Unit 2 Organizer: “DIVINE DECIMALS” · Web viewWhen multiplying or dividing decimals, students...

Grade 5 Mathematics Frameworks

Unit 2Decimal Understanding and Operations

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Georgia Performance Standards FrameworkFifth Grade Mathematics Unit 2 1st Edition

Unit 2DECIMAL UNDERSTANDING AND OPERATIONS

(7 weeks)

TABLE OF CONTENTS

Overview..............................................................................................................................3 Key Standards & Related Standards....................................................................................4 Enduring Understandings.....................................................................................................6 Essential Questions..............................................................................................................6

Concepts & Skills to Maintain.............................................................................................7

Selected Terms and Symbols...............................................................................................7

Classroom Routines...........................................................................................................10

Strategies for Teaching and Learning................................................................................10 Evidence of Learning.........................................................................................................11 Tasks..................................................................................................................................12

High Roller Revisited............................................................................................13 Patterns-R-Us.........................................................................................................21 Base Ten Activity..................................................................................................26 How Much Money?...............................................................................................33 Super Slugger Award.............................................................................................38 Number Puzzle.......................................................................................................43 What’s My Rule?...................................................................................................48 Do You See an Error?............................................................................................52 Road Trip...............................................................................................................56

Culminating Tasks Teacher for a Day..................................................................................................61 Bargain Shopping...................................................................................................66

Georgia Department of EducationKathy Cox, State Superintendent of Schools

MATHEMATICS GRADE 5 UNIT 2: DECIMAL UNDERSTANDING AND OPERATIONSJuly 2009 of 73

Copyright 2009 © All Rights Reserved


OVERVIEW

In this unit students will:

Understand place value from hundredths to one million Model and explain multiplication and division of decimals Multiply and divide decimals Understand the rules for multiplication and division of decimals Use formulas to represent the relationship between quantities Use variables for unknown quantities

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as add/subtract decimals and fractions with like denominators, whole number computation, angle measurement, length/area/weight, number sense, data usage and representations, characteristics of 2D and 3D shapes, and order of operations should be addressed on an ongoing basis.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.





STANDARDS ADDRESSED IN THIS UNIT

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

KEY STANDARDS

M5N2. Students will further develop their understanding of decimals as part of the base-ten number system.

a. Understand place value.b. Analyze the effect on the product when a number is multiplied by 10, 100, 1000, 0.1,

0.01, and .001.c. Use <, >, or = to compare decimals and justify the comparison.

M5N3. Students will further develop their understanding of the meaning of multiplication and division with decimals and use them.

a. Model multiplication and division of decimals.b. Explain the process of multiplication and division, including situations in which the

multiplier and divisor are both whole numbers and decimals. c. Multiply and divide with decimals including decimals less than one and greater than one.d. Understand the relationships and rules for multiplication and division of whole numbers

also apply to decimals.

M5A1. Students will represent and interpret the relationships between quantities algebraically.

a. Use variables, such as n or x, for unknown quantities in algebraic expressions.b. Investigate simple algebraic expressions by substituting numbers for the unknown.c. Determine that a formula will be reliable regardless of the type of number (whole

numbers or decimals) substituted for the variable.

RELATED STANDARDS

M5M1. Students will extend their understanding of area of geometric plane figures. a. Estimate the area of geometric plane figures. d. Find the areas of triangles and parallelograms using formulae.

M5D2 Students will collect, organize, and display data using the most appropriate graph.

M5P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.b. Solve problems that arise in mathematics and in other contexts.c. Apply and adapt a variety of appropriate strategies to solve problems.d. Monitor and reflect on the process of mathematical problem solving.





M5P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.c. Develop and evaluate mathematical arguments and proofs.d. Select and use various types of reasoning and methods of proof.

M5P3. Students will communicate mathematically.a. Organize and consolidate their mathematical thinking through communication.b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and

others.c. Analyze and evaluate the mathematical thinking and strategies of others.d. Use the language of mathematics to express mathematical ideas precisely.

M5P4. Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.b. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole.c. Recognize and apply mathematics in contexts outside of mathematics.

M5P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.c. Use representations to model and interpret physical, social, and mathematical

phenomena.





ENDURING UNDERSTANDINGS

Students will understand that the location of a decimal determines the size of a number. Students will understand that the placement of the decimal is determined by multiplying

or dividing a number by 10 or a multiple of 10. Students will understand that multiplication and division are inverse operations of each

other. Students will understand that rules for multiplication and division of whole numbers also

apply to decimals.

ESSENTIAL QUESTIONS

How does the location of digit in the number affect the size of a number? Why does placement or position of a number matter? How is place value different from digit value? How can we use models to demonstrate decimal values? How can we use models to demonstrate multiplication and division of decimals? What happens when we multiply decimals by powers of 10? How do the rules of multiplying whole numbers relate to multiplying decimals? How are multiplication and division related? How are factors and multiples related to multiplication and division? What happens when we multiply a decimal by a decimal? What happens when we divide a decimal by a decimal? What are some patterns that occur when multiplying and dividing by decimals? How do we compare decimals? How do we find the highest batting average? How do we best represent data in a graph? How can we efficiently solve multiplication and division problems with decimals? How can we multiply and divide decimals fluently? What are some patterns that occur when multiplying and dividing by decimals? What strategies are effective for finding a missing factor or divisor? How can we check for errors in multiplication or division of decimals? What are the various uses of decimals? How do we solve problems with decimals? How are multiplication and division related?





CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

Number sense Order of Operations Whole numbers and decimal computation Add and subtract decimals Add and subtract common fractions with like denominators Angle measurement Length, area, and weight Characteristics of 2-D and 3-D shapes Data usage and representation

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

Array: A rectangular arrangement of objects or numbers in rows and columns.

Associative Property of Multiplication: The product of a set of numbers is the same regardless of how the numbers are grouped. Example: If (3 x 5) x 2 = 15 x 2 = 30, and 3 x (5 x 2) = 3 x 10 = 30, then (3x5) x 2 = 3 x (5 x 2).

Commutative Property of Multiplication: The product of a group of numbers is the same regardless of the order in which the numbers are arranged. Example: If 8 x 6 = 48 and 6 x 8 = 48, then 8 x 6 = 6 x 8.

Distributive Property: A product can be found by multiplying the addends of a number separately and then adding the products. Example: 4 x 53 = (4 x 50) + (4 x 3) = 200 + 12 = 212

Dividend: A number that is divided by another number.Example: dividend ÷ divisor = quotient

Division: An operation in which a number is shared or grouped into equal parts.Georgia Department of Education

Kathy Cox, State Superintendent of SchoolsMATHEMATICS GRADE 5 UNIT 2: DECIMAL UNDERSTANDING AND OPERATIONS

July 2009 of 73Copyright 2009 © All Rights Reserved


Divisor:(1) In a fair sharing division problem, the divisor is the number of equal groups. In a measurement (repeated subtraction) division problem, the divisor indicates the size of each group.(2) A number by which another number is to be divided.Example: dividend ÷ divisor = quotient

Factor: (1) n. A number that is multiplied by another number to get a product(2) v. To “factor" means to write the number or term as a product of its factors

Hundred Thousands: The digit that tells you how many sets of one hundred thousand are in the number. Example: The number 432,895 has four hundred thousands.

Hundreds: The digit that tells you how many sets of one hundred are in the number. Example: The number 784 has seven hundreds.

Hundredths: The digit that tells you how many sets of hundredths there are in the number. Example: The number 0.6495 has four hundredths.

Identity Property of Multiplication: Any number that is multiplied by 1 results in the number itself. Example: 1 x 5 = 5 x 1 = 5

Measurement Division (or repeated subtraction): Given the total amount (dividend) and the amount in a group (divisor), determine how many groups of the same size can be created (quotient).

Millions: The digit that tells you how many sets of one million are in the number. Example: The number 3,901,245 has three millions.

Multiple: The product of a given whole number and an integer.

Multiplier: The number in a multiplication equation that represents the number of (equal-sized) groups.

Ones: The digit that tells you how many sets of ones are in the number. Example: The number 784 has four ones.





Partial Products: The products that result when ones, tens, or hundreds within numbers are multiplied separately. Example: When multiplying 63 x 37 = 1800 + 420 + 90 + 21 = 2,331

60 x 30 = 180060 x 7 = 42030 x 3 = 903 x 7 = 21

The resulting partial products are 1800, 420, 90, and 21.

Partition Division (or fair-sharing): Given the total amount (dividend) and the number of equal groups (divisor), determine how many/much in each group (quotient).

Place Value: The use in number systems of the position of a digit in a number to indicate the value of the digit.

Product: A number that is the result of multiplication.

Quotient: A number that is the result of division (without remainders)Example: dividend ÷ divisor = quotient

Remainder: The number left over when a number cannot be divided evenly.

Ten Thousands: The digit that tells you how many sets of ten thousand are in the number. Example: The number 43,987 has four ten thousands.

Tens: The digit that tells you how many sets of ten are in the number. Example: The number 784 has eight tens.

Tenths: The digit that tells you how many sets of tenths are in the number. Example: The number 0.6495 has six tenths.

Thousands: The digit that tells you how many sets of thousand are in the number. Example: The number 5,321 has five thousands.




Partial Products


CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include such obvious activities such as taking attendance and lunch count, doing daily graphs, and daily question and calendar activities as whole group instruction. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, and how to access classroom technology such as computers and calculators. An additional routine is to allow plenty of time for children to explore new materials before attempting any directed activity with these new materials. The regular use of routines is important to the development of students’ number sense, flexibility, fluency, collaborative skills, and communication. All of which will support students’ performances on the tasks in this unit and throughout the school year.

STRATEGIES FOR TEACHING AND LEARNING

Students should be actively engaged by developing their own understanding. Mathematics should be represented in as many ways as possible by using graphs, tables,

pictures, symbols, and words. Appropriate manipulatives and technology should be used to enhance student learning. Students should be given opportunities to revise their work based on teacher feedback,

peer feedback, and metacognition which includes self-assessment and reflection. Students need to write in mathematics class to explain their thinking, talk about how they

perceive topics, and justify their work to others.

Math Literature ConnectionsMillions of Cats. (2006/1928) by Wanda Ga’gHow Much is a Million? (1997) by David M. SchwartzIf You Made a Million. (1994) by David M. SchwartzOn Beyond a Million: An Amazing Math Journey. (2001) by David M. SchwartzCount to a Million: 1,000,000. (2003) by Jerry Pallotta

Useful web pages http://arcytech.org/java/b10blocks/b10blocks.html An interactive Base-10 website that

allows the student to use some virtual manipulatives. http://www.aaamath.com/plc51b-placevalues.html This website contains some

information and activities dealing with place value and decimals. http://www.enchantedlearning.com/math/decimals/ This website contains some

information and activities dealing with decimals. http://argyll.epsb.ca/jreed/math7/strand1/1201.htm This website has some decimal

activities using pattern blocks. http://www.shodor.org/interactivate/lessons/MultiplyingDecimals/ This website provides

information that is designed to reinforce skills associated with multiplying decimals and allow students to visualize the effects of multiplying by a decimal.

http://home.att.net/~teaching/filecab.htm This website has different activities on various mathematical concepts.




http://home.att.net/~teaching/filecab.htm

http://www.shodor.org/interactivate/lessons/MultiplyingDecimals/

http://argyll.epsb.ca/jreed/math7/strand1/1201.htm

http://www.enchantedlearning.com/math/decimals/

http://www.aaamath.com/plc51b-placevalues.html

http://arcytech.org/java/b10blocks/b10blocks.html

http://search.barnesandnoble.com/Count-to-a-Million/Jerry-Pallotta/e/9780439389150/?itm=65

http://search.barnesandnoble.com/On-beyond-a-Million/David-M-Schwartz/e/9780440411772/?itm=14


http://nlvm.usu.edu/en/nav/vlibrary.html This website for the National Library of Virtual Manipulatives has lots of different interactive manipulatives for teachers and students to use.

http://mathforum.org/library/ This website for The Math Forum Internet Mathematics Library provides a variety if mathematical content information as well as other useful math website links.

http://www.internet4classrooms.com/ This website contains helpful classroom ideas for teachers to use with their classroom instruction.

Web pages with links to many mathematics topics: http://members.shaw.ca/dbrear/mathematics.html http://teacher.scholastic.com/maven/triplets/index.htm http://www1.center.k12.mo.us/edtech/resources/money.htm http://teacher.scholastic.com/maven/daryl/index.htm http://www.amblesideprimary.com/ambleweb/numeracy.htm

EVIDENCE OF LEARNING

Students should demonstrate a conceptual understanding of operations with decimals as opposed to a purely procedural knowledge. For example, students should understand that if they are multiplying tenths by tenths, the product must be expressed as hundredths. (i.e., 1/10 x 1/10 = 1/100). Students should also know to round to the nearest whole number and estimate to place the decimal, using the estimate to determine the reasonableness of an answer, rather than only knowing to count the digits after the decimal point to place the decimal point in the answer.

By the conclusion of this unit, students should be able to demonstrate the following competencies:

understand place value compare decimals model multiplication and division of decimals multiply and divide decimals by powers of 10 use estimation when multiplying and dividing decimals multiply and divide decimals with fluency determine relationship between quantities algebraically recognize student errors in multiplication and division of decimals use decimals to solve problems




http://www.amblesideprimary.com/ambleweb/numeracy.htm

http://teacher.scholastic.com/maven/daryl/index.htm

http://www1.center.k12.mo.us/edtech/resources/money.htm

http://teacher.scholastic.com/maven/triplets/index.htm

http://members.shaw.ca/dbrear/mathematics.html

http://www.internet4classrooms.com/

http://mathforum.org/library/

http://nlvm.usu.edu/en/nav/vlibrary.html


TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all fifth grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

Task Name Task Type/Grouping Strategy Content Addressed

High Roller Revisited Learning TaskPartner/Small Group Task

Understanding place value, comparing decimals

Patterns-R-Us Learning TaskPartner Task

Multiplying and dividing decimals by powers of 10

Base Ten Activity Learning TaskPartner Task

Modeling multiplication and division of decimals

How Much Money? Learning TaskIndividual/Partner Task Dividing decimals by decimals less than one

Super Slugger Award Performance TaskIndividual/Partner Task

Dividing to find decimals less than one, comparing decimals

Number Puzzle Performance TaskIndividual/Partner Task Multiplying and dividing decimals

What’s My Rule? Learning TaskClass/Small Group Task

Logically deduce a multiplication or division rule

Do You See an Error? Performance TaskIndividual/Partner Task

Find student errors when multiplying or dividing decimals

Road Trip Performance TaskIndividual/Partner Task Determine cost of gasoline for a family trip

Culminating Task #1:Teacher for a Day

Performance TaskIndividual/Partner/Small

Group Task

Explain multiplication and division of decimals

Culminating Task #2:Bargain Shopping

Performance TaskIndividual/Partner Task Determine best prices for school supplies





LEARNING TASK: High Roller RevisitedAdapted from the Grade 2 Unit 2 Frameworks

STANDARDS ADDRESSED


a. Understand place value.c. Use <, >, or = to compare decimals and justify the comparison.

M5P1. Students will solve problems (using appropriate technology).a. Build new mathematical knowledge through problem solving.d. Monitor and reflect on the process of mathematical problem solving.

M5P2. Students will reason and evaluate mathematical arguments.a. Recognize reasoning and proof as fundamental aspects of mathematics.b. Make and investigate mathematical conjectures.




coherent whole.

ESSENTIAL QUESTIONS

How does the location of digit in the number affect the size of a number? Why does placement or position of a number matter? How is place value different from digit value? How can we use models to demonstrate decimal values?

MATERIALS

“High Roller Revisited” Recording Sheet One die (6-sided, 8-sided, or 10-sided); or a deck of number cards (4 sets of 0-9)

GROUPING

Partner/Small Group Task





TASK DESCRIPTION, DEVELOPMENT, AND DISCUSSION

In this task students will play games using different place value charts to create the largest possible number by rolling a die and recording digits on the chart one at a time.

CommentsThese games should be played multiple times for students to begin to develop strategies

for number placement. Students should discuss their strategies for playing the game and any problems they encountered. For example, students may roll several smaller (or larger) numbers in a row and must decide where to place them. Or, they may need to decide where to place any given number such as a 3.

Variations: Students could also try to make the least number by playing the game “Low Roller.” Players could keep score of who created the greatest or least number during the game. Students could be required to write the word name, read the number aloud, or write

the number in expanded notation. These games can also be played with the whole class. The class can be divided into two

teams and a student from each team can take turns rolling the die or drawing a card. Students from each team would complete the numbers on a chart. Alternatively, the students can play individually against each other and the teacher. The teacher can play on the white board and use a think-aloud strategy when placing digits on the board. This provides students with an opportunity to reflect on the placement of digits.

Background KnowledgeYou may want to share the chart below with your students to show the multiple

representations for place value, fractions, and decimals.

Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths

1,000 100 10 1 . 1__10

1__

100

1__

1,000

.1,000 wholes

100 wholes

10wholes

1 whole . 1 part of 1 whole that has been

divided into 10 parts1 part of 1 whole that has been

divided into 100 parts1 part of 1 whole that has been

divided into 1,000 parts

Task DirectionsStudents will follow the directions below from the two versions of High Roller Revisited

and for The Smallest Difference Game.





High Roller Revisited – Version 1Directions:

Players will work in groups to play the game “High Roller Revisited.” Roll the die 9 times. After each roll decide where to record the digit on the place

value chart. Use the 9 digits to make the greatest number possible. Once a digit is recorded, players may not make changes to their number. Pass the die to the next player and continue to play. Compare numbers. The player with the higher number wins the round. Play 5 rounds. The player who wins the most rounds wins the game.

Millions , Hundred Thousands

Ten Thousands

Thousands , Hundreds Tens Ones . Tenths Hundredths

1. , , .2. , , .3. , , .4. , , .5. , , .

High Roller Revisited – Version 2Directions:

Players will work in groups to play the game “High Roller Revisited.” Roll the die 4 times. After each roll decide where to record the digit on the place

value chart. Use the 4 digits to make the greatest number possible. Once a digit is recorded, players may not make changes to their number. Pass the die to the next player and continue to play. Compare numbers. The player with the highest number wins the round. Play 5 rounds. The player who wins the most rounds wins the game.

Round Tens Ones . Tenths Hundredths

1. .2. .3. .4. .5. .

Smallest Difference Game – High Roller Revisited – Version 3Georgia Department of Education




Students set up a game board as shown below:

1. _____ _____ . _____ _____ > _____ _____ . _____ _____

2. _____ _____ . _____ _____ > _____ _____ . _____ _____

3. _____ _____ . _____ _____ > _____ _____ . _____ _____

4. _____ _____ . _____ _____ > _____ _____ . _____ _____

5. _____ _____ . _____ _____ > _____ _____ . _____ _____Directions:

The object of each round is to make a true number sentence (the first number is greater than the second number). Then subtract the smaller number from the greater number. Note: If a player ends up with a false statement (i.e. the first number is not greater than the second number), then the player needs to switch the sign so that the number sentence is correct and subtract the two numbers. But that student cannot win the smallest difference for that round.

Roll the die 8 times. After each roll decide where to record the digit on the game board.

Once a digit is recorded, players may not make changes to their number. Pass the die to the next player and continue to play. When the two numbers are subtracted, the player with the smallest difference

wins the round. Play five rounds. The player who wins the most rounds wins the game.

Example:

_____ _____ . _____ _____ > _____ _____ . _____ _____




9 2 3 1 8 4 7 6 92.31

- 84.76 7.55

If 7.55 is the smallest difference, then this player wins the round.


Questions/Prompts for Formative Student Assessment What do you do with a 1 if the hundredths place is already filled? How do you decide where to place a three or four (when using a six-sided die)? How do you decide where to place a 6 (when using a six-sided die)?

Questions for Teacher Reflection Are students placing digits in a number based on an understanding of place value? Are students able to articulate their thinking when deciding where to place a digit?

DIFFERENTIATION

ExtensionHave students write about “winning tips” for one of the games. Encourage them to write

all they can about what strategies they use when they play.

InterventionPrior to playing the game, give students 9 number cards at once and have them make the

largest number they can. Let them practice this activity a few times before using the die and making decisions about placement one number at a time.

TECHNOLOGY CONNECTION

http://www.321know.com/dec44ax3.htm Interactive practice writing the decimal for a decimal number given in word form (decimal numbers to the thousandths place). Note: web site contains advertising.

http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html Virtual manipulatives that may be used to represent decimals up to the thousandths.




http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html

http://www.321know.com/dec44ax3.htm


Name ________________________________________ Date ___________________________

High Roller RevisitedVersion 1

Materials: 1 die (can be 6-sided, 8-sided, or 10 sided); Recording SheetNumber of Players: 2 or moreDirections:

Roll the die 9 times. After each roll decide where to record the digit on the place value chart.

Use the 9 digits to make the greatest number possible. Once a digit is recorded, you may not make changes to your number. Pass the die to the next player and continue to play. Compare numbers. The player with the higher number wins the round. Play five rounds, the player who wins the most rounds wins the game.


Ten Thousands


1. , , .2. , , .3. , , .4. , , .5. , , .

Game 1:

Game 2:


Ten Thousands


1. , , .2. , , .3. , , .4. , , .5. , , .





Name ________________________________________ Date ___________________________

High Roller RevisitedVersion 2

Materials: 1 die (can be 6-sided, 8-sided, or 10-sided, numbered 0-9); Recording SheetNumber of Players: 2 or moreDirections:

Roll the die 4 times. After each roll decide where to record the digit on the place value chart.

Use the 4 digits to make the greatest number possible. Once a digit is recorded, you may not make changes to your number. Pass the die to the next player and continue to play. Compare numbers. The player with the higher number wins the round. Play 5 rounds. The player who wins the most rounds wins the game.

Game 1:


1. .2. .3. .4. .5. .

Game 2:


1. .2. .3. .4. .5. .





Name ________________________________________ Date ___________________________

Smallest Difference GameMaterials: 1 die (can be 6-sided, 8-sided, or 10-sided, numbered 0-9)Number of Players: 2 or moreDirections:

The object of each round is to make a true number sentence (the first number is greater than the second number). Then subtract the smaller number from the greater number.Note: If a player ends up with a false statement (i.e. the first number is not greater than the second number), then the player needs to switch the sign so that the number sentence is correct and subtract the two numbers. But that student cannot win the smallest difference for that round.

Roll the die 8 times. After each roll decide where to record the digit on the game board.

Once a digit is recorded, you may not make changes to your number. Pass the die to the next student and continue to play. When the two numbers are subtracted, the player with the smallest difference

wins the round. Play five rounds. The player who wins the most rounds wins the game.

Example:

Game Board:

1. _____ _____ . _____ _____ > _____ _____ . _____ _____

2. _____ _____ . _____ _____ > _____ _____ . _____ _____

3. _____ _____ . _____ _____ > _____ _____ . _____ _____

4. _____ _____ . _____ _____ > _____ _____ . _____ _____

5. _____ _____ . _____ _____ > _____ _____ . _____ _____





LEARNING TASK: Patterns-R-Us

STANDARDS ADDRESSED


a. Understand place value.b. Analyze the effect on the product when a number is multiplied by

10, 100, 1000, 0.1, 0.01, and .001.


c. Multiply and divide with decimals including decimals less than one and greater than one.d. Understand the relationships and rules for multiplication and division of whole numbers












ESSENTIAL QUESTIONS

What happens when we multiply decimals by powers of 10?

How do the rules of multiplying whole numbers relate to multiplying decimals?

MATERIALS

“Patterns-R-Us” Recording Sheet Calculators (one per team)

GROUPING

Partner Task


In this task, students are asked to identify, describe, and explain any patterns they notice when multiplying or dividing numbers by 1000, 100, 10, 0.1, and 0.01.

CommentsThis task is designed to serve as a discovery opportunity for the students. Students

should notice that a pattern is created when a number is multiplied or divided by a power of 10. While students may notice patterns in each individual part of the task, encourage them to look for a pattern when considering the overall task. Students should be able to explain and defend their solutions through multiple representations. For example, students should try several numbers for each part to verify that each number follows the same pattern. This activity lends itself to working in pairs for reinforcement.

An introduction for this task could be a round of “What’s My Rule?” See the “What’s My Rule?” on page 48 of this unit. The rule could be x1000, x100, x10, x0.1, or x0.01. Also, the rule could be ÷1000, ÷100, ÷10, ÷0.1, or ÷0.01.

Background KnowledgeStudents should develop an understanding that when a number is multiplied by a number

less than 1, the product is less than the original number, and when a number is divided by a decimal number less than 1, the quotient will be greater than the dividend. This is important, yet often difficult for students to understand because it is counterintuitive based on students’ previous experiences with multiplication and division.

Calculators are optional for this investigation. However, students will be more likely to explore a variety of numbers and be able to recognize patterns more efficiently with the use of a calculator. Require students to record what they put into the calculator and the result. If students could benefit from some practice with multiplication and/or division, require them to solve the problems in part one without a calculator and allow students to use a calculator for the rest of the task.





Task DirectionsStudents will follow the directions below from the “Patterns-R-Us” Recording Sheet.

A statistician is interested in finding out what pattern is created, if any, under certain situations. Your mission is to help come up with concrete rules for certain mathematical situations. Record all of your work and explain your thinking in order to defend your answer. Good luck!

PART ONE1. Start with any whole number, for example 18.2. Multiply that number by 1000, 100, 10, 0.1, and 0.01.3. What is happening?4. Is there a pattern?5. What do you think would happen if you multiplied your number by 1,000,000?

0.00001?

PART TWO1. Pick any decimal as your number, for example 12.3.2. Multiply that number by 1000, 100, 10, 0.1, and 0.01.3. What is happening?4. Is there a pattern?5. What do you think would happen if you multiplied your number by 1,000,000?

0.00001?

PART THREE1. Start with any whole number, for example 18.2. Divide that number by 1000, 100, 10, 0.1, and 0.01.3. What is happening?4. Is there a pattern?5. What do you think would happen if you divided your number by 1,000,000?

0.00001?

PART FOUR1. Pick any decimal as your number, for example 10.8.2. Predict what will happen when you divide that number by 1000, 100, 10, 0.1, and

0.01.3. After working out the problem, is your prediction correct? Why or why not?4. Is there a similar pattern that you recognize?

Questions/Prompts for Formative Student Assessment How did you get your answer? How do you know your answer is correct? What would happen if you started with a different number? What patterns are you noticing? Can you predict what would come next in the pattern?

Questions for Teacher Reflection Georgia Department of Education




Are students able to consider all four parts to find an overall pattern? Are students able to persevere and think in different ways if they are not able to find a

pattern quickly? Can students accurately describe the patterns? Are students able to generalize the pattern into a mathematical conjecture? Are students able to defend/prove their conjectures?

DIFFERENTIATION

ExtensionHave students multiply a number by 0.1. Now ask them to divide that same number

by 10. What happened? Repeat this with several numbers. Can a conjecture be made based on the results? Have students write their conjecture. Now, share their conjecture with a partner. Are the two conjectures the same? (You may also use 0.01 and 100 as another example.)

InterventionPair students who may need additional time together so that they will have time needed to

process this task.


http://www.mathagonyaunt.co.uk/INTERACTIVE/mult_divide/mult_div_lorry.html - Mathagony Aunt: Interactive mathematical practice opportunities

http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks1/maths/dice/ - Virtual 6-, 8-, and 10-sided dice




http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks1/maths/dice/

http://www.mathagonyaunt.co.uk/INTERACTIVE/mult_divide/mult_div_lorry.html


Name______________________________________ Date______________________________

Patterns-R-UsA statistician is interested in finding out what pattern is created, if any, under certain situations. Your mission is to help come up with concrete rules for certain mathematical situations. Record all of your work and explain your thinking in order to defend your answer. Good luck!

PART ONE1. Start with any whole number, for example 18.2. Multiply that number by 1000, 100, 10, 0.1, and 0.01.3. What is happening?4. Is there a pattern?5. What do you think would happen if you multiplied your number by 1,000,000?

0.00001?

PART TWO1. Pick any decimal as your number, for example 12.3.2. Multiply that number by 1000, 100, 10, 0.1, and 0.01.3. What is happening?4. Is there a pattern?5. What do you think would happen if you multiplied your number by 1,000,000? 0.00001?

PART THREE1. Start with any whole number, for example 18.2. Divide that number by 1000, 100, 10, 0.1, and 0.01.3. What is happening?4. Is there a pattern?5. What do you think would happen if you divided your number by 1,000,000? 0.00001?

PART FOUR1. Pick any decimal as your number, for example 10.8.2. Predict what will happen when you divide that number by 1000, 100, 10, 0.1, and 0.01.3. After working out the problem, is your prediction correct? Why or Why Not?4. Is there a similar pattern that you recognize?





LEARNING TASK: Base Ten Activity

STANDARDS ADDRESSED












phenomena.





ESSENTIAL QUESTIONS

How are multiplication and division related? What happens when we multiply a decimal by a decimal? What happens when we divide a decimal by a decimal? How can we use models to demonstrate decimal values? How can we use models to demonstrate multiplication and division of decimals?

MATERIALS

Base-ten blocks (or virtual base-ten blocks) Grid paper (or plain paper) to record work Colored pencils, crayons, or markers

GROUPING

Partner Task


In this task, students will create rectangular arrays as a representation of multiplication and division of decimals.

CommentsWhen making rectangles to represent decimal multiplication, you are actually using the

length and the width of each block to represent the factors. Therefore, a flat is actually 1 unit by 1 unit, a long is 1 unit by 0.1 unit, and a unit block is 0.1 unit by 0.1 unit.

Background KnowledgeThis activity will also allow you to work with decimal numbers by using the place value

representation provided. In this case, the red block (flat) represents 1 square unit, the blue (long) represents one tenth of the square unit or 0.1 square units, and the green (unit) represents one hundredth or 0.01 square units. You can use this system to introduce decimal numbers and the place value with decimal numbers. Also, students can explore addition and subtraction of decimals using base-ten blocks in a manner similar to adding and subtracting whole numbers using base ten blocks.




Decimal Block Values1 (flat) 0.1 (long) 0.01 (unit)


In addition to the example shown below, you can do multiplication as repeated addition. For example, to multiply 3 x 0.4 you pull out 3 sets of 0.4, represented by 3 sets of 4 longs. Arrange them in a 3 by 0.4 array as shown below. There are a total of 12 longs. 10 longs can be traded for a flat, with 2 longs left over. Or you can think of 12 tenths as being one whole and 2 tenths, therefore, 3 x 0.04 = 1.2.

Division can be represented by using the dividend as the total area and arranging the blocks in groups according to the divisor. The number of groups created is the quotient. For the fifth problem, 3.6 ÷ 1.2, ask the students how many groups of 1 flat and 2 longs can be made with a group of 3 flats and 6 longs.

In the example below (4.83 ÷ 2.1) we are dividing a total of 4.83 blocks into equal groups of 2.1. The quotient or number of groups is 2.3. One dimension of this array is the divisor, or 2.1 given in the problem. The other dimension is 2.3 which is the quotient.”




2.1 units across the top

How manyunits along this side?

4.83 square units


Example:To illustrate the product 3.2 x 2.4 you will need to combine blocks in a rectangular array. Start with the length and the width.

length = 3.2 units

width = 2.4 units





Then complete the rectangle.

Area = 3.2 x 2.4 = (3 x 2) + (3 x 0.4) + (0.2 x 2) + (0.2 x 0.4) = 6 + 1.2 + 0.4 + 0.08 = 7.68Therefore, 3.2 units x 2.4 units = 7.68 units2 as shown above.

Task DirectionsStudents will follow the directions below from the “Base 10 Activity” Recording Sheet.

Your task is to use the base-ten blocks to represent multiplication and division with decimals. Use the decimal block values below to help you find the product or quotient of each decimal problem.

1. 4.8 times 32. 2.1 x 5.43. 0.6 x 1.94. 12 ÷ 0.35. 3.6 divided by 1.2




3.2 units

2.4 units


3 x 2 = 6 0.2 x 2 = 0.4

3 x 0.4 = 1.2

0.2 x 0.4 = 0.08


Questions/Prompts for Formative Student Assessment If the flat is 1 unit, what does a long represent? What does a unit block represent? How many groups do you need to represent? How many do you have in all? How can you create an array using the dividend? How do you represent the divisor in the

array? Where is the quotient represented?

Questions for Teacher Reflection Do students understand an area model for division of decimals using base-ten blocks? Are students able to identify and describe any patterns?

DIFFERENTIATION

ExtensionHave students create their own practice problems with solutions and then switch them

with a partner. Have the partner work the problems using the base-ten blocks. When finished, students can compare solutions.

InterventionHave students work with a partner or with a teacher in small groups to help develop these

concepts. Scaffold student understanding by initially providing arrays for students to use to find the product or quotient. Then provide a partially completed array or the outline of an array. Slowly remove scaffolding as students become more independent with finding a product or quotient using the base ten blocks.


http://www.arcytech.org/java/b10blocks/b10blocks.html Virtual Base Ten blocks http://www.arcytech.org/java/b10blocks/description.html Provides more information

on this lesson http://www.hbschool.com/activity/elab2004/gr6/1.html Interactive site for

multiplying decimals less than or equal to one.

This task was adapted from the following website http://argyll.epsb.ca/jreed/math7/strand1/1201.htm.




http://argyll.epsb.ca/jreed/math7/strand1/1201.htm

http://www.hbschool.com/activity/elab2004/gr6/1.html

http://www.arcytech.org/java/b10blocks/description.html

http://www.arcytech.org/java/b10blocks/b10blocks.html


Name ______________________________________ Date _____________________________

Base Ten Activity

Your task is to use the base-ten blocks to represent multiplication and division with decimals. Use the decimal block values below to help you find the product or quotient of each decimal problem.

1. 4.8 times 32. 2.1 x 5.43. 0.6 x 1.94. 12 ÷ 0.35. 3.6 divided by 1.2






LEARNING TASK: How Much Money?

STANDARDS ADDRESSED



0.01, and .001.


b. Explain the process of multiplication and division, including situations in which the multiplier and divisor are both whole numbers and decimals.














phenomena.

ESSENTIAL QUESTIONS

What happens when we divide a decimal by a decimal? How do the rules of multiplying whole numbers relate to multiplying decimals?

MATERIALS

“How Much Money?” Recording Sheet

GROUPING

Individual/Partner Task


In this task, students will use division or other basic operations to find the number of coins equal to a given dollar value.

Background KnowledgeWe want students to understand that when they are trying to determine the number of coins

of specific values, they are indeed performing division (measurement division). Then, when they are solving 34.50 divided by 0.25, one way to do this is to think of everything in terms of cents and determine how many group of 25 cents can be made with 3450 cents. In this process, we changed both decimal numbers into whole numbers. Dividing to find the number of dimes might lead to an idea with efficiency. For example, instead of thinking about 3450 divided by 10, students can find the same answer by dividing 345.0 by 1.0.

Answers to the chart are shown below.

Final Daily Balance # of Pennies ($0.01)

# of nickels ($0.05)

# of dimes ($0.10)

# of quarters ($0.25)

Example $34.50 3450 690 345 138Day 1 -- $21.00 2100 420 210 84Day 2 -- $35.50 3550 710 355 142Day 3 -- $69.00 6900 1380 690 276Day 4 -- $121.00 12100 2420 1210 484Day 5 -- $234.50 23450 4690 2345 938





Task DirectionsStudents will follow the directions below from the “How Much Money?” Recording Sheet.

You have been asked by your school principal to help count the money at your school store. Your job is to determine how many pennies ($0.01), nickels ($0.05), dimes ($0.10), and quarters ($0.25) you have at the end of the day.

Complete the chart below and determine the maximum number of each type of coin that can be found in the final daily school store balance.



# of dimes ($0.10)


Example $34.50 3450 690 345 138Day 1 -- $21.00Day 2 -- $35.50Day 3 -- $69.00Day 4 -- $121.00Day 5 -- $234.50

After you have completed the chart, answer the questions below. Be prepared to share your answers with the class.

1. How can you be assured your answers are correct? Students should be able to “prove” their solutions make sense and be able to show how they arrived at their solution.

2. How did you find your individual solutions?3. Can you think of another method to find the number of coins in your daily

balance?4. Do you see any patterns? If so, what are they? Various responses may be noted

here. One possible response could be that “dimes are half the number of nickels each time.” Another response could be that the “number of nickels is five times the number if pennies.”

Questions/Prompts for Formative Student Assessment How did you think about this problem? What operation did you use to find the number of pennies? Nickels? Dimes? Quarters? Is there another way you could have found the number pennies? Nickels? Dimes?

Quarters? Which method is more efficient? Which method is easier to do in your head? Do you notice any patterns as you look down each column? As you look across each

row?

Questions for Teacher Reflection Are students flexible when thinking about how to solve this problem? Are students able to find efficient ways to find the number of needed coins?





DIFFERENTIATION

ExtensionHave the students determine how many ways they can create the final daily balance with

different combinations of coins. (Example: $21.00 = 2100 pennies, or 80 quarters and 400 pennies, or…)

InterventionUse manipulatives and/or real coins to model smaller amounts of money before moving

to the recording sheet.


http://www.kokogiak.com/megapenny/default.asp This web page demonstrates the size of different collections of pennies from one to one quintillion.




http://www.kokogiak.com/megapenny/default.asp


Name ______________________________________ Date _____________________________

How Much Money?

You have been asked by your school principal to help count the money at your school store. Your job is to determine how many pennies ($0.01), nickels ($0.05), dimes ($0.10), and quarters ($0.25) you have at the end of the day.

Complete the chart below and determine the maximum number of each type of coin that could be found in the final daily school store balance.



# of dimes ($0.10)


Example $34.50 3450 690 345 138

Day 1 -- $21.00

Day 2 -- $35.50

Day 3 -- $69.00

Day 4 -- $121.00

Day 5 -- $234.50





PERFORMANCE TASK: Super Slugger Award

STANDARDS ADDRESSED


a. Understand place value.


d. Understand the relationships and rules for multiplication and division of whole numbers also apply to decimals.


a. Use variables, such as n or x, for unknown quantities in algebraic expressions.b. Investigate simple algebraic expressions by substituting numbers for the unknown.

M5D2 Students will collect, organize, and display data using the most appropriate graph.












phenomena.

ESSENTIAL QUESTIONS

How do we compare decimals? How do we find the highest batting average? How do we best represent data in a graph?

MATERIALS

“Super Slugger Award” Recording Sheet

GROUPING



In this task, students work to determine baseball players’ batting averages, represent the data in a graph, and then determine the recipient of a “Super Slugger Award.”

CommentsAnswers are shown below along with sample student responses.

1. Figure out the batting averages of each player on the team. Round each batting average to the thousandths decimal place. Remind students that in order to find a hitter’s batting average you divide the number of hits (h) a player gets by the number of times they have been to bat (b); h ÷ b.

2. Use the information that you collected to organize and display the data using the most appropriate graph.




Player Number of Hits Number of Times at Bat Batting AverageK. Smith 25 76 0.328T. Rusch 33 80 0.412A. Patrick 51 96 0.531

K. Waldrop 18 59 0.305P. Corbett 29 62 0.467J. Mark 29 64 0.453C. Mudd 42 71 0.591C. Cohen 38 67 0.567

D. Kirkland 37 61 0.606


3. Explain who should receive the Super Slugger Award from the team and why you feel the player deserves it. Overall best batting average would go to D. Kirkland. Students should note mention that 0.606 is the largest number of all the batting averages.

Background KnowledgeRounding to the nearest whole number is introduced to students in Grade 4. However,

students were not introduced to rounding to the tenth, hundredth, and thousandth places. You may want to introduce students to rounding to different place values for exposure before completing the activity.

Task DirectionsStudents will follow the directions below from the “Super Slugger Award” Recording

Sheet.

Congratulations! Your baseball team has made it to the World Series Little League Baseball Championship. The team has won a record-setting twelve games in a row, with an overall record of 23 wins and only 4 losses.

Your coach needs your help. He is having a hard time figuring out which player on the team has the best batting average and has asked for your assistance.

Your job has three parts. 1. Determine the batting averages of each player on the team. (Round each batting

average to the thousandths decimal place.) Remember that in order to find a hitter’s batting average you divide the number of hits (h) a player gets by the number of times they have been to bat (b); h ÷ b.


3. Explain who should receive the Super Slugger Award from the team and why you feel the player deserves it.





Questions/Prompts for Formative Student Assessment To how many place values do you need to divide? Why? What is an appropriate graph for this data? Why do you think so? What are the disadvantages of using this type of graph? What are the advantages? What criteria did you use to determine the student who deserves the Super Slugger

Award? Who else could deserve this award?

Questions for Teacher Reflection Are students able to identify and construct the best type of graph to use to represent the

data in this task? Do students divide to the ten-thousandths’ place in order to round to the thousandths’

place? Are students rounding accurately? Able to compare numbers accurately and with ease? Are students able to clearly communicate their reasons for their choice of the Super

Slugger Award recipient?

DIFFERENTIATION

Extension Have the students look in the sports page for the batting averages of a minor or major

league baseball team and construct graphs to represent the data. Have the students see if they can find the number of hits for a player, given only the

batting average and the number of times at bat.

Intervention Teachers can create their own line-up with fewer players to meet the needs of their

students.


http://atlanta.braves.mlb.com/stats/sortable_player_stats.jsp?teamPosCode=atl&statType=1&noHighlight=true&section1=1&timeFrame=1&c_id=atl&statSet1=1&subScope=teamCode&=&timeSubFrame=2009&&sortByStat=POS This link is to “Sortable Player Stats” for the Atlanta Braves. There are links to all Major League baseball teams from this page.




http://atlanta.braves.mlb.com/stats/sortable_player_stats.jsp?teamPosCode=atl&statType=1&noHighlight=true&section1=1&timeFrame=1&c_id=atl&statSet1=1&subScope=teamCode&=&timeSubFrame=2009&&sortByStat=POS




Name_________________________________________ Date ___________________________

Super Slugger Award Recording Sheet

Congratulations! Your baseball team has made it to the World Series Little League Baseball Championship. The team has won a record-setting twelve games in a row, with an overall record of 23 wins and only 4 losses.

Your coach needs your help. He is having a hard time figuring out which player on the team has the best batting average and has asked for your assistance.

Your job has three parts. 1. Determine the batting averages of each player on the team. (Round each batting

average to the thousandths decimal place.) Remember that in order to find a hitter’s batting average you divide the number of hits (h) a player gets by the number of times they have been to bat (b); h ÷ b.


3. Explain who should receive the Super Slugger Award from the team and how you would justify your choice.

Player Number of Hits (h)

Number of Times at Bat (b) Batting Average

K. Smith 25 76

T. Rusch 33 80

A. Patrick 51 96

K. Waldrop 18 59

P. Corbett 29 62

J. Mark 29 64

C. Mudd 42 71

C. Cohen 38 67

D. Kirkland 37 61





PERFORMANCE TASK: Number Puzzle

STANDARDS ADDRESSED





ESSENTIAL QUESTIONS

How can we efficiently solve multiplication and division problems with decimals?

MATERIALS

“Number Puzzle” Recording Sheet “Create-a-Number Puzzle” Recording Sheet

GROUPING



In this task, students solve problems using multiplication and division of decimals and find the answers in a number-search puzzle. After students solve the puzzle on the “Number Puzzle” Recording sheet, they create a number puzzle of their own using problems they make and solve. Students then trade papers allowing other students to find the solutions hidden within their number-search puzzle.

CommentsStudents may use a calculator to check their work before creating their puzzle to ensure

accuracy.

Background KnowledgeIf your students are not familiar with this type of activity, you may want to share an

example of a word search and/or number search with them prior to assigning the task.The number puzzle and solutions are shown below.





1. 0.12 • 0.35 2. 0.2 • 1.33. 10.5 • 2.84. 0.69 • 0.025. 19.87 x 0.46. 12.3 ÷ 3.27. 200.5 ÷ 2.5

8.

9.

10. 15.9 ÷ 0.6

Task DirectionsStudents will follow the directions below from the “Number Puzzle” and “Create-a-Number

Puzzle” Recording Sheets.




.0 4 2 1 4 8 3.

1 8 1 8 0. 2 8

3 8 .2 3 7 8. 4

8 .0 6 2 9. 4 3

4 2 6. 5 5 6 7

9 0 7. 9 4 8 5

.0 4 2 8 7 2 3

.0 4 2 1 4 8 3.

1 8 1 8 0. 2 8

3 8 .2 3 7 8. 4

8 .0 6 2 9. 4 3

4 2 6. 5 5 6 7

9 0 7. 9 4 8 5

.0 4 2 8 7 2 3


Part IFind the answers to the given problems hidden in the number search.1. 0.12 • 0.35 =2. 0.2 • 1.3 =3. 10.5 • 2.8 =4. 0.69 • 0.02 =5. 19.87 x 0.4 =6. 12.3 ÷ 3.2 =7. 200.5 ÷ 2.5 =

8. =

9. =

10. 15.9 ÷ 0.6 =

Encourage students to use estimation and mental math to find as many answers as they can.

Part IIStudents will create their own number puzzles using the “Create-a-Puzzle” Recording Sheet.

Directions are as follows:1. Make up five multiplication problems and five division problems. All

multiplication/division problems must include decimals.2. Using the forty-nine squares on your game board, make a number search puzzle with the

ten answers to your multiplication/division problems. The answers can be hidden horizontally or vertically in the grid. Fill in any unused spaces with random numbers.

3. Switch your number search puzzle with a partner and try to solve each other’s number puzzles.

Questions/Prompts for Formative Student Assessment Can you explain your process for computing with decimals? What are your strategies for creating your own number puzzle? How can you use estimation to help you find the product? Quotient?

Questions for Teacher Reflection Are students able to correctly solve the Number Puzzle? Are they able to successfully create their own puzzle? How fluently can students multiply and divide decimals? Which students are having difficulty with the computation and how will I address their

needs?




.0 4 2 1 4 8 3.

1 8 1 8 0. 2 8

3 8 .2 3 7 8. 4

8 .0 6 2 9. 4 3

4 2 6. 5 5 6 7

9 0 7. 9 4 8 5

.0 4 2 8 7 2 3


DIFFERENTIATION

ExtensionEncourage students to try to create overlaps with their hidden solutions so that a given

number appears in both a vertical and horizontal solution. Intervention

Have students work with a partner or in small groups to solve and create the number puzzles.





Name _________________________________________ Date __________________________

Number Puzzle

Find the answers to the given problems hidden in the number search.

11. 0.12 • 0.35 =

12. 0.2 • 1.3 =

13. 10.5 • 2.8 =

14. 0.69 • 0.02 =

15. 19.87 x 0.4 =

16. 12.3 ÷ 3.2 =

17. 200.5 ÷ 2.5 =

18. =

19. =

20. 15.9 ÷ 0.6 =




.0 4 2 1 4 8 3.

1 8 1 8 0. 2 8

3 8 .2 3 7 8. 4

8 .0 6 2 9. 4 3

4 2 6. 5 5 6 7

9 0 7. 9 4 8 5

.0 4 2 8 7 2 3


Name _________________________________________ Date __________________________

Create - a - Number Puzzle

1. Make up five multiplication problems and five division problems. All multiplication/division problems must include decimals.

2. Using the forty-nine squares on your game board, make a number search puzzle with the ten answers to your multiplication/division problems. The answers can be hidden horizontally or vertically in the grid. Fill in any unused spaces with random numbers.

3. Switch your number search puzzle with a partner and try to solve each other’s number puzzles.

Write your number problems below. Create your number puzzle below.

1. ______________________________

2. ______________________________

3. ______________________________

4. ______________________________

5. ______________________________

6. ______________________________

7. ______________________________

8. ______________________________

9. ______________________________

10. ______________________________





LEARNING TASK: What’s My Rule?

STANDARDS ADDRESSED





a. Use variables, such as n or x, for unknown quantities in algebraic expressions.b. Investigate simple algebraic expressions by substituting numbers for the unknown.c. Determine that a formula will be reliable regardless of the type of number (whole

numbers or decimals) substituted for the variable.





M5P4. Students will make connections among mathematical ideas and to other disciplines.a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

ESSENTIAL QUESTIONS

What are some patterns that occur when multiplying and dividing by decimals? What strategies are effective for finding a missing factor or divisor?





MATERIALS

Overhead projector, white/chalk board, or computer projector Scratch paper “Guess My Rule” problems (pre-determined)

GROUPING

Class/Small Group Task


In this task, students will work to discover an unstated rule that the teacher is using to find outcomes.

CommentsThis activity is similar to the “What’s My Rule?” activities done at earlier grade levels, but

with appropriate rules and numbers for fifth grade. When creating problems to present to the class, think about the level of difficulty of each problem and the order in which you present the problems.

Students will need time to consider number relationships after each entry, so encourage students to be patient and quiet during this time. When first doing this activity, you may want to begin with simpler rules such as adding or subtracting a number and then work toward using multiplication and division.

The teacher will want to decide on a predetermined rule for an input/output table. The students are asked to give numbers and the teacher will complete the chart using the predetermined rule. This process continues until several entries are in the table. When students think they’ve discovered the rule, they should keep it to themselves while other students are still trying to determine the rule.

Here are some sample “rule tables.” NOTE: When recording the table for students to see, the rule will not be displayed initially.

Rule: x • 0.25 In Out3 0.755 1.257 1.7510 2.512 3





Rule: n ÷ 0.013 3006 60010 1,00015 1,5000.1 10

Background KnowledgeStudents may soon realize that by using 1 as their suggested number, the result will

clearly show the rule. Since one is the multiplicative identity, any number times one will equal itself. For example, using the sample table above, if the rule is n • 0.25 and students suggest the number 1, the rule is seen immediately in the result of 1 • .25 = .25. Once students start using this strategy for uncovering the rule, you may need to eliminate 1 as a number that students can suggest.

The second sample table gives the rule n ÷ 0.01. This rule gives the same result as n x 100. This is a great problem to get students to rethink their understanding of the relationship between multiplication and division. If a student suggests n x 100, be clear that could be the rule, but isn’t the one you had planned. Push students to determine another possibility to elicit the actual rule for the table. They should have had enough experience with dividing decimals that it will be apparent to them.

Task Directions Draw an input/output table for students to see. Tell students you’re thinking of a rule for the table and ask them to give you numbers to

add to the table. You will then complete the chart for each number they give you. After several numbers are entered, students will likely begin to discover the rule. When

students think they know the correct rule, give them a number and ask them to use the rule to give you the correct number to record in the table, without actually stating the rule. This will allow you to check their thinking and still allow other students to think independently.

When students have discovered the rule, have a discussion about the strategies they used to determine the rule.

Questions/Prompts for Formative Student Assessment What strategies are you using to determine the rule? At what point did you know your prediction for the rule was correct? How did you know for sure your rule was correct?





Questions for Teacher Reflection Are students suggesting numbers that make it easier to determine the rule? Do students discover the advantage of using the multiplicative identity (1) to determine

the rule? If so, how does the game need to change once they use this strategy? Are students able to express the rule in algebraic terms, using a variable (n) for the input

value?

DIFFERENTIATION

ExtensionHave students develop their own rule tables to share with partners or in small groups.

InterventionBegin with simpler rules and have students explain their thinking so any misconceptions

can be addressed immediately.


http://ims.ode.state.oh.us/ODE/IMS/Lessons/Content/ CMA_LP_S04_BF_L04_I03_01.pdf Extensive description of and extensions for “Guess My Rule.”

http://www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines.html A student-created input-output machine with which students can try to guess the rule.




http://www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines.html

http://ims.ode.state.oh.us/ODE/IMS/Lessons/Content/CMA_LP_S04_BF_L04_I03_01.pdf

http://ims.ode.state.oh.us/ODE/IMS/Lessons/Content/CMA_LP_S04_BF_L04_I03_01.pdf


PERFORMANCE TASK: Do you see an Error?

STANDARDS ADDRESSED



0.01, and .001.


b. Explain the process of multiplication and division, including situations in which the multiplier and divisor are both whole numbers and decimals.



M5P1. Students will solve problems (using appropriate technology).e. Build new mathematical knowledge through problem solving.f. Solve problems that arise in mathematics and in other contexts.g. Apply and adapt a variety of appropriate strategies to solve problems.h. Monitor and reflect on the process of mathematical problem solving.

M5P2. Students will reason and evaluate mathematical arguments.e. Recognize reasoning and proof as fundamental aspects of mathematics.f. Make and investigate mathematical conjectures.g. Develop and evaluate mathematical arguments and proofs.h. Select and use various types of reasoning and methods of proof.

M5P3. Students will communicate mathematically.e. Organize and consolidate their mathematical thinking through communication.f. Communicate their mathematical thinking coherently and clearly to peers, teachers, and

others.g. Analyze and evaluate the mathematical thinking and strategies of others.h. Use the language of mathematics to express mathematical ideas precisely.

M5P4. Students will make connections among mathematical ideas and to other disciplines.d. Recognize and use connections among mathematical ideas.e. Understand how mathematical ideas interconnect and build on one another to produce a

coherent whole.f. Recognize and apply mathematics in contexts outside of mathematics.

M5P5. Students will represent mathematics in multiple ways.d. Create and use representations to organize, record, and communicate mathematical ideas.





e. Select, apply, and translate among mathematical representations to solve problems.f. Use representations to model and interpret physical, social, and mathematical

phenomena.

ESSENTIAL QUESTIONS

What happens when we multiply a decimal by a decimal? What happens when we divide a decimal by a decimal? How can we check for errors in multiplication or division of decimals?

MATERIALS

“Do You See an Error?” Recording Sheet

GROUPING



In this task, students will look for errors in student work, find the correct solution, and write a letter to the students regarding the error and how it is corrected.

CommentsFinding and explaining errors is a powerful way for students to further develop their thinking

about their own strategies for solving problems. It not only causes them to closely examine the solutions, but also may prevent them from making the same mistakes.

Using student work from your classroom as an “error analysis warm-up” on a regular basis can help students begin to find and appropriately address errors in their work with greater success. NOTE: When using actual student work, always conceal student names to protect their identity. Whenever possible, use work from a different class or a previous class with the student’s identity concealed.

Background KnowledgeStudents should be able to check the placement of the decimal point quickly by estimating

the product, 24 x 11 = 264. Therefore, students should find that the decimal is not placed correctly in the first example. For the second example, both the dividend and the divisor were multiplied by 100 to create an easier problem, 1468 ÷ 2. Again, the student placed the decimal incorrectly, because the estimate would be 1500 ÷ 2 = 750.

Task DirectionsStudents will follow the directions below from the “Do You See an Error?” Recording Sheet.





Below are samples of two students’ work. Your job is to determine if they have the correct solution to the problem. If a student has the wrong solution, your next assignment is to:

1. Find the student error in each example.2. Re-work the problem showing the correct solution.3. Write a note to the student explaining where they went wrong and the steps they

need to take to correct their mathematical thinking.Sample student work:

Questions/Prompts for Formative Student Assessment How did you know that there is an error in the student work? Did you estimate the product or quotient first? How will you explain the error and how to correct it? How can doing this task keep you from making the same errors?

Questions for Teacher Reflection Are students using their estimation skills with which to determine the reasonableness of

the students work? Are students able to determine the errors in the student examples? Are students able to accurately explain in writing how to correct the errors?

DIFFERENTIATION

ExtensionGive students sample problems with multiple errors that involve not only decimal

placement, but also computational errors. Have them explain all errors.

InterventionRevisit the “Patterns-R-Us” task on page 21 of this unit. Encourage students to use the

patterns found from that task to estimate answers for the students’ work.


http://illuminations.nctm.org/LessonDetail.aspx?id=L252 Activity 3 in the “Too Big or Too Small?” lesson gives a related exploration.




http://illuminations.nctm.org/LessonDetail.aspx?id=L252


Name________________________________________ Date ____________________________

Do You See an Error? Recording Sheet

Below are samples of two students’ work. Your job is to determine if they have the correct solution to the problem.

If a student has the wrong solution, your next assignment is to:1. Find the student error in each example.2. Re-work the problem showing the correct solution.3. Write a note to the student explaining their error and the steps they

need to take to correct their mathematical thinking.

Student #1 (Hannah) Student #2 (Randy)





PERFORMANCE TASK: Road Trip

STANDARDS ADDRESSED


c. Multiply and divide with decimals including decimals less than one and greater than one.

d. Understand the relationships and rules for multiplication and division of whole numbers also apply to decimals.








phenomena.





ESSENTIAL QUESTIONS

What happens when we multiply a decimal by a decimal? What happens when we divide a decimal by a decimal? What are the various uses of decimals? How do we solve problems with decimals?

MATERIALS

“Road Trip” Recording Sheet Maps, atlases, and/or internet access Markers or highlighters

GROUPING



In this task, students determine fuel costs for a trip through the southeast of the United States using a given cost of fuel and the number of miles per gallon the family car gets.

CommentsAs a class, determine the price you will be using for gasoline. You may want to use the local

price while working on the project. For consistency, it will be easier for students to use the same price for the entire trip.

If the primary goal of this task is using decimal numbers in a problem solving setting, you may consider having students use an online mapping service because they will give distances to the closest hundredth of a mile.

Background KnowledgeYou may want to begin this task with a short class discussion about trips, costs of gasoline,

and budgets for vacation.

How many miles is it from the starting point to the stopping point on each day?

TRAVELING DAYS

Starting Point Stopping Point Total Number of Miles

DAY ONE Your Home Town Birmingham, AL will varyDAY TWO Birmingham, AL Dallas, TX 641.75

DAY THREE Dallas, TX Memphis, TN 452.56DAY FOUR Memphis, TN Chattanooga, TN 343.49DAY FIVE Chattanooga, TN Atlanta, GA 118.39DAY SIX Atlanta, GA Your Home Town will vary





Make sure that students have access to maps, an atlas, or the Internet to determine the total number of miles between the cities. Also, be sure students are careful when they carry their daily miles over from one day to the next.

Task DirectionsStudents will follow the directions below from the “Road Trip” Recording Sheet.

You and your family want to take a road trip. However, before your family can begin this wonderful adventure, you must first determine how much your trip is going to cost. You are responsible for figuring out how much money you will need for gas.

Using different resources in your classroom, determine how many miles it is to each day’s planned stopping point from the starting point.

Your family car you can travel 21.2 miles on one gallon of gasoline.

1. How many miles is it from each day’s starting point to the stopping point?2. How much will it cost your family to make the trip each day? What is the

total cost of gasoline for the entire trip? Assume that when your family reaches the stopping point, the car is not driven until the next day.

3. Use the information you collected to organize and display the data using the most appropriate graph.

4. Show at least two of your computations and explain your thinking in words.5. If your family car’s gas tank holds 19.7 gallons of fuel, how far (in miles) can

your family car travel on one tank of gasoline? Two tanks? Five tanks? 6. How long will your family spend traveling between cities if your average

speed is 65 miles per hour?

Questions/Prompts for Formative Student Assessment Explain your strategy for determining daily mileage. How are you organizing your work for this task? How does this task help you understand the importance of budgeting for a trip?

Questions for Teacher Reflection Are students organizing their work in a way that limits careless errors? Are students clearly organizing their work to aid their parents in understanding the costs? Are students using strategies that allow them to find the cost of fuel correctly?





DIFFERENTIATION

Extension Have students share other cities they could visit at the end of their road trip if they left

Atlanta, GA before they had to fill up with gas again. Have them explain why they would/could visit those cities?

You may want to allow students to use a calculator to check their work after they have completed the task to check for accuracy.

Have students create a trip they would want to take and determine mileage and cost of gas.

Have students create a budget for an entire trip that includes cost of gasoline, lodging, and an estimate for food and entertainment.

Intervention Give students one city at a time to compute the distance. Provide an organizer on which students can record the information required to find costs.


www.mapquest.com Online distance calculator and directions for desired trips http://www.driveusa.net/maps&tripplans.htm List of online auto travel services




http://www.driveusa.net/maps&tripplans.htm

http://www.mapquest.com/


Name _____________________________________ Date _____________________

Road Trip Recording Sheet

You and your family want to take a road trip. However, before your family can begin this wonderful adventure you must first determine how much your trip is going to cost. You are responsible for figuring out how much money you will need for gas.

Using different resources in your classroom, determine how many miles it is to each day’s planned stopping point from the starting point.

Your family car you can travel 21.2 miles on one gallon of gasoline.

Be sure to answer all the questions below.

1. How many miles is it from each day’s starting point to the stopping point?

2. How much will it cost your family to make the trip each day? What is the total cost of gasoline for the entire trip? Assume that when your family reaches the stopping point, the car is not driven until the next day.

3. Use the information you collected to organize and display the data using the most appropriate graph.

4. Show at least two of your computations and explain your thinking in words.

5. If your family car’s gas tank holds 19.7 gallons of fuel, how far (in miles) can your family car travel on one tank of gasoline? Two tanks? Five tanks?

6. How long will your family spend traveling between cities if your average speed is 65 miles per hour?





UNIT TWO CULMINATING TASK #1

This culminating task represents the level of depth, rigor and complexity expected of all fifth grade students to demonstrate evidence of learning.

PERFORMANCE TASK: Teacher for a Day

STANDARDS ADDRESSED


a. Understand place value.










coherent whole.





c. Recognize and apply mathematics in contexts outside of mathematics.


phenomena.

ESSENTIAL QUESTIONS

How are multiplication and division related? What happens when we multiply a decimal by a decimal? What happens when we divide a decimal by a decimal? How do the rules of multiplying whole numbers relate to multiplying decimals?

MATERIALS

Access to computer to create a PowerPoint or similar computer project; alternately, poster paper and markers to create visual displays

GROUPING

Individual/Partner/Small Group Task


In this task, students represent and explain their understanding about how to multiply and divide decimals, their understanding of the place value system, and their knowledge of how decimals are used throughout life. Also, students are asked to explain and communicate why decimals are important. Finally, students are asked to teach the process of multiplying and dividing decimals. They can either create a lesson on PowerPoint or create visual displays with posters and markers.

CommentsWhile this task may serve as a summative assessment, it also may be used for teaching and

learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them.

Background KnowledgeStudents should demonstrate a conceptual understanding of operations with decimals as

opposed to a purely procedural knowledge. For example, students should understand that if they are multiplying tenths by tenths, the product must be expressed as hundredths. (i.e., 1/10 x 1/10 = 1/100). Students should also know to round to the nearest whole number and estimate to place the decimal, using the estimate to determine the reasonableness of an answer, rather than only knowing to count the digits after the decimal point to place the decimal point in the answer.





When multiplying or dividing decimals, students should demonstrate an understanding of place value and digit values rather than only showing the algorithmic steps one could use to divide. Students should show the operations using base ten blocks and/or partial products or quotients (rather than procedural steps), demonstrating an understanding of the operations modeled.

Task DirectionsStudents will follow the directions below from the “Teacher for a Day” recording sheet.

Your fifth grade teacher has asked for your help. A teacher friend who teaches at Old Mill Elementary School has a fifth grade class that is having difficulty with multiplying and dividing decimals.

Your teacher knows that you are an expert at multiplying and dividing decimals. With your partner, create a presentation that addresses the following items:

1. Explanation of the place value system2. How to multiply decimals3. How to divide decimals4. When good multiplication and division skills with decimals are needed and why

they are importantYour presentation should include examples and visual models to help the fifth

grade class at Old Mill Elementary School have a clear understanding of how to work with decimals. Remember, your presentation needs to have multiple representations using pictures, numbers, and/or words.

Be prepared to give a practice presentation to your classmates.

Questions/Prompts for Formative Student Assessment What models are you going to use to show examples of multiplying and dividing

decimals? How do you know when to multiply and when to divide with decimals? When would you use multiplication and division of decimals in your daily life?

Questions for Teacher Reflection Are students able to demonstrate a conceptual understanding of the operations with

decimals? Are students clearly organizing their work to aid students in understanding computation

with decimals? Did students complete all parts of the task successfully? Have students used pictures, numbers, and words to explain their thinking?

DIFFERENTIATION

Extension Have students include in their presentation an explanation of the relationship between

multiplication and division of decimals. Have students include examples of when these skills are used in daily life.

InterventionGeorgia Department of Education




Have students give their responses to the portions of the task verbally before asking students to put their explanations in writing.


Students may want to use any of the web pages listed throughout the unit to enhance their presentation.





Name __________________________________________ Date _________________________

Teacher for a Day Task Directions

Your fifth grade teacher has asked for your help. A teacher friend who teaches at Old Mill Elementary School has a fifth grade class that is having difficulty with multiplying and dividing decimals.

Your teacher knows that you are an expert at multiplying and dividing decimals. With your partner, create a presentation that addresses the following items:

1. Explanation of the place value system2. How to multiply decimals3. How to divide decimals4. When good multiplication and division skills with decimals are needed and why they

are importantYour presentation should include examples and visual models to help the fifth grade

class at Old Mill Elementary School have a clear understanding of how to work with decimals. Remember, your presentation needs to have multiple representations using pictures, numbers, and/or words.

Be prepared to give a practice presentation to your classmates.





UNIT TWO CULMINATING TASK #2

This culminating task represents the level of depth, rigor and complexity expected of all fifth grade students to demonstrate evidence of learning.

PERFORMANCE TASK: Bargain Shopping*Adapted from A Collection of Performance Tasks and Rubrics: Upper Elementary School Mathematics by C. Danielson.

STANDARDS ADDRESSED










M5P5. Students will represent mathematics in multiple ways.a. Create and use representations to organize, record, and communicate mathematical ideas.b. Select, apply, and translate among mathematical representations to solve problems.





c. Use representations to model and interpret physical, social, and mathematical phenomena.

ESSENTIAL QUESTIONS

How do the rules of multiplying whole numbers relate to multiplying decimals? How can we multiply and divide decimals fluently? How do we solve problems with decimals?

MATERIALS

“Bargain Shopping” Recording Sheet Sale Papers for Canton Supplies and Cherokee Discounts

GROUPING



In this task, students will determine the best store to buy certain items on their list of needed school supplies. Students will determine how much money they will have left after making appropriate purchases. With the remaining money from their original purchases, students are asked whether or not they can purchase school supplies for their siblings.

CommentsWhile this task may serve as a summative assessment, it also may be used for teaching and

learning. It is important that all elements of the task be addressed throughout the learning process so that students understand what is expected of them. Students may want to practice their estimation skills before participating in this activity.

You may want to have students bring in copies of local advertisements from the newspaper to use for a similar introductory activity. Also, you may want to collect advertisements during the first week of school and save them until you are ready to use them. This way, students could purchase supplies from the store they think offers a better buy.

Background KnowledgeYou may want to have some class discussions about what constitutes a good value before

beginning this task. Students may have different answers for Question 3 based on prior experiences of what “better” means to them. Have students justify their answers.

Task DirectionsStudents will follow the directions below from the “Bargain Shopping” Recording Sheet.

It is time to go shopping for school supplies for next school year. You have ads from two local stores indicating their prices (which include sales tax) for different school supplies. Your mother has given you $45.50 to spend, and wants you to find the best prices on things you will need.





School Supplies Needed

Eight foldersOne calculator

Four packs of regular lined notebook paper (pack of 500 sheets)Three rulers

Three bottles of glueThirty-three pencils

Ten pensFive spiral notebooks

Two backpacks

1. Determine the cheapest price for each item on your list of school supplies needed. From which store would you buy each school supply? How much money will you have to spend? How much of the $45.50 will you have left over?

2. With the remaining money, you decide to buy each of your three siblings the same school supply item. What item would you purchase?

3. Is one store better than the other? Why or why not?

4. If you could choose only one store, which would it be? Explain your reasoning by writing an overall summary of your findings.

See sale papers from two different stores shown below.

Questions/Prompts for Formative Student Assessment What is your strategy for completing this task? What factors influence your decision about which store has better values? How would you determine the store in which you would make your purchases if you had

to choose only one store?





Questions for Teacher Reflection Are students clearly organizing their work to aid accuracy? Are students fluent with multiplying and dividing decimals? Are students able to clearly explain their thinking and justify their answers?

DIFFERENTIATION

Extension Have students research prices on websites such as Staples, Wal-Mart, Office

Max, Office Depot, etc. and compare total costs with one another. Have students plan ways to earn the money necessary for school supplies and

other back to school items such as clothes, shoes, computer, etc. Have students plan ways to earn money to provide school supplies for

students in the community who are unable to purchase back to school supplies.

Intervention Have students make an organized list of the items and the prices at both stores to help

them make comparisons.


Students may view local store websites as an alternative to using the sale papers provided.





Name ________________________________________ Date ___________________________

Bargain Shopping Recording Sheet

It is time to go shopping for school supplies for next school year. You have ads from two local stores indicating their prices (which include sales tax) for different school supplies. Your mother has given you $45.50 to spend, and wants you to find the best prices on the things you will need.

School Supplies Needed

Eight foldersOne calculator

Four packs of regular lined notebook paper (pack of 500 sheets)Three rulers

Three bottles of glueThirty-three pencils

Ten pensFive spiral notebooks

Two backpacks

1. Determine the cheapest price for each item on your list of school supplies needed. From which store would you buy each school supply? How much money will you have to spend? How much of the $45.50 will you have left over?

2. With the remaining money, you decide to buy each of your three siblings the same school supply item. What item would you purchase?

3. Is one store better than the other? Why or why not?

4. If you could choose only one store, which would it be? Explain your reasoning by writing an overall summary of your findings.





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Erasers – large60¢ each

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Backpack $8.37

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Ruler$1.14 each

Spiral Notebooks75¢ each




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Glue2 oz. bottle95¢

Backpack$8.97

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Ruler$1.45 each

Spiral Notebooks99¢ eachBUY FIVE GET ONE FREE

Unit 2 Organizer: “DIVINE DECIMALS” · Web viewWhen multiplying or dividing decimals, students...

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Transcript of Unit 2 Organizer: “DIVINE DECIMALS” · Web viewWhen multiplying or dividing decimals, students...