# Lesson 20

Dividing Decimals by Decimals

## 20.1: Number Talk: Dividing by 4 (5 minutes)

### Warm-up

The purpose of this number talk is to help students use the structure of base-ten numbers and the distributive property to solve a division problem involving decimals.

### Launch

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

### Student Facing

Find each quotient mentally.

$$80 \div 4$$

$$12 \div 4$$

$$1.2 \div 4$$

$$81.2 \div 4$$

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone have 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?”

Highlight the use of the distributive property in finding $$81.2 \div 4$$. Students should recognize that since $$81.2=80+1.2$$, we have $$81.2\div 4=(80\div 4)+(1.2\div 4)$$. To make this clear, consider explaining that the division could be equivalently represented by $$81.2\boldcdot \frac14=(80+1.2)\boldcdot\frac14=(80\boldcdot \frac14)+(1.2\boldcdot\frac14)$$.

Speaking: MLR8 Discussion Supports. Provide sentence frames to support students with explaining their strategies. For example, “I noticed that ______, so I ______.” or “First, I ________ because ________.” When students share their answers with a partner, prompt them to rehearse what they will say when they share with the full group. Rehearsing provides opportunities to clarify their thinking.
Design Principle(s): Optimize output (for explanation)

## 20.2: Dividends and Divisors (15 minutes)

### Activity

In this activity, students study some carefully chosen quotients where the dividends are decimal numbers. The key goal here is to notice that there are other quotients of whole numbers that are equivalent to these quotients of decimals. In other words, when the dividend is a terminating decimal number, we can find an equivalent quotient whose dividend is a whole number. In combination with the previous task, this gives students the tools they need to divide a decimal number by a decimal number.

This activity strongly supports MP7. Students notice that, when working with a fraction, multiplying the numerator and denominator in a fraction by 10 does not change the value of the fraction. They use this insight to develop a way to divide decimal numbers in subsequent activities. This work develops students’ understanding of equivalent expressions by emphasizing that, for example, $$8\div 1= (8\boldcdot 10) \div (1\boldcdot 10)$$. Eventually, students will recognize the equivalence of $$8\div 1$$ to statements such as $$(8\boldcdot y)\div (1\boldcdot y)$$. However, in this activity, students only examine situations where the dividend and divisor are multiplied by powers of 10.

Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 10 minutes.

### Launch

Display the following image of division calculations for all to see.

Ask students what quotient each calculation shows. ($$8 \div 1$$, $$800 \div 100$$, and $$80,\!000 \div 10,\!000$$).  Give students 1–2 minutes to notice and wonder about the dividends, divisors, and quotients in the three calculations. Ask them to give a signal when they have at least one observation and one question. If needed, remind students that the 8, 800, and 80,000 are the dividends and the 1, 100, and 10,000 are the divisors.

Invite a few students to share their observations and questions. They are likely to notice:

• Each calculation shows that the value of the corresponding quotient is 8; it is the same for all three calculations.
• All calculations have an 8 in the dividend and a 1 in the divisor.
• All calculations take one step to solve.
• Each divisor is 100 times the one to the left of it.
• Each dividend is 100 times the one to the left of it.
• Each dividend and each divisor have 2 more zeros than in the calculation immediately to their left.

They may wonder:

• Why are the quotients equal even though the divisors and dividends are different?
• Would $$80 \div 10$$ and $$8,\!000 \div 10$$ also produce a quotient of 8?
• Are there other division expressions with an 8 in the dividend and a 1 in the divisor and no other digits but zeros that would also produce a quotient of 8?

Without answering their questions, tell students that they’ll analyze the sizes of dividends and divisors more closely to help them reason about quotients of numbers in base ten.

Give students 7–8 minutes of quiet work time to answer the four questions followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

### Student Facing

Analyze the dividends, divisors, and quotients in the calculations, and then answer the questions.

1. Complete each sentence. In the calculations shown:

• Each dividend is ______ times the dividend to the left of it.

• Each divisor is ______ times the divisor to the left of it.

• Each quotient is _____________________ the quotient to the left of it.

2. Suppose we are writing a calculation to the right of $$72,\!000 \div 3,\!000$$. Which expression has a quotient of 24? Be prepared to explain your reasoning.

1. $$72,\!000 \div 30,\!000$$
2. $$720,\!000 \div 300,\!000$$
3. $$720,\!000 \div 30,\!000$$
4. $$720,\!000 \div 3,\!000$$
3. Suppose we are writing a calculation to the left of $$72 \div 3$$. Write an expression that would also give a quotient of 24. Be prepared to explain your reasoning.
4. Decide which of the following expressions would have the same value as $$250 \div 10$$. Be prepared to share your reasoning.

1. $$250 \div 0.1$$
2. $$25 \div 1$$
3. $$2.5 \div 1$$
4. $$2.5 \div 0.1$$
5. $$2,500 \div 100$$
6. $$0.25 \div 0.01$$

### Activity Synthesis

Ask students to write a reflection using the following prompt:

What happens to the value of the quotient when both the divisor and the dividend are multiplied by the same power of 10? Use examples to show your thinking.

The goal of this discussion is to make sure students understand that the value of a quotient does not change when both the divisor and the dividend are multiplied by the same power of ten. Ask students to explain why $$\frac{25}{20} = \frac{250}{200}$$. Possible responses include:

• Both the numerator and denominator of $$\frac{250}{200}$$ have a factor of 10, so the fraction can be written as $$\frac{25}{20}$$.
• Both fractions are equivalent to $$\frac{5}{4}$$.
• Dividing 250 by 200 and 25 by 20 both give a value of 1.25.

Tell students that their observations here will help them divide decimals in upcoming activities.

Speaking, Listening: MLR7 Compare and Connect. After students have answered the four questions in this activity, ask them to work in groups of 2–4 and identify what is similar and what is different about the approaches they used in analyzing the dividends, divisors, and quotients for the last question. Lead a whole-class discussion that draws students attention to what worked or did not work well when deciding which expressions have the same value as $$250÷10$$ (i.e., using powers of 10, long division, unbundling, etc). Look for opportunities to highlight mathematical language and reasoning involving multiplying or dividing by powers of 10. This will foster students’ meta-awareness and support constructive conversations as they develop understanding of equivalent quotients.
Design Principles(s): Cultivate conversation; Maximize meta-awareness

## 20.3: Placing Decimal Points in Quotients (15 minutes)

### Activity

The goal of this task is to show that we can calculate quotients of two decimals by “moving the decimal point” (multiplying both numbers by an appropriate power of 10) and, as a result, work only with whole numbers. Students can calculate the quotient of whole numbers using long division or another method of their choice. Students also have an opportunity to evaluate and critique another’s reasoning (MP3).

Students use the structure of base-ten numbers (MP7) to move the decimal point (through multiplication by an appropriate power of 10), and they use their understanding of equivalent expressions to know that multiplying both the numbers in a division by the same factor does not change the value of the quotient. Both pieces of knowledge allow students to replace a quotient of decimal numbers with a quotient of whole numbers.

Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 10 minutes.

### Launch

Arrange students in groups of 2. Give students 3 minutes of quiet time to consider how to find the first quotient. Encourage them to think of more than one way to do so, if possible. Then, give partners 2–3 minutes to discuss their methods and another 2–3 minutes to find the second quotient together. Follow with a brief whole-class discussion, reviewing the first two questions. If not brought up by a student, discuss the equivalent expressions $$300 \div 12$$ and $$1,\!800 \div 4$$. Consider bringing up the first expression and asking students to find an analogous expression for the second problem.

Ask students to finish the last problem and follow with a whole-class discussion.

Engagement: Develop Effort and Persistence. Connect a new concept to one with which students have experienced success. For example, remind students about the quotients that have the same value as $$5.04 \div 7$$ in the previous activity. Explain to students that multiplying both the numbers in a division by the same factor does not change the value of the quotient. Ask students how they can use this idea to find a quotient that has the same value as $$3 \div 0.12$$.
Supports accessibility for: Social-emotional skills; Conceptual processing
Speaking, Listening: MLR7 Compare and Connect. After students have 3 minutes of quiet time to consider how to find the first quotient, ask them to create a visual display that shows their strategy and a brief explanation. Give students time to meet with 2–3 partners, to share discuss connections they notice between their different approaches. Follow up with a whole-class discussion to identify and highlight correspondences between different approaches or representations you observe in the room. Listen for and amplify key phrases such as “multiply both numbers by 10,” “move the decimal point,” “use whole numbers,” or “create an equivalent expression.” This will help students make sense of mathematical strategies by relating and connecting other approaches to their own.
Design Principles(s): Cultivate conversation; Maximize meta-awareness

### Student Facing

1. Think of one or more ways to find $$3 \div 0.12$$. Show your reasoning.
2. Find $$1.8 \div 0.004$$. Show your reasoning. If you get stuck, think about what equivalent division expression you could write.
3. Diego said, “To divide decimals, we can start by moving the decimal point in both the dividend and divisor by the same number of places and in the same direction. Then we find the quotient of the resulting numbers.”

Do you agree with Diego? Use the division expression $$7.5 \div 1.25$$ to support your answer.

### Student Facing

#### Are you ready for more?

Can we create an equivalent division expression by multiplying both the dividend and divisor by a number that is not a multiple of 10 (for example: 4, 20, or $$\frac12$$)? Would doing so produce the same quotient? Explain or show your reasoning.

### Activity Synthesis

The goal of this discussion is to help students recognize when division expressions are equivalent.

Ask students to write a division expression that looks like it might be equivalent to either $$3 \div 0.12$$ or $$1.8 \div 0.004$$ but has different decimal point locations. Select a few students to share their expression with the class.

## 20.4: Practicing Division with Decimals (15 minutes)

### Activity

In this activity, students practice calculating quotients of decimals by using any method they prefer. Then, they extend their practice to calculate the division of decimals in a real-world context. While students could use ratio techniques (e.g., a ratio table) to answer the last question, encourage them to use the division of decimal numbers. The application of division to solve real-world problems illustrates MP4.

As students work on the first three problems, monitor for groups in which students have different strategies used on the same question.

### Launch

Arrange students in groups of 3–5. Give groups 5–7 minutes to work through and discuss the first three questions. Ask them to consult with you if there is a disagreement about a correct answer in their group. (If this happens, let them know which student’s work is correct and have that student explain their thinking so all group members are in agreement.)

After all group members have answered the first three questions and have the same answer, have them complete the last question. Follow with a whole-class discussion.

### Student Facing

Find each quotient. Discuss your quotients with your group and agree on the correct answers. Consult your teacher if the group can't agree.

1. $$106.5 \div 3$$
2. $$58.8 \div 0.7$$
3. $$257.4 \div 1.1$$

4. Mai is making friendship bracelets. Each bracelet is made from 24.3 cm of string. If she has 170.1 cm of string, how many bracelets can she make? Explain or show your reasoning.

### Anticipated Misconceptions

Some students might have trouble calculating because their numbers are not aligned so the place-value associations are lost. Suggest that they use graph paper for their calculations. They can place one digit in each box for proper decimal point and place-value alignment.

### Activity Synthesis

The purpose of this discussion is to highlight the different strategies used to answer the division questions. Select a previously identified group that used different strategies on one of the first three questions. Ask each student in the group to explain their strategy and why they chose it. For the fourth question, ask students:

• “Could the answer be found by calculating the quotient of this expression: $$24.3 \div 170.1$$?” (No, because the question is asking how many pieces of string of length 24.3 are in the long string of length 170.1. This is equivalent to asking how many groups of 24.3 are in 170.1 or $$170.1\div 24.3$$.)
• “What would the quotient $$24.3 \div 170.1$$ represent in the context of the problem?” (This quotient would represent what fraction the length of bracelet string is of the full length of string.)
Writing, Speaking, Conversing: MLR3 Clarify, Critique, and Correct. Before students share their answers, present an incorrect solution that uses long division to find the quotient of $$106.5÷3$$. Consider using this statement to open the discussion: “Sam believes the quotient of $$106.5÷3$$ is 355 because he multiplied 106.5 by 10 and then got his answer.” Ask pairs to identify the ambiguity or error, and critique the reasoning presented in the statement. Guide discussion by asking, “What do you think Sam was thinking when performing this division problem?”, “What strategy did he use to find the quotient?” and/or “What is unclear?” Invite pairs to offer a response that includes a correct version of Sam’s long division (e.g., the long division work for $$1,\!065÷30$$). This will help students explain how to use multiplication by powers of 10 in both the divisor and dividend to create equivalent expressions.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we saw that we can divide decimals by decimals by first making the decimals whole numbers. As long as we multiply both numbers by the same number, the value of the quotient will not change.

• What equivalent expression can we write to help us find $$18.4 \div 0.2$$? (We can multiply both numbers by 10 to get $$184 \div 2$$, which is equivalent to the original expression.)
• How might we find the quotient of $$184 \div 2$$? (We can use any methods learned so far: base-ten diagrams, partial quotients, or long division.)
• Do we always multiply the dividend and divisor by 10? For example, what number should we multiply to enable us to find $$1.25 \div 0.005$$? (We can multiply by any power of 10. In this example, we should multiply both numbers by 1,000 to turn the 0.005 into 5, so that we can find $$1,\!250 \div 5$$.)
• Why is it helpful to multiply by a power of 10 instead of another number that is not a power of 10? (Because we are working with base-ten numbers, multiplying by a power of 10 allows us to easily “remove” the decimal point from a decimal so that we end up with a whole number.)

## Student Lesson Summary

### Student Facing

One way to find a quotient of two decimals is to multiply each decimal by a power of 10 so that both products are whole numbers.

If we multiply both decimals by the same power of 10, this does not change the value of the quotient. For example, the quotient $$7.65 \div 1.2$$ can be found by multiplying the two decimals by 10 (or by 100) and instead finding $$76.5 \div 12$$ or $$765 \div 120$$

To calculate $$765 \div 120$$, which is equivalent to $$76.5 \div 12$$, we could use base-ten diagrams, partial quotients, or long division. Here is the calculation with long division: