# Lesson 19

Dividing Numbers that Result in Decimals

## 19.1: Keep Dividing (5 minutes)

### Warm-up

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).

### Launch

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.

### Student Facing

Mai used base-ten diagrams to calculate $$62 \div 5$$. She started by representing 62.

She then made 5 groups, each with 1 ten. There was 1 ten left. She unbundled it into 10 ones and distributed the ones across the 5 groups.

Here is Mai’s diagram for $$62 \div 5$$.

1. Mai should have a total of 12 ones, but her diagram shows only 10. Why?
2. She did not originally have tenths, but in her diagram each group has 4 tenths. Why?
3. What value has Mai found for $$62 \div 5$$? Explain your reasoning.

### Activity Synthesis

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

## 19.2: Using Long Division to Calculate Quotients (25 minutes)

### Activity

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).

Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 15 minutes. Consider skipping the third and fourth sets of questions in this activity to save time for the next activity, where students will continue practicing these skills with problems where the dividend is a decimal.

### Launch

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.

Representation: Access for Perception. Display and read Lin’s method for calculating the quotient of $$62 \div 5$$ aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language; Conceptual processing

### Student Facing

Here is how Lin calculated $$62 \div 5$$.

• Lin put a 0 after the remainder of 2. Why? Why does this 0 not change the value of the quotient?
• Lin subtracted 5 groups of 4 from 20. What value does the 4 in the quotient represent?
• What value did Lin find for $$62 \div 5$$?
2. Use long division to find the value of each expression. Then pause so your teacher can review your work.

1. $$126 \div 8$$
2. $$90 \div 12$$

3. Use long division to show that:

1. $$5 \div 4$$, or $$\frac 54$$, is 1.25.

2. $$4 \div 5$$, or $$\frac 45$$, is 0.8.

3. $$1 \div 8$$, or $$\frac 18$$, is 0.125.

4. $$1 \div 25$$, or $$\frac {1}{25}$$, is 0.04.

4. Noah said we cannot use long division to calculate $$10 \div 3$$ because there will always be a remainder.

1. What do you think Noah meant by “there will always be a remainder”?
2. Do you agree with him? Explain your reasoning.

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to help students improve their written responses to the question, “Noah said we cannot use long division to calculate $$10÷3$$ because there will always be a remainder. Do you agree or disagree with this statement?” Give students time to meet with 2–3 partners to share and get feedback on their responses. Provide listeners with prompts that will help their partners strengthen their ideas and clarify their language (e.g., “Can you explain why there is always a remainder?”, “What number always remains?”, and “Could you try to explain this using a different example?”). Give students 1–2 minutes to revise their writing based on the feedback they received.​ Communicating their reasoning with a partner will help students understand situations in which long division does not work
Design Principle(s): Optimize output (for explanation); Cultivate conversation

## 19.3: Using Diagrams to Represent Division (15 minutes)

### Activity

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 $$0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1+0.1$$.

### Launch

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.

### Student Facing

To find $$53.8 \div 4$$ using diagrams, Elena began by representing 53.8.

She placed 1 ten into each group, unbundled the remaining 1 ten into 10 ones, and went on distributing the units.

This diagram shows Elena’s initial placement of the units and the unbundling of 1 ten.

1. Complete the diagram by continuing the division process. How would you use the available units to make 4 equal groups?

As the units get placed into groups, show them accordingly and cross out those pieces from the bottom. If you unbundle a unit, draw the resulting pieces.

2. What value did you find for $$53.8 \div 4$$? Be prepared to explain your reasoning.

3. Use long division to find $$53.8 \div 4$$. Check your answer by multiplying it by the divisor 4.

4. Use long division to find $$77.4 \div 5$$. If you get stuck, you can draw diagrams or use another method.

### Student Facing

#### Are you ready for more?

A distant, magical land uses jewels for their bartering system. The jewels are valued and ranked in order of their rarity. Each jewel is worth 3 times the jewel immediately below it in the ranking. The ranking is red, orange, yellow, green, blue, indigo, and violet. So a red jewel is worth 3 orange jewels, a green jewel is worth 3 blue jewels, and so on.

A group of 4 craftsmen are paid 1 of each jewel. If they split the jewels evenly amongst themselves, which jewels does each craftsman get?

### Anticipated Misconceptions

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.

### Activity Synthesis

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.

Representing, Speaking, Listening: MLR2 Collect and Display. As students discuss how the division problem $$53.8÷4$$ is related to $$62÷5$$ from a previous lesson, create a two-column display with “similarities” and “differences” for the headers. Circulate through the groups and record student language in the appropriate column. Look for phrases such as “whole-number dividend,” “decimal dividend,” and “place value.” This will help students compare and contrast different types of division problems by recognizing the structure of the numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we saw that the quotient of two whole numbers can result in a decimal value. We can observe how this happens with base-ten blocks and by calculating with long division.

• When using base-ten blocks to divide, how can we work with a remainder of ones? (Unbundle the ones into tenths.)
• When calculating with long division, how can we keep dividing when there is a remainder? (Put a decimal point after the dividend and follow it with zeros. This allows you to bring down a zero and continue the calculation.)
• How do we divide a whole number that is smaller than the divisor, for instance $$4 \div 5$$? (Start by writing a 0 for the quotient and follow it with a decimal point. In the example of $$4 \div 5$$, the 0 means there is not enough ones to divide equally into 5 groups. Then, unbundle the number into ten of the next smaller unit so that it can be divided. In this example, 4 ones can be unbundled into 40 tenths, which can then be divided by 5.)

## Student Lesson Summary

### Student Facing

Dividing a whole number by another whole number does not always produce a whole-number quotient. Let’s look at $$86 \div 4$$, which we can think of as dividing 86 into 4 equal groups.

We can see in the base-ten diagram that there are 4 groups of 21 in 86 with 2 ones left over. To find the quotient, we need to distribute the 2 ones into the 4 groups. To do this, we can unbundle or decompose the 2 ones into 20 tenths, which enables us to put 5 tenths in each group.

Once the 20 tenths are distributed, each group will have 2 tens, 1 one, and 5 tenths, so $$86 \div 4 = 21.5$$.

We can also calculate $$86 \div 4$$ using long division.

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 tenth 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 $$86 \div 4 = 21.5$$.