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# Where Does the Decimal Point Go?

## Overview

Students use estimation or other methods to place the decimal points in products and quotients. They review the algorithms for the four basic decimal operations and solve multistep word problems involving decimal operations.

# Key Concepts

The algorithms for whole-number operations can be extended to decimal operations. Students learned the algorithms for decimal operations in Grade 5. By the end of Grade 6, they should be fluent with these operations.

For decimal addition and subtraction, once the decimal points of the addends are aligned (which aligns like place values), the algorithms are the same as for whole numbers. The decimal point in the sum or difference goes directly below the decimal point in the numbers that were added or subtracted.

For decimal multiplication and division, one method is to ignore the decimal points and apply the whole-number algorithms. Then use estimation or some other method to place the decimal point in the answer.

# Goals and Learning Objectives

- Review and practice the algorithms for all four decimal operations.
- Solve real-world problems involving decimal operations.

# Where Does the Decimal Point Go?

Explain to students that the answers to the problems in the Opening have the correct digits but may be missing decimal points.

# Mathematics

For the multiplication problems, students are likely to use one of two methods:

- Estimate. For example, for 81.2 × 0.65, the product must be between half of 81.2 and all of 81.2, because 0.65 is more than half and less than 1. Therefore, it is 52.780.
- Count decimal places. For 81.2 × 0.65, there is one decimal place in 81.2, and there are two decimal places in 0.65. So, the product must have three decimal places. Therefore, it is 52.780.

The rule can be misleading in some cases. For example, students may think the equation 81.2 × 0.65 = 52.78 is incorrect because it appears to violate the rule: the factors have a total of 3 decimal places, while the product has only 2. However, ignoring the decimal points and multiplying, gives 812 × 65 = 52,780. Placing the decimal point to get three places gives 52.780, which is equivalent to 52.78.

For the division problems, students are likely to use one of two methods:

- Estimate. For example, for 5.8 ÷ 0.4, 5.8 is more than 10 × 0.4, which is 4, and less than 20 × 0.4, which is 8. Therefore, the answer must be between 10 and 20, and so the answer is 14.5 .
- Change the divisor to a whole number. Move the decimal points the same number of places in the dividend and divisor to get a whole number divisor (This is the same as multiplying both numbers by the same power of 10). The decimal point in the quotient will then go directly above the decimal point in the dividend.

SWD: Using approximation or estimation can be challenging for some students with disabilities. Review and reinforce this skill with students prior to having them start the tasks in which they will be expected to approximate or estimate.

# Answers

26 × 2.6 = 67.6

408 × 0.07 = 28.56

81.2 × 0.65 = 52.780

5.8 ÷ 0.4 = 14.5

22.57 ÷ 61 = 0.37

378 ÷ 0.12 = 3,150

## Opening

# Where Does the Decimal Point Go?

The decimal points are missing from these products and quotients.

- Work with your partner to place the decimal points in the correct locations.

# Generalize from Repetitions

# Mathematical Practices in Action

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

Have students watch the video that shows Jan and Carlos engaged in Mathematical Practice 8: Look for and express regularity in repeated reasoning. After students watch the video, give partners time to talk about the discussion questions. Then lead a whole-class discussion. Focus the discussion on how Jan’s observation about where to place the decimal point in decimal multiplication problems led her to a general rule. Discuss how Jan justified her rule and have students talk about if they think Jan’s rule works. You may want to try some examples as a class by placing the decimal using estimation and placing the decimal using Jan’s rule to see if you get the same answer either way.

In the discussion, elicit that in mathematics, it is important to observe and look for patterns and things that repeat over and over again.

## Opening

# Generalize from Repetitions

Watch the video that shows Jan and Carlos using a general rule that they developed after seeing something happen over and over again.

- What do you think Jan noticed over and over again as she solved decimal multiplication problems?
- How did this repetition help her come up with her general rule?
- How did Jan justify that her rule works?
- Do you think Jan’s rule works?

VIDEO: Mathematical Practice 8

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will solve mathematical and real-world problems involving multidigit decimals.

## Opening

Solve mathematical and real-world problems involving multidigit decimals.

# Perform Operations with Multidigit Decimals

# Lesson Guide

These problems review the four operations. Give students about 5 minutes to solve the problems, working individually.

SWD: Individual work is intended to be a time for students to grapple with the mathematics. It is crucial that you only help students to understand the tasks and that you not solve the problem.

# Mathematics

**[common error]** Some students may forget to align the decimal points in the addition and subtraction problems. Remind them that they must add digits with the same place value. Aligning decimal points will ensure that the digits are in the correct place-value columns.

# Answers

- 26.19 + 378.5 = 404.69
- 56.4 − 5.22 = 51.18
- 7.5 × 0.57 = 4.275
- 19.76 ÷ 5.2 = 3.8

## Work Time

# Perform Operations with Multidigit Decimals

Solve each problem and show your steps.

- 26.19 + 378.5
- 56.4 – 5.22
- 7.5 × 0.57
- 19.76 ÷ 5.2

Ask yourself:

- How can you use estimation to help you determine where to place the decimal point?

# Prepare a Presentation

# Preparing for Ways of Thinking

Look for students to share their solutions to each of the four problems during Ways of Thinking. Choose both incorrect and correct solutions. You can use the incorrect solutions to highlight common errors and clear up any misconceptions students may have.

Identify students who:

- Correctly align the decimal points in the addition and subtraction problems.
- Forget to align the decimal points in the addition and subtraction problems.
- Use estimation to correctly place the decimal point in multiplication and division problems.
- Use another method to correctly place the decimal point in multiplication and subtraction problems.
- Identify ways the problems are similar and different.

## Work Time

# Prepare a Presentation

Explain how you found the solution to each problem. What was similar about the processes? What was different?

# Make Connections

# Mathematics

Present, or have a student present, the solutions to each of the Work Time problems, explaining each step. Emphasize that the algorithms are essentially the same as for whole-number operations. However, for addition and subtraction, students must make sure to align like place values, and for multiplication and division, they must use estimation or some other method to place the decimal point.

ELL: Have students create a graphic organizer that incorporates all conventions for adding, subtracting, multiplying, and dividing decimals. Include the “estimation” and “counting decimal places” methods.

## Performance Task

# Ways of Thinking: Make Connections

Take notes on your classmates’ methods for performing operations with multidigit decimals.

As your classmates present, ask questions such as:

- How are operations with whole numbers similar to operations with decimals?
- How can you use estimation to check whether your solution is reasonable?
- Why do you need to align the decimal points when adding and subtracting with decimals?
- In a multiplication problem with decimals, how do you know where to place the decimal point in the product?
- In a division problem with decimals, why do you move the decimal point the same number of places in both the dividend and divisor?
- Why does your solution method make sense?

# Real-World Problems with Multidigit Decimals

# Lesson Guide

Students should work with a partner and choose one problem in this Work Time section. These problems give students practice using decimal operations in a real-world context. All of the problems require more than one operation to solve. Encourage students to check their answer in the context of the problem situation to make sure it is reasonable.

For students working on the Challenge Problem, suggest that one of their operations could be about calculating the sales tax for their purchases.

Look for a variety of solution strategies. Choose at least two solutions for each problem to be presented during Ways of Thinking. Choose correct solutions as well as incorrect solutions.

Some students may have interesting ways to simplify some of the computations or to do them mentally. Have students share their methods in Ways of Thinking.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Some of the problems are quite complex and all of them require several steps to solve. Students will need to analyze each problem to determine what information they know and what they need to find and then devise a plan for finding a solution.

**Mathematical Practice 4: Model with mathematics.**

Students may use expressions, equations, or diagrams to model each problem. If students find that an answer does not make sense in the problem context, they will need to revisit their model to see if they made an error.

**Mathematical Practice 6: Attend to precision.**

Students are asked to show all their steps so others can follow their work. Encourage students to use correct mathematical notation and to label any diagrams they create.

# Interventions

**Student has difficulty starting a problem.**

- Describe the problem in your own words to your partner.
- What do you know?
- What are you trying to find?
- About how big will the answer be?

**Student has an incorrect solution.**

- Have you checked your work?
- Does your answer seem reasonable in the problem context?

ELL: Explain the meaning of suntan lotion, flip-flops, ice pop, and any other terms students are not familiar with. You might show photographs or make drawings to illustrate some of these terms.

# Answers

- $3.16

Total cost of items is 3 × $2.79 + $8.47 = $16.84

Change is $20 − $16.84 = $3.16 - 5 ice pops

He spent $24.35 − $19.95, or $4.40 on ice pops.

$4.40 ÷ $0.88 = 5 ice pops - 73 quarters; $0.21 change

The total price is $16.75 + $1.29 = $18.04.

73 quarters is $18.25. The change is $18.25 − $18.04 = $0.21.

# Challenge Problem

## Answers

- Problems will vary.
- Solutions will vary.

## Work Time

# Real-World Problems with Multidigit Decimals

Choose one problem to solve. Use the table to solve the problem. Clearly show each step in your solution.

- Mia bought 1 pair of flip-flops and 3 beach balls. If she paid with a $20 bill, how much change did she get?
- Carlos spent $24.35 on a pair of sunglasses for himself and ice pops for his nieces and nephews. How many ice pops did he buy?
- Martin bought a towel and a bottle of water. He paid for the items using quarters. How many quarters did he use? How much change did he get?

# Challenge Problem

- Write your own problem about the items in the table. Your problem should require using at least two operations to solve.
- Show the solution to your problem.

Ask yourself:

- What is the price of the flip-flops that Mia bought?
- What is the price of a beach ball? How much do 3 beach balls cost?
- How much money did Carlos spend on the sunglasses? How much money altogether did Carlos spend on ice pops? How can you find how many ice pops Carlos bought?
- What is the price of the towel that Martin bought? What is the price of the bottled water? How much money did Martin spend altogether? How can you find how many quarters Martin used?

# Make Connections

# Mathematics

The algorithms for decimal operations are very similar to those for whole number operations. Prompt students to discuss the differences.

- For addition and subtraction, decimal points must be aligned so that digits with the same place value are added and subtracted.
- For multiplication and division, the decimal points can initially be ignored and the whole-number algorithm used. To place the decimal point correctly, estimation or some other method must be used.

ELL: As ELLs explain their reasons verbally and in writing, their answers may have language errors. Remember, language mistakes are natural.

- Focus on the content being communicated.
- Allow processing time.
- Do not emphasize grammar.
- Model standard English.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

Students should present their solutions, showing and explaining each step. Encourage other students to ask questions and critique reasoning.

**Mathematical Practice 6: Attend to precision.**

Work as a class to correct mistakes and errors in reasoning.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' approaches to writing and solving real-world problems involving operations with decimals.

As your classmates present, ask questions such as:

- What information from the table did you use to solve the problem?
- What do you need to find?
- What operations do you need to use to solve the problem? Why?
- How does this expression, equation, or model match the problem situation?
- Why does your solution method make sense? Do you show all the steps?
- Is your answer reasonable?

# Operations with Multidigit Decimals

# Lesson Guide

Have students discuss the summary with a partner before turning to a whole-class discussion. Use this opportunity to correct or clarify misconceptions.

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Students should be able to connect the summaries to performing operations with whole numbers and to the models and numerical methods from previous lessons.

## Formative Assessment

# Summary of the Math: Operations with Multidigit Decimals

**Read and Discuss**

- Adding or Subtracting Decimals
- Line up the digits according to place value.
- Line up the decimal points.
- “Carry” regrouped numbers from one column to the next, just as you do when adding or subtracting whole numbers.

- Multiplying Decimals
- Multiply decimals using the same method that you use to multiply whole numbers. Ignore the decimal points (do not line up the decimal points).
- After multiplying, use estimation to place the decimal point. Or, count the total number of decimal places in the numbers you multiplied. In your product, start from the right and move one place to the left for each of the decimal places you counted. Place the decimal point at the resulting location.

- Dividing Decimals
- Multiply the divisor and the dividend by the same power of 10. The power of 10 should be large enough so that the resulting numbers are whole numbers. Then divide using the long division method. Use estimation to make sure your answer makes sense.

Can you:

- Explain how to add and subtract decimals?
- Explain how to multiply and divide decimals?
- Explain how estimating can help when you are performing operations with decimals?

# Reflect On Your Work

# Lesson Guide

Have students write a brief reflection before the end of class. Review the reflections to find out what still confuses students about operations with decimals.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**Something that confuses me about operations with decimals is …**