This Math Talk focuses on division of multi-digit numbers by a single-digit divisor. It encourages students to think about place value and to rely on what they know about base-ten numbers and properties of operations to mentally solve problems. The reasoning elicited here will be helpful later in the lesson when students use long division to divide decimals by whole numbers.
To find the value of the last two expressions, students need to look for and make use of structure (MP7). In explaining their reasoning, students need to be precise in their word choice and use of language (MP6).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Find the value of each quotient mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
Highlight how partial quotients are used in finding . Students recognized 81.2 as , so they added and to find .
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class. Advances: Speaking, Representing
19.2
Activity
Standards Alignment
Building On
Addressing
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
In this activity, students use long division to divide whole numbers whose quotient is not a whole number. Students begin by analyzing a finished long-division calculation and explaining the steps they see. Along the way, they notice that the same reasoning can be applied to divide a remainder in the ones place and continue into the tenths place and beyond. Then they go on to use long division to divide whole numbers, including those that produce quotients less than 1.
As students make sense of the digits in the calculation and their placement, interpreting them in terms of place value and operations, they reason abstractly and quantitatively (MP2).
During the Launch, consider displaying Mai’s method for students to analyze and discuss before Lin’s method. This can help students make the connection between decomposing base-ten units and continuing the long division to the right of the decimal point.
Mai used base-ten diagrams to calculate . She started by representing 62.
She then made 5 groups. Here is Mai’s diagram for .
Ask students:
“Mai should have a total of 12 ones, but her diagram shows only 10. Why?” (Mai decomposed 2 ones to make 20 tenths. So instead of 12 ones, her diagram has 10 ones and 20 tenths.)
“She did not originally have tenths, but in her diagram each group has 4 tenths. Why?” (Mai took the 20 tenths and placed 4 tenths in each of the equal groups.)
“What value has Mai found for ?” (12.4. Each group has 1 ten, 2 ones, and 4 tenths, so .)
Launch
Arrange students in groups of 2. Give students 4–5 minutes to analyze and discuss, with a partner, Lin’s work. Pause for a class discussion before completing the rest of the questions.
Invite students to share their responses to the first set of questions. Consider asking students:
“How is this division process different from the long division you did in an earlier lesson?” (There is a remainder after subtracting the last group of 3. The quotient is a decimal.)
“Why is it okay to add a zero after the 2? Wouldn’t that change the value of the dividend?” (The 0 is in the tenths place and has a value of 0 tenths, so 62.0 is still 62.)
“What value does the 4 in the quotient represent?” (4 tenths) “How do you know?” (It is located to the right of the decimal point.)
If not made explicit in students’ explanations, point out that up until reaching the decimal point, long division works the same for as it does for . In , there is a remainder of 2 ones, but we can decompose them into 20 tenths, and then divide these into 5 equal groups of 4 tenths. Clarify that a vertical line can help us keep track of the location of the decimal point and separate the decimal values from the ones—both in the quotient at the top and as we work further down in the calculation.
Give students time to complete the rest of the activity independently or with a partner. Provide access to graph paper. Leave at least a few minutes for discussion.
Representation: Access for Perception. Provide access to base-ten blocks or paper cutouts of base-ten representations. Ask students to make connections between each step of the completed long division for (or ) and the corresponding action they can take with base-ten blocks. Supports accessibility for: Visual-Spatial Processing, Organization
Activity Synthesis
Ask a few students to display their long division calculations for all to see and to explain their steps. Invite the rest of the class to ask clarifying questions.
Focus the discussion on the last set of division problems. Ask questions such as:
“When you calculated , what was your first step?” (Put a 0 above the dividend 1 and add a decimal point for the quotient because there are 0 groups of 5 in 4.)
“What did you do next?” (Bring down the 4 and add a 0 to the right of it.)
“What is the value of the 40?” (40 tenths)
“What is the value of the 8 in the quotient?” (8 tenths)
19.3
Activity
Standards Alignment
Building On
Addressing
6.NS.B.2
Fluently divide multi-digit numbers using the standard algorithm.
In this activity, students use long division to divide a decimal by a whole number. They notice that the steps in the division process are the same as when dividing a whole number by a whole number, but they need to think even more carefully about place value and where the decimal point goes in the quotient.
The second question involves dividing a decimal dollar amount by a whole number. The monetary context can further reinforce the idea of decomposing ones into tenths (and tenths into hundredths) as needed to divide. For example, along the way of finding by long division, students need to divide 2 dollars by 5. Because there aren’t enough whole dollars to divide by 5, we can think of the 2 dollars as 20 dimes, combine those with the 4 dimes in \$77.40, and distribute 24 dimes into 5 groups. The remaining 4 dimes need to be traded for 40 pennies to be distributed into 5 groups equally.
During the Activity Synthesis, consider displaying Elena’s method for students to analyze and discuss. This can help students make the connection between decomposing base-ten units and continuing the long division to the right of the decimal point.
To find using diagrams, Elena
She placed 1 ten into each group, decomposed the remaining 1 ten into 10 ones, and went on distributing the units.
4 groups of base-ten diagrams. Group 1, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 2, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 3, 1 rectangle labeled, tens, 1 square labeled, ones, 2 small rectangles labeled, tenths. Group 4, 1 rectangle labeled, tens, 2 small rectangles labeled, tenths. Below the groups, one rectangle decomposed into 10 squares.
Here’s Elena’s finished diagram, showing the quotient of .
Ask students:
“What did Elena do after decomposing the 1 ten into 10 ones? How did she get to the last diagram?” (Elena put 9 of the 10 ones into the 4 groups so that each group now has 3 ones. There is 1 one left, which she decomposed into 10 tenths. She put 8 tenths into the 4 groups, 2 tenths in each group. There are 2 tenths left, which she decomposed into 20 hundredths. She put 5 hundredths into each of the 4 groups. There are no pieces left.)
“Based on Elena’s work, what is the value of $53.8 \div 4$?” (13.45, because there are 1 ten, 3 ones, 4 tenths, and 5 hundredths in each group.)
Launch
Keep students in groups of 2. Tell students that they will now use long division to divide a decimal by a whole number. Give students 5–6 minutes to work independently or with a partner. Provide access to graph paper in case students wish to use a grid to align the digits as they divide.
Activity
None
Use long division to answer each question.
What is the value of ?
Five students raised \$77.40 for a charity. If everyone raised the same amount, how much money did each student raise?
Activity Synthesis
Invite 1–2 students to display and explain their work. Ask if others performed the division the same way, and discuss any disagreements. Ask students to share challenges, if any, that they encountered when carrying out long division.
Lesson Synthesis
Focus the discussion on how the long-division calculations are alike and different when the dividend is a whole number and when it is a decimal. Display the long-division calculations for and .
Discuss questions such as:
“How is the process of calculating like the process of finding ?” (When we get to the final place value, there is still a remainder. It is necessary to decompose the remainder and to add a zero after the last digit of the dividend.)
“How are they different?” (There is already a decimal in the dividend 53.8, but the dividend 62 is a whole number, so we needed to add a decimal point and a zero after it. The quotient of goes to the tenths place, or there are no more remainders after the tenths are divided. The quotient of goes to the hundredths place, so there is an extra step.)
“In general, when using long division, how can we keep dividing when there is a remainder? (Think of that number in terms of 10 of the next smaller place-value unit. This might mean adding a zero—like when seeing 2 ones as 20 tenths—or doing away with a decimal point—like when seeing 1.8 ones as 18 tenths.)
“How do we know where to put the decimal point in the quotient?” (We can line it up with the decimal point in the dividend and make sure that each digit in the quotient represents the correct place value.)
Student Lesson Summary
We can use long division to find quotients even when the numbers involved are not whole numbers. Here is the long-division calculation of , which results in a decimal quotient.
Long division calculation of 86 divided by 4. 8 rows. First row: 21 point 5, Second row: 4, long division symbol with 86 inside. Third row: minus 8. Horizontal line. Fourth row: 6. Fifth row: minus 4. Horizontal line. Sixth row: 2 point 0. Seventh row: minus 2 point 0. Horizontal line. Eighth row: 0.
The calculation shows that, after removing 4 groups of 21, there are 2 ones remaining. We can continue dividing by writing a 0 to the right of the 2 and thinking of that remainder as 20 tenths, which can then be divided into 4 groups.
To show that the quotient we are working with now is in the tenths place, we put a decimal point to the right of the 1 (which is in the ones place) at the top. It may also be helpful to draw a vertical line to separate the ones and the tenths.
There are 4 groups of 5 tenths in 20 tenths, so we write 5 in the tenths place at the top. The calculation likewise shows .
Standards Alignment
Building On
5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
In the third step, Lin drew a vertical dashed line to the right of the 2 in 62. What do you think that line is for?
She also wrote a point and a 0 to the right of 62. Then she put a 0 after the remainder of 2. What do you think the zeros are for?
Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?
What value did Lin find for ?
Use long division to find the value of each expression. Then pause so your teacher can review your work.
Use long division to show that:
, or , is 1.25.
, or , is 0.8.
, or , is 0.125.
Student Response
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Building on Student Thinking
Students may be unsure what to do when the divisor is too large for the digit being divided, such as when dividing the 9 in or the 4 in . Encourage students to think of these cases in terms of equal-size groups. Consider asking: “How many 12s can go into 9?” or “How many groups of 5 can be made with 4?” Remind them that a zero can be written in the quotient when a whole group cannot be made.
