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# Dividing by Decimals

Lesson 18 of 19

## Objective: SWBAT divide a decimal number by a decimal number by multiplying the divisor and dividend by a multiple of ten in order to create an equivalent expression with a whole number divisor

## Big Idea: To divide a decimal by a decimal, we make an equivalent expression with a whole number divisor. The easiest way to do this is to multiply the dividend and divisor by powers of ten until the divisor is a whole number.

*60 minutes*

#### Think About It

*7 min*

Students complete the Think About It problem in pairs. Although students have not yet learned how to divide with a decimal divisor, they will be able to solve this problem. Most students will use repeated addition and/or mental math to determine that Autumn bought 4 stickers.

After 2-3 minutes of student work time, I call on a student to share how (s)he determined the answer. I'll then ask students how this problem is different than the division problems we've been working on.

I then frame the lesson by telling students that to solve a division problem where we have to divide by a decimal, we can write an equivalent expression that has the same value but uses a whole number divisor.

#### Resources

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#### Intro to New Material

*15 min*

To start the Intro to New Material section, I model how to solve the Think About It problem, using equivalent expressions. The steps that I follow:

**Steps for Dividing Decimal Numbers**

1) Rewrite the division problem as a fraction with the dividend over the divisor.

2) Create an equivalent division problem by multiplying both the dividend and the divisor by a power of 10 that makes the divisor a whole number.

3) Rewrite the division problem using the equivalent division problem.

4) Divide using the standard algorithm.

I ask students why 1.60/.40 is the same as 16/4. I am looking for students to articulate that because we are multiplying by a fraction equal to 1 (10/10) which does not change the value of the original expression (Identity Property of Multiplication).

Together, we fill in the first **key idea**: To divide a decimal by a decimal, we can make an equivalent expression by multiplying the dividend and the divisor by powers of ten so that the divisor is a whole number.

I have students guide us through completing problems 1-3, and then have everyone write a response to the prompt. This prompt is one that the students have seen in previous lessons in this unit, and aligns to the second **key idea **of this lesson. Students are able to fill in the blanks without my help, as this is a focus of all of the division work we've done together. When the divisor is less than the dividend, the quotient will be larger than the dividend. When the divisor is greater than the dividend, the quotient will be smaller than the dividend.

#### Resources

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#### Partner Practice

*15 min*

Students work in pairs on the Partner Practice problem set. As they are working, I circulate around the room and check in with each group. I am looking for:

- Are students explaining their thinking to their partner?
- Are students writing their work in the work space?
- Are students using multiplication to check their work?
- Are students making equivalent expressions with whole number divisors?
- Are students answering in a full sentence, when appropriate?

I'm asking:

- Explain you determined this quotient.
- How did you use multiplication to check your answer?
- How did you know where to place the decimal point in your quotient?
- How did you make an equivalent expression?

After 10 minutes of partner work time, I bring the class back together for a conversation. I have a student present his/her thinking for the first problem. The student is likely to have multiplied both the numerator and denominator by 10. I ask if it would be correct to multiply the numerator and denominator by 100. My goal here is to reiterate that as long as we multiply the divisor and the dividend by the same number, our expressions will be equivalent.

#### Resources

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#### Independent Practice

*15 min*

Students work on the Independent Practice problem set.

For Problem 2 (and any word problem in this lesson), I do expect students to annotate and draw a model to represent the problem. These are sustained expectations throughout the year.

During this lesson, some students will be able to make generalizations about how to work with decimal divisors. Rather than set up equivalent fractions, some students will see the pattern of 'moving' the decimal the same number of times in the divisor and the dividend. When I see students making this connection, I encourage them to test it out along side the power of ten work. I'll let them know that the ability to make equivalent fractions and divide using the standard algorithm is what they're working on mastering today, but I'll praise them (heavily) for their insightful observations.

#### Resources

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#### Closing and Exit Ticket

*8 min*

After independent work time, I have students turn to their partners and share their thinking for Problem 6. Estimation is an important tool, and I want to be sure my students are building really strong number sense during their year with me.

Students then work on the Exit Ticket independently to close the lesson. Some exit ticket exemplars are included here.

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- UNIT 1: Number Sense
- UNIT 2: Division with Fractions
- UNIT 3: Integers and Rational Numbers
- UNIT 4: Coordinate Plane
- UNIT 5: Rates and Ratios
- UNIT 6: Unit Rate Applications and Percents
- UNIT 7: Expressions
- UNIT 8: Equations
- UNIT 9: Inequalities
- UNIT 10: Area of Two Dimensional Figures
- UNIT 11: Analyzing Data

- LESSON 1: Performance Task Lesson - GCF
- LESSON 2: Finding Factors
- LESSON 3: Finding Greatest Common Factors (GCF) Using T-Charts
- LESSON 4: Finding Greatest Common Factors (GCF) Using Prime Factorization
- LESSON 5: Performance Task Lesson - LCM
- LESSON 6: Multiples and Least Common Multiples (LCM)
- LESSON 7: Factors and Multiples in the Real World
- LESSON 8: Distributive Property
- LESSON 9: Division Bar Models
- LESSON 10: Estimating Quotients Using Compatible Numbers
- LESSON 11: Division: One-digit Divisors
- LESSON 12: Division: Two-Digit Divisors, Part 1
- LESSON 13: Division: Two-Digit Divisors, Part 2
- LESSON 14: Solve and Interpret Division Problems
- LESSON 15: Adding and Subtracting Decimals
- LESSON 16: Multiplying Decimals
- LESSON 17: Decimal Quotients
- LESSON 18: Dividing by Decimals
- LESSON 19: Performance Task Lesson - Decimal Operations