Grade 5 - PBworks

Revised 7/04/04

SAN DIEGO CITY SCHOOLS

Instructional Module to Enhance the Teaching of

Grade 5 Module 6 − Modified

Divide Whole Numbers and Decimals

H A R C O U R T

Math California Edition

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Harcourt Math−Grade 5 MODULE 6

Revised 7/04 2

Grade Five Traditional Calendar – 2004-2005 Order of Units and Pacing Guide

Month Module Number of Days

September 19 instructional days

Module 1: Data and Graphing Module 2: Place Value and Addition and

Subtraction of Whole Numbers and Decimals

11 days

8 days

October 21 instructional days

Module 2: Place Value and Addition and Subtraction of Whole Numbers and Decimals

Module 3: Algebra: Use Addition and

Multiplication; Integers

7 days

14 days

November 18 instructional days

Module 3: Algebra: Use Addition and Multiplication; Integers

Module 4: Geometry

8 days

10 days

December 13 instructional days

Winter Break 12/20 – 12/31

Module 4: Geometry Module 5: Multiply Whole Numbers and

Decimals; Percent

2 days

11 days

January 20 instructional days

Module 5: Multiply Whole Numbers and Decimals; Percent

Module 6: Divide Whole Numbers and Decimals

3 days

17 days

February 17 instructional days

Module 7: Number Theory; Fraction Concepts and Addition and Subtraction of Fractions

17 days

March 18 instructional days Spring Break 3/21 – 3/25


Module 8: Geometry: Area, Perimeter, and

Volume

5 days

12 days

April 20 instructional days

Module 9: Operations with Fractions: Multiplication and Division

Module 10: Measurement, Probability and Ratio

11 days

9 days

May 21 instructional days

STAR 4/26 – 5/17 DMT 5/9 – 5/13

Module 10 Measurement, Probability and Ratio

12 days

June 13 instructional days

Review Grade 5 concepts Preview Grade 6 Concepts

13 days


Revised 7/04 3

Grade Five – Year Round Calendar – 2004-2005 Order of Units and Pacing Guide

Month Module Number of Days

September 19 instructional days

Module 1: Data and Graphing Module 2: Place Value and Addition and

Subtraction of Whole Numbers and Decimals

11 days 8 days

October 21 instructional days

Module 2: Place Value and Addition and Subtraction of Whole Numbers and Decimals


7 days

14 days

November 18 instructional days


Module 4: Geometry

8 days

10 days

December 13 instructional days

Winter Break 12/22 – 1/17

Module 4: Geometry Module 5: Multiply Whole Numbers and

Decimals; Percent

2 days 11 days

January 10 instructional days

Module 5: Multiply Whole Numbers and Decimals; Percent

Module 6: Divide Whole Numbers and Decimals

3 days

7days

February 17 instructional days

Module 6: Divide Whole Numbers and Decimals Module 7: Number Theory; Fraction Concepts and

Addition and Subtraction of Fractions

10 days

7 days

March 11 instructional days Spring Break 3/16 – 4/8


11 days

April 15 instructional days


Module 8: Geometry: Area, Perimeter, and Volume

4 days

11 days

May 21 instructional days

Module 8: Geometry: Area, Perimeter, and Volume Module 9: Operations with Fractions: Multiplication

and Division Module 10: Measurement, Probability and Ratio

1 days

11 days

9 days

June 22 instructional days

STAR 5/27 – 6/17 DMT 6/13 – 6/17

Module 10: Measurement, Probability and Ratio 12 days

July 14 instructional days

Review Grade 5 concepts Preview Grade 6 Concepts

14 days

Revised 7/04/04

San Diego City Schools

Instruction and Curriculum Division

MATHEMATICS CURRICULUM MAP – GRADE 5

MODULE 6 – Divide Whole Numbers and Decimals Modules represent individual units of study that lead to essential learnings

THREADS THROUGHOUT THE YEAR: The threads represent ongoing learning opportunities in which students should be actively engaged throughout all units of inquiry during the entire school year. These items should not be isolated to any one particular unit of inquiry. Students will: • Develop understanding of numbers and the number system and use their understanding to solve problems and recognize reasonable results. • Develop understanding of and fluency in basic computation and procedural skills. • Use mathematical reasoning to solve problems. • Communicate their mathematical thinking by using words, numbers, symbols, graphs and charts and translate between different representations. • Use equations and variables to express generalizations of patterns and relationships. • Develop logical thinking to analyze evidence and build arguments to support or refute a hypothesis. • Make connections among mathematical ideas and between other disciplines • Develop and use strategies, skills, and concepts to solve problems. • Use appropriate tools, including technology, as vehicles to learn mathematical concepts.

These are essential learnings that represent bigger ideas/concepts: • Students understand and use the two models of division (sharing and grouping) when solving division problems. • Students use a variety of strategies to estimate and find quotients.

These are essential questions that learners ask themselves in order to achieve the essential learnings: • *How do I use counters to review key division ideas and vocabulary? • How do I estimate quotients using patterns and compatible numbers? • How do I use patterns and basic facts to write quotients for decimals divided by whole numbers? • How do I extend estimation strategies to larger dividends? • *How do I use the context to interpret remainders? • *How do I use different strategies to solve division problems (basic facts, the inverse relationship of multiplication, division patterns of multiples of ten, partial products and partial quotients)? • How do I predict when the quotient will be a decimal less than one? • How do I know when a fraction represents a division problem? • How do I use patterns to find quotients in decimal division? • How does “clearing” the decimal point in the divisor connect to patterns of multiplying and dividing by powers of ten?

* Presented in previous grades

Resources: Van de Walle: Chapter 10 (143-153), Ch 11 (173-174), Ch 14 (235-236), Ch 15 (243), Ch 17 (294-295); Mathematics Source Book, (pp. 49, 51, 57-58)


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Harcourt Mathematics

Grade 5

Unit 4: Divide Whole Numbers and Decimals

MODULE 6 NOTES

• The numbers that are used in this module are based on the standards for

the grade. They may or may not reflect your students’ number sense understanding. Adjust the numbers where necessary to give all students access to the learning. It is sometimes necessary to give different numbers to different students so that each student is challenged within their “zone of proximal development.”

• Many of the Explores in this module pose questions with multiple answers.

Your students will develop their higher level thinking skills by solving these problems. It is your role to encourage the exploration needed to do these tasks. To do this, you will need to pose many questions to the students throughout the lesson in order to push their thinking.

• The order of Lessons 13.1 and 13.2 and Lessons 14.1 and 14.2 has been

switched. • Chapter 14 is inserted within the lessons for Chapter 13. • Students are encouraged to use strategies that make sense to them

including partial quotients. Using such methods negates any reason for differentiating between 1-digit and 2-digit divisors. Therefore, the first 10 days of this module use both size divisors.

• Consider using parts of Unit 4 Test for Assessment

Revised 7/04/04

Harcourt Mathematics 17 Days of Instruction – Unit 4

CHAPTER 11: “Divide by 1-Digit Divisors” Day 1 Lesson 11.1 Estimate Quotients Using Compatible Numbers

Day 2 Lesson 11.2 Divide Large Numbers

Day 3 Lesson 11.3 Understand and Use Strategies forDividing Including Zeros in Division

Day 4 Lesson 11.4 Divide Large Numbers by 1-and 2-Digit Divisors

Day 5 Lesson 11.6 Using Context to Interpret the Remainder

CHAPTER 12: “Divide by 2-Digit Divisors” Day 6 Lesson 12.1 Algebra: Patterns in Division

Day 7 Lesson 12.2 Estimate Quotients

Day 8 Lesson 12.3 Divide by 2-Digit Divisors

Day 9 Lesson 12.4 Practice Division

Day 10 Lesson 12.6 Practice Division

CHAPTER 13: “Divide Decimals by Whole Numbers” Day 11 Lesson 13.2 Hands on: Decimal Division

Day 12 Lesson 13.1 Algebra: Patterns inDecimal Division

CHAPTER 14: “Divide Decimalsby Decimals” Day 13 Lesson 14.2 Hands on: Divide with Decimals

Day 14 Lesson 14.1 Algebra: Patterns in Decimal Division

Day 15 Lesson 13.3/14.3 Divide Decimals by Whole Numbers/Divide Decimals by Decimals

Day 16 Lesson 13.5 Divide to Change a Fraction to a Decimal

Day 17 Assessment


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DAY: 1 Divide Whole Numbers and Decimals

Unit 4: Chapter 11 LESSON 11.1, pp. 181-183

MATERIALS: 26 counters (cubes, beans, tiles, etc.) for each pair or small group. LESSON FOCUS: Estimate quotients using compatible numbers. CALIFORNIA STANDARDS:

Number Sense: 1.1: Estimate, round, and manipulate very large and very small numbers. 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning: 2.1: Use estimation to verify the reasonableness of calculated results. 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.

PURPOSE OF LESSON/ ESSENTIAL QUESTIONS:

How do I estimate quotients using patterns and compatible numbers? How do I use different strategies to solve division problems?

LAUNCH: Compatible numbers are numbers that are easy to compute mentally.

To connect to prior learning: Students use 26 counters. • Ask students how many groups of 4 are in 26. • Have them show the groups on their desks. • Ask students to record the problem:

26 ÷ 4 4 26) 264

• Ask which multiples of 4 does 26 come between. (students might count by fours or recognize six 4s = 24; seven 4s = 28, so 26 comes between 24 and 28. These are compatible numbers because they can be divided by 4 mentally. Using compatible numbers is one strategy for estimating in division. Highlight this process and encourage students to use compatible numbers to estimate that the answer is between 6 and 7.

To connect to the Explore, write the following problem for the class to see: Ten years ago, Mr. Jiminy started 5 stores that sold video games. He ordered 489 video games during that first year. He wanted each store to get the same number of video games to sell. How many did each store receive?

• Ask students what numbers would make this an “easy” problem to solve in their heads. (e.g., 500 ÷ 5 or 450 ÷5)

• Ask: What would be a reasonable number of video games for each store to get?

• Ask students that if we were dividing by 5 what “easy” number would they think of if the dividend was 324? 2,477?

Note: Different students will have different numbers they think are “easy” to work with.


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EXPLORE:

Write the following problems for the class to see: This year Mr. Jiminy owns 72 stores that sell video games. He has ordered 24,051 video games this year. He still wants each store to get the same number of video games to sell. How many will each store receive? This year Mr. Jiminy is still selling video games. As a special offer, he is packaging the games in 24-packs. If he has 24,051 video games, how many 24-pack boxes can he make?

• To be sure all students understand the problems, have volunteers restate both problems in their own words.

• Have students work with a partner to estimate and solve both problems.

Note: Circulate during work time looking for estimation and solution strategies that are connected to the purpose to share (i.e., using compatible numbers to estimate, using partial quotients as a division strategy, using multiplication to help solve division problems). Note: Students used the partial quotient algorithm in both 3rd and 4th grades. See examples to the left.

SUMMARIZE:

Focus the conversation around the 2 Essential Questions for this lesson: “How do I estimate quotients using patterns and compatible numbers?” “How do I use different strategies to solve division problems?” • Share estimation and division strategies. • Chart the strategies you have chosen to be shared as the students share

them. • Help students make connections between the strategies by recognizing how

the strategies are alike and how they are different. While several students may use the partial quotient method, the groups they pull out each time may be different. Other students may use strategies other than partial quotients.

PRACTICE/HOME-WORK:

Suggestion: Have students choose 2 – 3 problems from p. 183 and write a story problem for each of them. They should include an estimate and a solution for their story problems. Note: Save these problems as future practice and homework.

Examples of Partial Quotient Method 72 24,051 - 7,200 100 16,851 -7,200 100 9,651 -7,200 100 2,451

- 720 10 1,731

- 720 10 1,011 - 720 10 291 - 144 2 147

- 144 + 2 3 334 R.3 24 24,051 -24,000 1,000 51 -48 + 2 3 1,002 R.3


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Unit 4: Chapter 11-13

LESSON 11.2, pp. 184-185

MATERIALS: LESSON FOCUS: Divide large numbers by 1- and 2-digit divisors. CALIFORNIA STANDARDS:



How do I use different strategies to solve division problems? How do I estimate quotients using patterns and compatible numbers? How do I use the context to interpret remainders?

LAUNCH:

Note: Collect the homework problems to be used in Lesson 11.3. Write the following problem for the class to see: Zack has 65 candies. If he shares them equally with 7 friends, how many candies will each friend get? • Have students estimate the answer. • Ask students what compatible numbers they used to estimate the answer.

(e.g., 63 ÷ 7 or 70 ÷ 7) • Ask students what is the exact answer. Does the estimate show the

reasonableness of the actual answer? • Discuss the fact that there is a remainder. What did the students do with the

remainder and why did they make those choices (e.g., “I left the quotient the same. Zack just didn’t share them.” “I divided the leftover candies into fractional parts so that each friend got 9 & 2/7 pieces of candy.”)


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EXPLORE:

• Write the following problems for the class to see: The Early Risers Egg Company gathered 5,120 eggs this morning. How many cartons of eggs will they ship to the stores today if they pack them in cartons that hold a dozen eggs? The Early Risers Egg Company sells their eggs to 12 stores. If they gathered 5,120 eggs this morning and sell the same number of eggs to each store, how many eggs would each store receive today? Scaffold: To ensure the math is accessible to all students, adjust the number of eggs to smaller or larger numbers as needed. For example, 512 and 51,200. Note: The divisor in both of these problems has 2 digits. Some students may estimate the divisor to be 10 while others may keep 12 and find compatible numbers for it in the dividend. To be sure all students understand the problems, have volunteers restate both problems in their own words. Have students work in pairs to:

• Estimate the quotients • Find the exact answer using a method that makes sense to them • Compare their estimates to the exact answers to see if the actual answers

were reasonable Extension: If the Early Risers Egg Company changed the cartons they used to cartons that hold 18 eggs each, how would it change the number of cartons of eggs they would be shipping? Explain your thinking using actual examples (do the math!!). Note: Circulate during work time looking for estimation and solution strategies to share (i.e., using compatible numbers to estimate, using partial quotients as a division strategy, using multiplication to help solve division problems).

SUMMARIZE:

Focus the conversation around the 3 Essential Questions for this lesson: “How do I use different strategies to solve division problems?” “How do I estimate quotients using patterns and compatible numbers?” “How do I use the context to interpret remainders?” • Share efficient estimation and division strategies the students are using with

meaning. • Chart the strategies you have chosen to be shared as the students share

them. Discuss how the students dealt with the remainders. Does more than one way of dealing with the remainders make sense in this context?

PRACTICE/ HOMEWORK:

Suggestion: Have students choose 2 – 3 problems from p. 185 and write a story problem for each of them. They should include an estimate and a solution for their story problems. If there is a remainder, have students explain how they interpreted the remainder (i.e., drop the remainder, add 1 more to the quotient, write the remainder as a fractional quantity). Note: These problems will be used as future practice and homework.


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MATERIALS: LESSON FOCUS: Understand and use strategies for division including zeros in division CALIFORNIA STANDARDS:

Number Sense: 1.1: Estimate, round, and manipulate very large and very small numbers. 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning: 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.

PURPOSE OF LESSON/ ESSESNTIAL QUESTIONS:

How do I estimate quotients using patterns and compatible numbers? How do I use different strategies to solve division problems?

LAUNCH: Benchmark numbers are familiar numbers used as a point of reference.

• Collect homework problems to be used for Practice/Homework in Lesson 11.4.

• Discuss with students that sometimes benchmark numbers are used to give approximations of actual numbers. For example, if we say 20,000 people were at the ballgame, it’s very unlikely that exactly 20,000 people were there. But 20,000 gives an easy number to think about as a reference of the number of people that were at the game.

• Write the following problem for the class to see: About 30 people were at the party last night. Four people sat at each table, and there were no empty seats. How many people could have been at the party? How many tables could there have been? • Discuss possible numbers for the people at the party and the number of tables.


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EXPLORE:

Write the following problems for the class to see: Problem 1: A huge car transport ship docked in the Port of San Diego yesterday. It was loaded with about 11,000 cars from Japan. The cars were loaded on 15 monstrous decks. If the same number of cars were on each deck, how many cars could have been on each deck? How many cars could have been on the ship? Scaffold: To ensure that the math is accessible to all students, adjust the size of the possible number of cars on the ship. For example, 1,000 and 100. Problem 2: Another ship docked yesterday as well. But this ship was a container ship with about 14,000 containers stacked 6 high in the hold. How many stacks of containers could have been loaded on the ship? How many containers could have been on the ship? Scaffold: To ensure that the math is accessible to all students, adjust the size of the possible number of containers on the ship. For example, 1,000 and 200. Have students work in pairs to estimate then solve both problems. Note: In Problem 1, some students may use 20 as their divisor in the quotient while other students may leave the 15 and find compatible numbers to go with it.

SUMMARIZE:

Focus the conversation around the 2 Essential Questions for this lesson: “How do I use different strategies to solve division problems?” “How do I estimate quotients using patterns and compatible numbers?” • Have students share the number of cars/containers they found in each

problem and the strategies they used to find them. Accept all reasonable numbers of cars/containers.

PRACTICE/ HOMEWORK:

Give students several of the problems they created for homework for Lesson 11.1.


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DAY: 4

Divide Whole Numbers and Decimals Unit 4: Chapter 11

LESSON 11.4, pp. 188-189

MATERIALS: LESSON FOCUS: Divide large numbers by 1- and 2-digit divisors. CALIFORNIA STANDARDS:

Number Sense 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning: 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.

PURPOSE OF LESSON: ESSENTIAL QUESTIONS:

How do I use different strategies to solve division problems? How do I use the context to interpret remainders?

LAUNCH:

Write the following problem for the class to see: Joey wants to put his 24 baseball cards into small card-size boxes. He wants an equal number of cards in each box. How many boxes could he use and how many cards might be in each box? • Ask students to contrast this problem with the other problems they have

worked with during the module. (i.e., only the dividend is given) • Have students share some possible answers and the strategies they used

for finding them. Students should realize that more than one set of answers is possible. All answers will be factors of 24; there are no remainders possible in this context.


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EXPLORE:

Write the following problem for the class to see: Home Run Baseball Card Swap Shop has 51,208 cards in stock that they must sell. They want to put them into small card-size boxes to sell. If they put an equal number of cards in each box, how many boxes could they fill and how many cards might be in each box? Look at the possibilities you found. How does the number of cards in each box affect the number of boxes they can fill? Explain your thinking using actual examples (do the math!!!) (i. e., the more cards in each box, the fewer boxes). Scaffold: To ensure the math is accessible to all students, adjust the number of cards to larger or smaller numbers to meet the needs of all your students. For example, 5,120 and 512. Have students work in pairs to find possible number pairs that tell how many boxes they could fill and how many cards would be in each box.

SUMMARIZE Focus the conversation around the 2 Essential Questions for this lesson: “How do I use different strategies to solve division problems?” “How do I use the context to interpret remainders?” Note: This problem lends itself to using multiplication to solve division problems, divisibility rules (see p. 254 Harcourt Math), and patterns in multiplication and division. Include sharing these strategies during the Summarize.

PRACTICE/ HOMEWORK:

Give students several of the problems they did for Practice/Homework in Lesson 11.2.


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MATERIALS: LESSON FOCUS: Use the context of division problems to interpret the remainder. CALIFORNIA STANDARDS:



How do I use the context to interpret remainders?

LAUNCH:

Use the following problem to focus on the importance of context in interpreting the remainder. • Write 26 ÷ 4 on the board/overhead. • Ask students for the answer. They might respond 6 r 2 or 6 with 2 left over. • Ask students for the answers to the following problems.

1. Flower plants cost $4.00. You have $26.00. How many plants can you buy? (6)

2. 26 students are going on a field trip. 4 students fit in a car. How many cars will be needed? (7)

You have 26 brownies. If you share them equally between 4 students, how many brownies will each student get? (6 ) Remind students that the context of the problem helps us decide how to handle the remainder—either drop it, increase the answer by one, or write it as a fraction (or piece of a whole).

1 2


Revised 7/04 17

EXPLORE:

Write the following problems for the class to see: The Yummy Candy Company makes Gooey candy bars. Each day they produce 9,600 fun-size Gooey bars. They package them in equal sized bags that hold 36 bars. How many bags of fun-size Gooey bars can they make each day? The Stick’em Company makes masking tape. Each day they make 4,205 meters of 1-inch wide masking tape. If they make 72 rolls of 1-inch wide masking tape each day, how much tape is on each roll? I went to the marina over the weekend. A sign said that the marina had moorings for 1,560 boats. I noticed that each pier had 18 boats moored. The fog came in before I was able to count all the piers. How many piers would the marina have in order to moor the 1,560 boats? • Have students work in pairs to solve the problems. Note: Circulate during work time looking for division strategies to be shared during the Summarize. Notice the decisions they are making about the remainders. Are these decisions based on the context of the problem?

SUMMARIZE:

Focus the conversation around the Essential Question for this lesson. “How do I use the context to interpret remainders?” Call the students together to discuss their strategies for dividing as well as the decisions they made concerning the remainders in each context. Note: In the Stick’em problem the students can express the remainder as a fraction or some students may express the remainder as a decimal.

PRACTICE/ HOMEWORK:

Problem Solving Practice p. 195, # 1 – 4. Have students explain how the remainder affected the answer to each question in # 2 & 4.


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DAY: 6


LESSON 12.1, pp. 200-201

MATERIALS: LESSON FOCUS: Algebra: Patterns in Division. CALIFORNIA STANDARDS:

Number Sense 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning 1.1: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. 3.3: Develop generalizations of the results obtained and apply them in other circumstances.

PURPOSE OF LESSON/ESSENTIAL QUESTIONS:

How do I use patterns and basic facts to write quotients? How do I extend estimation strategies to larger dividends?

LAUNCH:

Review basic division facts with students and connect to related multiplication facts to reinforce concept of division as the inverse of multiplication: 8 ÷ 2 = How do you know? 15 ÷ 5 = How do you know? 48 ÷ 6 = How do you know? Have students identify, make and test conjectures, and discuss relationships between the dividends, divisors, and quotients in the following string. 8 ÷ 2 = ___4__ 80 ÷ 20 = ___4__ (This quotient is the same because both the 800 ÷ 20 = __40__ dividend and divisor were multiplied by 10.) 8,000 ÷ 20 = _400__ 80,000 ÷ 20 = 4,000__ (In this pattern the dividend increases 10 times in each step while the divisor stays the same. So the quotient will be 10 times greater as well.) Write Examples B & C on p. 200 for the class to see. Ask students to identify the pattern in the string and extend the pattern to the next equation.


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EXPLORE:

Write the following problem and extensions for the class to see: The Flaky Cookie Company makes Krumbly Kookies. They can produce 264,000 ounces of Krumbly Kookies each day. These cookies are packaged in 24-ounce boxes. How many boxes of Krumbly Kookies can they produce each day? 1. Find the solution to this problem using whatever strategy makes

sense to you. 2. Find the answers to the following questions and explain your

strategies: • By using patterns of division, tell how many boxes the company could

make if they made super-sized boxes that hold 240 ounces. • Again, by using patterns of division, tell how many boxes the

company could make if they went back to the regular sized boxes (24 ounces) but cut back on production and made only 26,400 ounces each day.

• What if the company used the regular sized boxes but increased their production to 2,640,000 ounces per day. How many boxes could they fill each day?

SUMMARIZE:

Focus the conversation around the Essential Questions for this lesson. “How do I use patterns and basic facts to write quotients”? “How do I extend estimation strategies to larger dividends?” Note: The generalization of the pattern in this problem is that if the divisor stays the same, the dividend and quotient are multiplied or divided by the same power of 10.

PRACTICE/ HOMEWORK:

Assign students to make 2 division strings that start with a basic math fact and go through dividends in the millions place and 2-digit divisors


Revised 7/04 20

DAY 7: Divide Whole Numbers and Decimals

Unit 4: Chapter 12 LESSON 12.2, pp. 202 - 203

MATERIALS: LESSON FOCUS: Estimate Quotients. CALIFORNIA STANDARDS:

Number Sense 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning: 1.1: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. 2.1: Use estimation to verify the reasonableness of calculated results. 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.


How do I extend estimation strategies to larger dividends? How do I use different strategies to solve division problems?

LAUNCH:

Students have textbooks closed (The students’ books need to be closed here because it distracts the students from doing their own thinking by showing answers and strategies that may not be meaningful to them.) Read “Wild Blue Yonder” p. 202, to the students. Have students estimate the answer. Some students may estimate as 3,600 ÷ 30 = 120 while other students may estimate as 3,600 ÷ 36 = 100. Note: Changing the divisor to a multiple of 10 and the dividend to a compatible number is another estimation strategy when there is a 2-digit divisor. Finding compatible numbers for both the dividend and divisor also continues to be useful.


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EXPLORE:

Write the following problems for the class to see: The IT’S COOL SHIRT COMPANY manufactured 2,526 shirts last month. They supply their shirts to 26 different clothing stores. Each store receives the same quantity. How many shirts did each store receive? Were there any shirts left over that were not sent? Estimate what you think the answer will be and explain why you think this answer makes sense. Check your estimate by figuring out the answer. If the IT’S COOL SHIRT COMPANY does not increase the number of shirts it produces each month, but does increase the number of clothing stores it supplies, how will this alter the number of shirts that it can supply each of its customers? Explain your thinking using actual examples (do the math!!!). Have students work in pairs to: • Estimate the quotients. • Find the exact answer using a method that makes sense to them. • Compare their estimates to the exact answers to see if the actual answers were reasonable.

SUMMARIZE:

Focus the conversation around the 2 Essential Questions for this lesson: “How do I extend estimation strategies to larger dividends?” “How do I use different strategies to solve division problems?” Note: Look for both estimation strategies (compatible numbers and multiples of 10) during the Explore time to share during Summarize (e.g., 2,600 ÷ 26 = ; 2,700 ÷ 30 = ; 2,400 ÷ 30 = ; 2,500 ÷ 25).

PRACTICE/ HOMEWORK:

P. 203: #6-8.


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DAY 8:


LESSON 12.3, pp. 204 - 207

MATERIALS: LESSON FOCUS: Divide by 2-digit divisors. CALIFORNIA STANDARDS:

Number Sense 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning 1.1: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns. 2.1: Use estimation to verify the reasonableness of calculated results. 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.


How do I estimate quotients using patterns and/or compatible numbers? How do I extend estimation strategies to larger dividends? How do I use different strategies to solve division problems? How do I use the context to interpret remainders?

LAUNCH:

Students have books closed. Write the Hank Aaron problem in “Learn” p. 204, for the class to see. • Have students estimate the answer. Ask them what strategy they

used to find their estimate (i.e., compatible numbers, making the divisor a multiple of 10).

• Discuss the different estimates the students have and the strategy that produced each estimate (i. e., 750 ÷ 25 = ; 800 ÷ 30 = ; 800 ÷ 20 = ).

• Have students find the actual answer. Discuss how they interpreted the remainder (i. e., Did they drop it? Did they make the answer 1 more since 19 is almost 23? Did they leave it as a remainder and didn’t think about how to interpret it?).


Revised 7/04 23

EXPLORE: Write the following problems for the class to see:

I saw a marina in Mission Bay that advertised that it had room for 1,560 boats. They have 65 piers at which the boats could be moored. If each pier held the same number of boats, how many boats would be moored at each pier? I saw an advertisement in the paper on Monday that the same marina was going to expand its facilities by adding room for another 1,200 boats. How many more piers will they be adding if the piers are the same size as the ones they already have? • To be sure all students understand the problems, have volunteers

restate both problems in their own words. • Have students work with a partner to estimate then solve both

problems. Note: Circulate during work time looking for estimation and solution strategies to share. (i.e., using compatible numbers to estimate, using multiples of 10 for the divisor to estimate, using partial quotients as a division strategy, using multiplication to help solve division problems) Note: Discuss why a remainder would not make sense in the context of this problem.

SUMMARIZE:

Focus the conversation around the 4 Essential Questions for this lesson: “How do I estimate quotients using patterns and/or compatible numbers?” “How do I extend estimation strategies to larger dividends?” “How do I use different strategies to solve division problems?” “How do I use the context to interpret remainders?” Choose 1 or 2 groups to share their estimation and/or solution strategies and their estimations.

PRACTICE/ HOMEWORK:

Assign the “Write” p. 207 TE. “Estimate and solve 286 ÷ 13. Compare your quotient to the estimate.”


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DAY 9:


LESSON 12.4, pp. 208 - 209

MATERIALS: LESSON FOCUS: Practicing Division. CALIFORNIA STANDARDS:


PURPOSE OF LESSON/ESSENTIAL QUESTIONS:

How do I estimate quotients using patterns and/or compatible numbers? How do I extend estimation strategies to larger dividends? How do I use different strategies to solve division problems?

LAUNCH:

Students have textbooks closed. Write “Dramatic Division” p. 208, for the class to see. • Ask students to estimate the number of rows. • Have students share their estimation strategies and estimations. • Ask students to solve the problem. Note: If students are using the partial quotient method of division, they will not need to know about correcting quotients. Correcting quotients is not the focus of this lesson. • Have students share their solution and division strategies. • Discuss how the remainder is used in this problem (the number of chairs in the balcony).


Revised 7/04 25

EXPLORE:

Write the following problem for the class to see: Adult tickets to the Chargers games cost $32. Student tickets cost $16. Jose spent $5,280 for tickets for a company outing. How many of each kind of ticket could he have bought? Is there more than one combination of student and adult tickets he could have bought for $5,280? • To be sure all students understand the problems, have volunteers

restate the problem in their own words.

Have students work in pairs to: • Estimate the quotients. • Find the exact answer using a method that makes sense to them. • Compare their estimates to the exact answers to see if the actual

answers were reasonable. SUMMARIZE: Focus the conversation around the Essential Questions for this

lesson: “How do I estimate quotients using patterns and/or compatible numbers?” “How do I extend estimation strategies to larger dividends?” “How do I use different strategies to solve division problems?” Note: There are many possible solutions to this problem. The conversation about the different strategies students used for finding possible solutions may be richer than the conversation about the actual answers.

PRACTICE/ HOMEWORK:

“Write About It” p. 209 # 34.


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DAY 10:


LESSON 12.6, pp.212 - 213

MATERIALS: LESSON FOCUS: Practice Division. CALIFORNIA STANDARDS:



How do I use different strategies to solve division problems? How do I use the context to interpret remainders?

LAUNCH:

Students have textbooks closed. Write the following problem (which is found on p. 213) for the class to see. The fifth-grade students had 181 paintings to be displayed in equal groups on walls throughout the school. No more than 30 paintings would fit on each wall. After the paintings were placed, 20 were left over and were placed on the office wall. How many groups were formed? How many paintings were in each group? Display the following questions. Ask students to discuss these 4 questions which are found under “Understand,” “Plan,” and “Solve” on p. 212: • “What are you asked to find?” • “What information will you use?” • “What strategy can you use to solve the problem?” • “How can you use the strategy to solve the problem?” • Have students solve the problem. • Discuss the “Check” question: “How can you decide if your answer is correct?”


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EXPLORE:

Students have textbooks closed. Write “Problem” p. 212 (and the follow up questions on p. 213) for the class to see. A collector has 158 world’s fair souvenirs. He wants to sell them in equal groups with no more than 25 in each group. After he divides them into equal groups, there are 20 souvenirs left over that he will sell individually. How many groups was he able to make? How many souvenirs are in each group? What if there were 18 souvenirs left over? How many groups would he be able to make? How many souvenirs would be in each group? Display the following questions. Ask students to discuss these 4 questions which are also found on p. 212 under “Understand,” “Plan,” and “Solve”: • “What are you asked to find?” • “What information will you use?” • “What strategy can you use to solve the problem?” • “How can you use the strategy to solve the problem?” • Have students solve the problem. • Discuss the “Check” question: “How can you decide if your answer is correct?”

SUMMARIZE:

Focus the conversation around the 2 Essential Questions for this lesson: “How do I use different strategies to solve division problems?” “How do I use the context to interpret remainders?”

PRACTICE/ HOMEWORK:

Mixed Strategy Practice p. 213; # 5-7


Revised 7/04 28



MATERIALS: Decimal Models Pg. TR9-10 - multiple pages per student

OR Base 10 blocks. LESSON FOCUS:

Hands On: Decimal Division.

CALIFORNIA STANDARDS:

Number Sense 2.1: Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. Mathematical Reasoning 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

PURPOSE OF LESSON/ ESSENTIAL QUESTION:

How do I predict when the quotient will be a decimal less than one?

LAUNCH: TR 9 – 10 and scissors OR Base 10 blocks

Students have textbooks closed. • Write the “Explore” problem (1.5 ÷ 3 =) from p. 220. 1. Ask students what question the problem could be asking (e. g., if I have

1.5 and divide it into 3 groups, how much will be in each group?) (see Mathematics Note below).

2. Ask students if they can tell before attempting to solve the problem if the answer is greater or less than 1. Discuss their predictions with a partner and why they think that way.

3. Ask students to model the problem using base 10 materials. 4. Have partners discuss how they modeled the problem, the answer they

got, and why they believe it is reasonable. Display the “Connect” problem (0.24 ÷ 6 =) from p. 221. 1. Ask students what question the problem could be asking (e. g., if I have

0.24 and divide it into 6 groups, how much will be in each group). 2. Ask students if they can tell before attempting to solve the problem if the

answer is greater or less than 1. Discuss their predictions with a partner and why they think that way.

3. Ask students to model the problem using base 10 materials.. 4. Have partners discuss how they modeled the problem, the answer they

got, and why they believe it is reasonable. Mathematics Note: While there are 2 models of division, finding the number of groups or finding the number in each group, finding the number in each group will make more sense to students when working with decimals. It’s hard to make sense of questions such as, “If I have 1.5, how many groups of 3 can I make?” It might make more sense to ask, “If I have 1.5 and divide it into 3 groups, how much will be in each group?”


Revised 7/04 29

EXPLORE: TR 9 – 10 and scissors OR Base 10 blocks

• Have students work in partners. • Have them choose 4 problems from p. 221:1-12. Partners will work

together to solve the problems and record their thinking process. Some things that can be included in their record include:

1. Write the question the problem could be asking. 2. Predict if the answer will be greater or less than 1 (including the reason

for their prediction). 3. Model the problem using base 10 materials. 4. Record the equation for the problem.

SUMMARIZE:

Focus the conversation around the Essential Question for this lesson. “How do I predict when the quotient will be a decimal less than one?” Call students together to share their strategies. Choose several groups to share one of the problems they chose to do in the Explore. Encourage the rest of the class to ask questions of the group. Focus the conversation on how the students made sense of division of decimals by whole numbers and how they made sense of the size of their answers. Guiding Questions: • Do you agree or disagree with the following statement? “Every time you

divide a decimal by a whole number, the answer will be less than one.” Why or why not?

• How is division of a decimal like division of whole numbers? • How is division of a decimal different from division of whole numbers?

PRACTICE/ HOMEWORK:

“A student said that 3 was the answer to 2.1 ÷ 7. Do you agree or disagree? How would you convince that student that you are right?” Students can draw or cut out decimal models to include in their explanation.


Revised 7/04 30



MATERIALS:

LESSON FOCUS: Algebra: Patterns in Decimal Division.

CALIFORNIA STANDARDS:

Number Sense 2.0: Students perform calculations and solve problems involving addition, subtraction, and simple multiplication and division of fractions and decimals. 2.1: Add, subtract, multiply, and divide with decimals: add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. Mathematical Reasoning 3.3: Develop generalizations of the results obtained and apply them in other circumstances.


How do I use patterns and basic facts to write quotients for decimals divided by whole numbers? How do I predict when the quotient will be a decimal less than one?

LAUNCH:

• Remind students of the division strings from the Launch on Day 6.

16 ÷ 2 = ___8_ 160 ÷ 2 = __80_ 1,600 ÷ 2 = _800 16,000 ÷ 2 = 8,000

• Ask students to identify the pattern in the relationship between the

dividends, divisors, and quotients. (The dividend is multiplied by 10 in each step while the divisor stays the same. So the quotient will be multiplied by 10 as well.)

• Tell the students you are going to do another string starting with a

large dividend. • Write the first 4 equations in the number string. • Ask them to predict what they think will happen in this string. • Partners discuss predictions. • Start string together as a class.

24,000 ÷ 3 = 2,400 ÷ 3 = 240 ÷ 3 = 24 ÷ 3 =

• Ask students, “Based upon the pattern, what will the next equation in the string look like and how do you know?”

• Partners discuss their predictions.


Revised 7/04 31

Continue the string: 2.4 ÷ 3 =

0.24 ÷ 3 = • Ask students to discuss what these 2 equations could be asking (to

divide 2.4 into 3 equal groups/to divide 0.24 into 3 equal groups) (see Mathematical Note in the Launch in Day 11).

• Have students think of a context for one of these 2 equations. Ask students to look once again at 2.4 ÷ 3 = . • Based on the pattern from the number string, they can see the

answer will be 0.8. Ask students to give a mathematically powerful defense other than patterns of how they know the answer will be less than 1.

EXPLORE:

Write the following problem and questions for the class to see: The Fast Relief Pharmaceutical Company makes Sniff-Free nose spray. They can produce 3.6 liters of Sniff-Free each day. If they can make 90 spray bottles each day, what size spray bottles do they fill with the nose spray? (Keep answer in liters.) 1. Determine if the answer will be greater or less than 1. 2. Find the solution to this problem using whatever strategy makes

sense to you. 3. Find the answers to the following questions and explain your

strategies: • Think about using patterns of division, to tell what size spray bottles

the company would be filling if they only made 9 spray bottles each day.

• Again, think about using patterns of division to tell what size spray bottles the company would be filling if they made 900 spray bottles each day.

• What if the company used the original size of spray bottle but increased their production to 36 L of Sniff-Free each day. How many spray bottles could they fill each day?

Extension: If a milliliter is 0.001 of a liter, how many milliliters would be in each spray bottle?

SUMMARIZE: Focus the conversation around the Essential Questions for this lesson. “How do I use patterns and basic facts to write quotients for decimals divided by whole numbers?” “How do I predict when the quotient will be a decimal less than one?” Call students together to share their strategies. Call on two or three groups of partners to explain their process of finding the value of the “4” in the various settings of the Explore problem. Encourage the rest of the class to ask questions of the group.


Revised 7/04 32

Guiding Questions: • What strategy helped you determine the value of the “4” in your

answer? • How did you know when the answer was greater or less than one? • How were all the problems alike? • How were all the problems different?

PRACTICE/ HOMEWORK:

Practice & Problem Solving, Pg. 218: #19 & 21.


Revised 7/04 33

Day 13 Divide Decimals by Decimals

Unit 4: Chapter 14 Lesson 14.2; p. 236-237

MATERIALS: TR 9 – 10 or base 10 materials LESSON FOCUS: Hands on: Divide with Decimals CALIFORNIA STANDARDS:

Number Sense 2.1: Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. Mathematical Reasoning 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning.

PURPOSE OF LESSON/ ESSENTIAL QUESTION:

How do I use patterns to find quotients in decimal division?

LAUNCH: TR 9 – 10 or base 10 materials

Write for the students to see: 2.4 ÷ 0.2 • Ask students what question this could be asking. (e.g. “How many groups

of 0.2 are in 2.4? If I have 2.4 and I make 0.2 of a group, how much is in that group?”) [The second question may be awkward for students to think about.]

• Ask students to predict whether the answer will be greater or less than 1 based on their understanding of the question.

• Have pairs of students use the base 10 materials or TR 9 to physically model the problem.

• Have groups discuss the question that is being asked, their prediction of the size of the answer, and their model.

Write for the students to see: 0.75 ÷ 0.25 • Ask students what question this could be asking. (e.g. “How many

groups of 0.25 are in 0.75? If I have 0.75 and I make 0.25 of a group, how much is in that group?”) [The second question may be awkward for students to think about.]

• Ask students to predict whether the answer will be greater or less than 1 based on their understanding of the question.

• Have students use the base 10 materials or TR 10 to physically model the problem.

• Have groups discuss the question that is being asked, their prediction of the size of the answer, and their model.

EXPLORE: Have students choose 4 problems from p. 237 to explore. Include in their task: 1. Write the question the problem could be asking. 2. Predict if the answer will be greater or less than 1 (including the reason for

their prediction). 3. Model the problem using base 10 materials. 4. Record the equation for the problem.


Revised 7/04 34

SUMMARIZE: Focus the discussion around the Essential Question for this lesson:

How do I use patterns to find quotients in decimal division? Call students together to share their strategies. Choose several groups to share one of the problems they chose to do in the Explore. Encourage the rest of the class to ask questions of the group. Focus the conversation on how the students made sense of division of decimals by decimals and how they made sense of the size of their answers. Be sure to continue to bring out the question being asked by division. Guiding Questions: • How is division of a decimal by a decimal like division of whole numbers? • How is division of a decimal by a decimal different from division of whole numbers?

PRACTICE/ HOMEWORK: TR 9 – 10 (optional)

0.8 ÷ 0.02 = • What question could it be asking? • Predict whether the answer is greater or less than 1. Draw a picture (use TR 9 – 10) to show the problem.


Revised 7/04 35

MATERIALS: LESSON FOCUS: Algebra: Patterns in Decimal Division. CALIFORNIA STANDARDS:

Number Sense 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning 1.1: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.


How do I use patterns to find quotients in decimal division? How does “clearing” the decimal point in the divisor connect to patterns of multiplying and dividing by powers of ten?

LAUNCH: Remind students of the division strings from the Launch on Day 12. 24,000 ÷ 3 = 2,400 ÷ 3 = 240 ÷ 3 =

24 ÷ 3 = 2.4 ÷ 3 = 0.24 ÷ 3 = • Ask students to identify the pattern in the relationship between the

dividends, divisors, and quotients. (The dividend is 10 times as small in each equation while the divisor stays the same. So the quotient will be 10 times as small as well.”

• Tell the students you are going to do another string with a different pattern. • Write the first 4 equations in the number string.

18,000 ÷ 2,000 = 1,800 ÷ 200 = 180 ÷ 20 = 18 ÷ 2 =

• Ask them to predict what they think will happen in this string. • Partners discuss predictions. • Do string together as a class, one equation at a time. • Ask students to identify the pattern in the relationship between the

dividends, divisors, and quotients. (The dividend and the divisor are both a tenth of the previous equation; therefore, the quotient stays the same.)

• Ask students, “Based on this new pattern, what do you think it will look like if we continue this string?”

• Partners discuss their predictions. • Do the next 2 equations in the string together as a class:

1.8 ÷ 0.2 =

Day 14 Dividing Whole Numbers and Decimals

Unit 4: Chapter 14 Lesson 14.1, p. 234-235


Revised 7/04 36

0.18 ÷ 0.02 = • Ask students to discuss what any of these equations could be asking. (to

divide 18 into groups of 2 to divide 1.8 into groups of 0.2 to divide 0.18 into groups of 0.02)

EXPLORE Write the following problem and questions for the class to see: Troy’s mom belongs to a crafts club. She gave Troy a roll of plastic lacing to cut for making bead bracelets. The roll had 12.6 meters of plastic lacing on it, and Troy’s mom asked him to cut it into lengths of 0.3 meter. How many pieces of lacing did Troy cut for the bracelets? 1. Find the solution to this problem using whatever strategy makes sense to

you. 2. Find the answers to the following questions and explain your strategies:

• Think about using patterns of division to tell how many lengths Troy would have cut if there had been 126 meters of lacing on the roll and he cut them into 3-meter lengths.

• Again, think about using patterns of division to tell how many lengths Troy would have cut if there had been 1.26 meters of lacing on the roll and he cut them into 0.03-meter lengths.

• What if Troy cut the original length for each bracelet (0.3) but the roll of plastic string held 126 meters of lacing. How many bracelets could he cut lacing for?

SUMMARIZE: Mathematical Note: Multiplying both the dividend and divisor by the same power of 10 until the divisor is a whole number is called “clearing the decimal.”

Focus the conversation around the Essential Questions for this lesson: “How do I use patterns to find quotients in decimal division?” “How does “clearing” the decimal point in the divisor connect to patterns of multiplying and dividing by powers of ten?” Call the students together to discuss the relationships they found between each part of the Explore problem. [Except for the last question, the relationship between the dividend and the divisor stays the same—as the dividend increases or decreases by a power of 10, the divisor increases or decreases by the same power of 10, so the quotient (relationship) stays the same.] Give students the opportunity to apply this relationship by asking them to solve 2.8 ÷ 0.04 =. (Some students may want to think about multiplying both dividend and divisor by 100 giving the equation 280 ÷ 4 = . Other students might want to work with number strings such as 2.8 ÷ 0.04 = 28 ÷ 0.4 = 280 ÷ 4 =) Have students discuss their strategies with each other. Be sure each student understands the strategy their partner is explaining to them. Guiding Questions: • How can knowing about the relationship between the dividend, divisor, and

quotient help you determine the answer to a division problem in which you are dividing by a decimal?

PRACTICE/ HOMEWORK:

Have students choose 2 problems from p. 235, #17-22, to solve using a strategy that makes sense to them that they can explain.


Revised 7/04 37


Unit 4: Chapter 13 & 14 LESSON 13.3 & 14.3, pp. 222-225, 238-241

MATERIALS: Decimal Models Pg. TR 9-10 - multiple pages per student LESSON FOCUS: Divide Decimals by Whole Numbers CALIFORNIA STANDARDS:

Number Sense 2.1: Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning 2.1: Use estimation to verify the reasonableness of calculated results. 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.


How do I use patterns and basic facts to write quotients for decimals divided by whole numbers? How do I predict when the quotient will be a decimal less than one?

LAUNCH: Students have textbooks closed. Write “Up, Up and Away” p. 222, for the students to see. • Ask students to determine if the answer will be greater or less than one. • Ask students to estimate the answer using a strategy that makes sense

to them. Share strategies and estimates with a partner. • Have partners solve the problem. • Look for students solving the problem using partial quotients. If no one

uses that method, ask “How would you demonstrate the division in this problem using the partial quotient method?”

• Give partners time to solve the problem using the partial quotient method.• Choose a student to share and discuss the partial quotient method of

division applied to decimals. EXPLORE:

Write the following problem for the class to see: Larry went to the store to buy some cereal. He found 4 kinds of cereal that he liked. He decided to find out which one would be the best buy per ounce. He made a quick chart to help him organize the information on each cereal he liked.

Example of Partial Quotient Method- Dividing Decimals 3e4.2 -3 1 1.2 -1.2 +0.4 0 1.4


Revised 7/04 38

Have students work in pairs to answer the following questions: • Which of these cereals would be the best for Larry to buy based on his

criteria to buy the one with the least cost per ounce? • Which cereal is the most expensive based on the cost per ounce? • What did you do with the remainder when you had one? • What strategies did you use to solve the problem?

SUMMARIZE:

Focus the conversation around the 2 Essential Questions for this lesson: “How do I use patterns and basic facts to write quotients for decimals divided by whole numbers?” “How do I predict when the quotient will be a decimal less than one?” Call the students together to share their strategies. Call on two or three strategically chosen groups to explain their thinking as they divided the decimals by whole numbers. Guiding Questions: • How did you know the answer would be greater or less than one? • How did you know your answer was a reasonable size? • Which strategy makes the most sense for you when you divide decimals

by whole numbers? PRACTICE/ HOMEWORK:

Assess, TE p. 225: DISCUSS: Erin paid $8.54 for a package of 7 greeting cards. What was the price of each card? Explain how you figured your answer.

Cereal Weight Price Super Sweet Crunch 15 oz. $3.39 Crunchy Munchies 14 oz. $3.08 Smackin’ Good Flakes 16.8 oz. $3.78 Yummy Chunkies 13.2 oz. $2.97


Revised 7/04 39



MATERIALS: LESSON FOCUS: Divide to Change a Fraction to a Decimal. CALIFORNIA STANDARDS:

Number Sense 1.2: Interpret percents as part of a hundred; find decimal and percent equivalents for common fractions and explain why they represent the same value; compute a given percent of a whole number. 2.2: Demonstrate proficiency with division, including division with positive decimals and long division with multi-digit divisors. Mathematical Reasoning 2.1: Use estimation to verify the reasonableness of calculated results. 2.3: Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. 2.4: Express the solution clearly and logically by using the appropriate mathematical notation and terms and clear language; support solutions with evidence in both verbal and symbolic work. 2.6: Make precise calculations and check the validity of the results from the context of the problem.


How do I use division to find the decimal equivalent for a fraction? How do I predict when the quotient will be a decimal less than one?

LAUNCH:

• Write on the overhead or chart. • Remind students that division can be represented as a fraction as well

as with ÷ or . • Ask students what question this division problem could be asking. [“If I

divide 1 into 4 equal groups, how much will be in each group?” or “If I have 1 and divide it into groups of 4, how many groups can I make?” (It is unlikely that students will give this second question because it’s hard to visualize this action, but it is still a valid question for the expression.]

• Ask students if the quotient will be greater or less than 1 and explain how they know.

• Write “1 ÷ 4 = ” on the overhead or chart. • Discuss with students that it’s difficult to think about this equation, but it

is related to another equation that is easier to think about 100÷ 4. • Start a written string: 100 ÷ 4 = 10 ÷ 4 = 1 ÷ 4 = • Discuss the pattern in this string. (It is the same pattern found in the

number strings in Days 6 and 12.) Using number strings can help us think about division situations that are hard to think about.

• Discuss with students that doing the division symbolized in fractional form gives a decimal representation that is equivalent to the fraction. Translating between these 2 forms is a skill they will use when working with fractional amounts expressed in both forms.

1 4


Revised 7/04 40

EXPLORE:

Write the following fractions for the class to see. Tell students to develop number strings that will end with the division shown in each fraction. • • • • Have students work with partners to develop the strings. After partners finish their number strings, ask them to discuss other strategies that could help them think about these division situations. Have them write down other strategies they come up with. Note: After a reasonable amount of time, if students need assistance, suggest a starting point such as 100 or 1,000. Mathematical Note: A strategy that a student might share is to make an equivalent fraction with a denominator of 10, 100, or 1000. For example: = or = then change the fraction into its decimal form 0.5 and 0.4. The most common strategy adults use for converting fractions into decimals is to place the decimal point, annex zeros, and divide until there is no remainder or the quotient repeats itself. Few adults know why that works! Mathematical Note: Because the strings that were used in the Launch and in Days 6 and 12 all followed the same pattern, that will likely be the pattern students use in their strings. However, an interesting string for could be: 1 ÷ 2 = 0.5 1 ÷ 4 = 0.25 1 ÷ 8 = 0.125 The divisor is doubled and the quotient is halved.

SUMMARIZE: Focus the conversation around the Essential Questions for this lesson. “How do I use division to find the decimal equivalent for a fraction?” “How do I predict when the quotient will be a decimal less than one?” Call students together to discuss the number strings they developed. Have 1 or 2 groups share their number strings. Ask students who developed other strategies for solving these division situations to share their strategies. Give students the opportunity to try one of these other strategies that makes sense to them by asking them to find the decimal form of . Guiding Questions: • How did you find the answers to division problems that were hard to think

about? • What strategies that other students presented made sense to you? Explain

your understanding of the strategy. PRACTICE/ HOMEWORK

Practice and Problem Solving p. 229: # 23

1 2

2 5

4 10

1 8

12

5 10

2 5

4 10

1 8

4 5

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