotient by the divisor. If the division was done correctly, then \(143 \boldcdot 7 = 1,\!001\), which is true).

• What happens if you check Priya’s answer for \(906 \div 3\)? (\(320 \boldcdot 3 = 960\), so this tells us that Priya’s answer is incorrect).

Make sure students notice that although checking an answer can tell you that you have made a mistake, it will not necessarily identify where the mistake is. It only works if you perform the multiplication correctly.