Lesson 13

Dividing Decimals by Decimals

13.1: Same Values (5 minutes)

Warm-up

In this warm-up, students continue the decimal division work from the previous lesson and do so in the context of money. The work reinforces the idea that the value of a quotient does not change if the numerator and denominator are both multiplied by the same power of 10.

Launch

Give students a moment to read the first question and to estimate whether the quotient will be less than 1 or more than 1. Ask them to give a signal when they have an estimate and can explain it. Ask one student from the “more than 1” group to explain their reasoning and another from the “less than 1” group to do the same. Clarify that the quotient will be less than 1 and give students a few minutes to complete the questions. If time is limited, ask students to work only on the second question. Follow with a whole-class discussion.

Student Facing

1. Use long division to find the value of $$5.04 \div 7$$.

2. Select all of the quotients that have the same value as $$5.04 \div 7$$. Be prepared to explain how you know.

1. $$5.04 \div 70$$
2. $$50.4 \div 70$$
3. $$504,\!000 \div 700$$
4. $$504,\!000 \div 700,\!000$$

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Focus the whole-class discussion on the second question. Select several students to explain why choices b and d are correct and why a and c are not. Students should see that $5.04 \div 70$ and $504,\!000 \div 700$ are not equivalent expressions to $5.04 \div 7$ because the dividend and divisors in each pair are not results of multiplying the 5.04 and 7 by the same factor or the same power of 10.

13.2: 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.

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 Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

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.

13.3: Two Ways to Calculate Quotients of Decimals (15 minutes)

Optional activity

This lesson demonstrates how the division of two equivalent expressions (e.g., $$48.78 \div 9$$ and $$4878 \div 900$$) result in the same quotient. By looking at worked-out calculations, students reinforce their understanding about what each part of the calculations represent. The advantage of representing a quotient with a different equivalent expression is so students can simplify problems involving division of decimals by rewriting expressions using whole numbers.

They use the structure of base-ten numbers (MP7) to move the decimal point (through multiplication by an appropriate power of 10). They also 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.

Launch

Arrange students in groups of 2. Give partners 5 minutes to discuss the first problem and then quiet work time for the second problem. Follow with a whole-class discussion.

Action and Expression: Develop Expression and Communication. To help get students started, display sentence frames such as “Both _____ and _____ are alike because…” and “_____ and _____ are different because…”
Supports accessibility for: Language; Organization

Student Facing

1. Here are two calculations of $$48.78 \div 9$$. Work with your partner to answer the following questions.

1. How are the two calculations the same? How are they different?
2. Look at Calculation A. Explain how you can tell that the 36 means “36 tenths” and the 18 means “18 hundredths.”
3. Look at Calculation B. What do the 3600 and 1800 mean?
4. We can think of $$48.78 \div 9=5.42$$ as saying, “There are 9 groups of 5.42 in 48.78.” We can think of $$4878 \div 900=5.42$$ as saying, “There are 900 groups of 5.42 in 4878.” How might we show that both statements are true?
1. Explain why $$51.2 \div 6.4$$ has the same value as $$5.12 \div 0.64$$.
2. Write a division expression that has the same value as $$51.2 \div 6.4$$ but is easier to use to find the value. Then, find the value using long division.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The goal of this discussion is for students to contrast the two division methods used in the task. Discuss:

• Why is the value of $$48.78 \div 9$$ the same as the value of $$4,\!878 \div 900$$? (The numbers in the second expression are both multiplied by 100, and this does not change the value of the quotient.)
• What are some advantages of calculating $$48.78 \div 9$$ with the decimal intact (the method on the left)? (It is fast, and we don’t need to deal with a bunch of 0’s. Also, if the numbers are from a contextual problem, we could better make meaning of them in their original form.)
• What are some advantages of calculating $$4,\!878 \div 900$$ with long division? (These are whole numbers, and we are familiar with how to divide whole numbers. Also, we could express this as a fraction and write an equivalent fraction of $$\frac{543}{100}$$, which then tells us that its value is 5.43.)

End the discussion by telling students that they will next look at quotients where both the divisor and the dividend are decimals. The method used here of multiplying both numbers by a power of 10 will apply in that situation as well.

Speaking: MLR8 Discussion Supports. As students discuss contrasts between the two calculations of $$48.78÷9$$, press for details and mathematical language in their explanations. Encourage students to elaborate on the idea that multiplying the dividend and divisor by the same factor does not change the value of the quotient. Provide these sentence frames to support the discussion: “The values are the same because ….”, “I can use powers of 10 by ….”, “Some advantages of Calculation A/Calculation B are ….”. This will help students make sense of the structure of base-ten numbers to move the decimal point through multiplication by an appropriate power of 10.
Design Principles(s): Support sense-making

13.4: 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.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

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.)

13.5: Cool-down - The Quotient of Two Decimals (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

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: