Lesson 19

This activity extends division techniques used to find whole-number quotients to divide whole numbers resulting in (terminating) decimal quotients. In these problems, students see remainders in the ones place. In order to continue the division process, the ones are broken into tenths. Conceptually, this is the same unbundling idea that is used when hundreds are broken into tens or when tens are broken into ones.

If students have trouble drawing the diagrams to represent unbundling, consider providing actual base-ten blocks or paper cut-outs of base-ten units (from the blackline master used earlier in the unit) so that they can physically trade the pieces (e.g., 2 ones for 20 tenths).

Arrange students in groups of 2. Provide access to graph paper. Give students 1 minute of quiet think time to analyze Mai’s work and 2–3 minutes to discuss their observations with their partner. Pause for a whole-class discussion, making sure that all students understand how Mai dealt with the remainder.

The goal of this discussion is for students to relate their earlier work on division, which resulted in whole-number quotients, to division involving decimal quotients. Below is an example of how the discussion may go, along with questions to ask students and some possible responses. Begin the discussion by reminding students of the work they have previously done to evaluate \(657 \div 3\).

How is computation for \(62 \div 5\) similar to that for \(657 \div 3\)?

• The method of division is the same: we divide a given number (62 or 657) into equal groups until everything is distributed.
• We divide by using place value, unbundling one unit into ten of a smaller unit as needed. For \(62 \div 5\), the 2 ones can be broken into 20 tenths, while in \(657 \div 3\), the 2 tens were unbundled into 20 ones.

How is computation for \(62 \div 5\) different from that for \(657 \div 3\)?

• There is no remainder for \(657 \div 3\), while there is a remainder of 2 for \(62 \div 5\).
• We need to write a decimal point and work with tenths in \(62 \div 5\).

It is important to stress that the methods and steps are the same in both computations. The big new idea here is that sometimes a division problem of whole numbers does not end when we get to the ones place. In these cases, we have to add a decimal because the number being divided involves tenths, hundredths, or smaller base-ten units.

Representing, Speaking: MLR7 Compare and Connect. Use this routine to give students an opportunity to compare approaches for finding the quotient of \(511÷5\). Ask students to share their approach with a partner. Invite groups to discuss what is the same and what it different about finding a quotient either by drawing base-ten diagrams or by using the partial quotients method. Listen for opportunities to highlight language such as “remainder,” and “tenths.” This will help students connect division techniques and extend them to find decimal quotients.
Design Principle(s): Optimize output; Cultivate conversation

In this activity, students use long division to divide whole numbers whose quotient is not a whole number. Previously, students found the quotient of \(62 \div 5\) using base-ten diagrams and the partial quotients method. Because the long division is a particular version of the partial quotients method, and because students have been introduced to long division, they have the tools to divide whole numbers that result in a (terminating) decimal. In this activity, students evaluate and critique the reasoning of others (MP3).

Arrange students in groups of 2. Provide access to graph paper. Give students 7–8 minutes to analyze and discuss Lin’s work with a partner and then complete the second set of questions.

Pause to discuss students’ analyses and at least one of the division problems. Students should understand that, up until reaching the decimal point, long division works the same for \(62 \div 5 = 12.4\) as it does for \(657 \div 3 = 219\). In \(62 \div 5\), however, there is a remainder of 2 ones, and we need to convert to the next smaller place value (tenths), change the 2 ones into 20 tenths, and then divide these into 5 equal groups of 4 tenths.

Prepare students to do the last set of questions by setting up the long division of \(5 \div 4\) for all to see. Discuss the placement of the decimal point. Reiterate that we can write extra zeros at the end of the dividend, following the decimal point, so that remainders can be worked with. Have students work through this problem and the others. Follow with a whole-class discussion.

Students may be perplexed by the repeating decimals in the last question and think that they have made a mistake. Ask them to compare their work with a partner’s, and then clarify during discussion that some decimals do repeat. Because this work comes into focus in grade 7, the goal here is simply for students to observe that not all decimals terminate.

Focus the whole-class discussion on the third and fourth sets of questions. Ask a few students to show their long division for all to see and to explain their steps. Some ideas to bring to uncover:

• Problems like \(1 \div 25\) are challenging because the first step is 0: there are zero groups of 25 in 1. This means that we need to introduce a decimal and put a 0 to the right of the decimal. But one 0 is not enough. It is not until we add the second 0 to the right of the decimal that we can find 4 groups of 25 in 100. Because we moved two places to the right of the decimal, these 4 groups are really \(0.04\), which is the quotient of 1 by 25.
• Problems like \(1 \div 3\) are not fully treated until grade 7. At this point, we can observe that the long division process will go on and on because there is always a remainder of 1.

Some questions to ask students that highlight these points include:

• When you found \(1 \div 8\), what was your first step? (I put a 0 above the dividend 1 and added a decimal because I cannot take any 8’s from 1.)
• When you found \(1 \div 25\), what were your first few steps? (I put a 0 above the dividend 1 and added a decimal because I cannot take any 25’s from 1. I needed to add another 0 to the right of the decimal because I still cannot take any 25’s from 10.)
• Why do we not write 0’s in advance to the right of the decimal? (We do not know in advance if we will need these 0’s or how many of them we might need, so we usually write them as needed.)

Another valuable link to make is to connect the values for the decimals in Problem 4 to percentages. For example, the fact that \(\frac{4}{5} = 0.8\) means that \(\frac{4}{5}\) of a quantity is 80% of that quantity. Similarly \(\frac{1}{25}\) of a quantity is the same as 4% of that quantity.

Students have learned several effective methods to divide a whole number by a whole number, including cases when there is a remainder. The goal of this task is to introduce a method for dividing a decimal number by a whole number. Students notice that the steps in the division process are the same as when dividing a whole number by a whole number, whether the division is done with base-ten diagrams, as in this task, or using partial quotients or the division algorithm as in future tasks. Here, students need to think even more carefully about place value and where the decimal point goes in the quotient.

Throughout this activity, students rely on their understanding of equivalent expressions to interpret the unbundling in Elena’s process. For example, to unbundle a one into ten tenths means going between the expressions 1 and \(0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1\).

Give students 1–2 minutes of quiet think time for students to analyze Elena’s work. Pause and discuss with the whole class. Select a couple of students to share their analyses of what Elena had done to divide a decimal by a whole number. Then, give students 8–9 minutes to complete the questions and follow with a whole-class discussion.

Some students may stop dividing when they reach a remainder rather than unbundling the remainder into smaller units. Remind them that they can continue to divide the remainder by unbundling and to refer to Elena’s worked-out example or those from earlier lessons, if needed.

Ask a student or two to display and explain their work. Ask if others performed the division the same way and if there are disagreements.

Then, focus the discussion on the connections between a division problem with a whole-number dividend (such as \(62 \div 5\)) and that with a decimal dividend (such as \(53.8 \div 4\)). Discuss:

• How is the division problem \(53.8 \div 4\) similar to \(62 \div 5\) from a previous lesson?
  • In both problems, when we get to the final place value (tenths for \(53.8 \div 4\) and one for \(62 \div 5\)), there is still a remainder.
  • In both problems, to complete the division and find the quotient we need to introduce a new place value (hundredths for \(53.8 \div 4\), and tenths for \(62 \div 5\)).
  • We have to unbundle at every step in both division problems.
• How is the division problem \(53.8 \div 4\) different to \(62 \div 5\) from a previous lesson?
  • There is already a decimal in \(53.8\): we had to write the decimal point for \(62 \div 5\).
  • The quotient \(53.8 \div 4\) goes to the hundredths place, so there is an extra step and an additional place value.

If we were to rewrite \(62 \div 5\) as \(62.0 \div 5\) (which is what is needed in order to complete the division), then the two division problems look similar. The biggest difference between \(53.8 \div 4\) and \(62 \div 5\) is that the former problem has an answer in the hundredths while the answer to the latter only has tenths.