Unit 3, Lesson 19
Dividing Numbers that Result in a Decimal
Accelerated 6
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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).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Find the value of each quotient mentally.
To involve more students in the conversation, consider asking:
Highlight how partial quotients are used in finding
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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
Ask students:
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:
If not made explicit in students’ explanations, point out that up until reaching the decimal point, long division works the same for
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.
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:
To Gather
Graph paper
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
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
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.
Ask students:
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.
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?
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.
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
Discuss questions such as:
We can use long division to find quotients even when the numbers involved are not whole numbers. Here is the long-division calculation of
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
Help us improve by sharing suggestions or reporting issues.
Here is how Lin calculated
Discuss with your partner:
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:
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