M A T H E M A T I C S
Grade 3
Mathematics
Frameworks
Unit 2
Multiplication and Division
of Whole Numbers
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Unit 2
MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
(5 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 ...............................................................................................9
Classroom Routines ...........................................................................................................11
Strategies for Teaching and Learning ..............................................................................11
Evidence of Learning .......................................................................................................11
Tasks..................................................................................................................................12
 Array Challenge ...............................................................................................13
 Multiplication Chart Mastery...........................................................................23
 Stuck on Multiplication....................................................................................28
 The Magic Money Machine .............................................................................34
 Armadillo Stories .............................................................................................40
 Change It Around! ...........................................................................................45
 Seating Arrangements ......................................................................................50
 Family Reunion ................................................................................................54
 Multiplication with Base-Ten Blocks ..............................................................59
 Array-nging our Fact Families .........................................................................65
 Division Patterns ..............................................................................................70
 Stuck on Division.............................................................................................76
 Making Cents of Division ................................................................................81
 Sharing Pumpkin Seeds ...................................................................................88
Culminating Task
 Ice Cream Scoops ............................................................................................94
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 2 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
OVERVIEW
In this unit, students will:
 begin to understand the concepts of multiplication and division
 learn the basic facts of multiplication and their related division facts
The understanding of and ability to use multiplication and division is the basis for all further
mathematics work and its importance cannot be overemphasized. As students move through
upper elementary grades and middle school, the foundation laid here will empower them to work
with fractions, decimals, and percents.
Although the units in this instructional framework emphasize key standards and big ideas at
specific times of the year, routine topics such as estimation, mental computation, and basic
computation facts should be addressed on an ongoing basis. Ideas related to the five process
standards, problem solving, reasoning, connections, communication, and representation, should
be addressed continually as well
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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 3 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
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
M3N3. Students will further develop their understanding of multiplication of whole
numbers and develop the ability to apply it in problem solving.
a. Describe the relationship between addition and multiplication, i.e. multiplication is
defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit
number.
d. Understand the effect on the product when multiplying by multiples of 10.
e. Apply the identity, commutative, and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3N4. Students will understand the meaning of division and develop the ability to apply it
in problem solving.
a. Understand the relationship between division and multiplication and between division
and subtraction.
b. Recognize that division may be two situations: the first is determining how many equal
parts of a given size or amount may be taken away from the whole as in repeated
subtraction, and the second is determining the size of the parts when the whole is
separated into a given number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
d. Explain the meaning of a remainder in division in different circumstances.
e. Divide a 2 and 3-digit number by a 1-digit divisor.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3A1. Students will use mathematical expressions to represent relationships between
quantities and interpret given expressions.
a. Describe and extend numeric and geometric patterns.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the
unknown in a number sentence.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 4 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
RELATED STANDARDS
M3P1. 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.
M3P2. 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.
M3P3. 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.
M3P4. 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.
M3P5. 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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 5 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ENDURING UNDERSTANDINGS
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Multiplication can be thought of as repeated addition.
Multiplication facts can be deduced from patterns.
The associative property of multiplication can be used to simplify computation.
The distributive property of multiplication allows us to find partial products and then
find their sum.
Patterns are evident when multiplying a number by ten or a multiple of ten.
Multiplication and division are inverses; they undo each other.
Multiplication and division can be modeled with arrays.
Multiplication is commutative, but division is not.
There are two common situations where division may be used.
o Partition (or fair-sharing) - given the total amount and the number of equal
groups, determine how many/much in each group
o Measurement (or repeated subtraction) - given the total amount and the amount in
a group, determine how many groups of the same size can be created.
As the divisor increases, the quotient decreases; as the divisor decreases, the quotient
increases.
There is a relationship between the divisor, the dividend, the quotient, and any
remainder.
ESSENTIAL QUESTIONS
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How are multiplication and addition alike?
How are multiplication and addition different?
What are strategies for learning multiplication facts?
How can we practice multiplication facts in a meaningful way that will help us remember
them?
How can we connect multiplication facts with their array models?
How is the commutative property of multiplication evident in an array model?
What patterns of multiplication can we discover by studying a times table chart?
How can we determine numbers that are missing on a times table chart by knowing
multiplication patterns?
What role can arithmetic properties play in helping us understand number patterns?
How can we model multiplication?
How are multiplication and addition related?
How can we write a mathematical sentence to represent a multiplication model we have
made?
Is there more than one way of multiplying to get the same product?
What patterns can be found when multiplying numbers?
What pattern is there when we multiply by ten or a multiple of ten? By one? By zero?
What math is involved in the study of Georgia animals?
How can multiplication help us repeatedly add larger numbers?
How does the order of the digits in a multiplication problem affect the product?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 6 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS (Continued)
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How does understanding the commutative property help us multiply?
How is multiplication like repeated addition?
How many different ways can you arrange 24 chairs?
How does drawing an array help us think about different ways to decompose a number?
How can multiplication and division be used to solve real world problems?
How can we use patterns to solve problems?
How can base-ten blocks help us understand how to multiply a two-digit number?
How does understanding the distributive property help us multiply large numbers?
How are multiplication and division related?
How can the same array represent both multiplication and division?
How do the parts of a division problem relate to each other?
What is the relationship between the divisor and the quotient?
What happens to the quotient when the dividend increases or decreases?
What do the parts of a division problem represent?
How can we model division?
How are multiplication and division related?
How are subtraction and division related?
How can we write a mathematical sentence to represent division models we have made?
How can we divide larger numbers?
What is the meaning of a remainder?
Does a remainder mean the same thing in every division problem?
How do estimation, multiplication, and division help us solve problems in everyday life?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 7 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
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.
 Odd and even numbers
 Skip counting by twos, threes, fives, and tens
 Determining reasonableness using estimation
 Addition and subtraction as inverse operations
 Multiplication of one-digit numbers
 Commutative, associative, and identity properties of addition
 Basic addition facts
 Making tens in a variety of ways
 Strategies to add quickly such as double, double plus one more, addition of tens, and
double minus one
 Basic subtraction facts
 Place value for ones, tens, hundreds, thousands, and tenths
 Modeling numbers using base 10 blocks and on grid paper
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 8 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
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.
 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
 Equal: Having the same value.
 Factor: A number that is multiplied by another number to get a product. To “factor"
means to write the number or term as a product of its factors.
 Identity Property of Multiplication: Any number that is multiplied by 1 results in the
number itself.
Example: 1 x 5 = 5 x 1 = 5
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 9 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
 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).
 Multiplicand: The number in a multiplication equation that represents the number of
objects in each (equal-sized) group.
 Multiplication: The operation of repeated addition of a number.
Example: 3 x 5 = 5 + 5 + 5 = 15
 Multiplier: The number in a multiplication equation that represents the number of
(equal-sized) groups.
 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 = 1800
60 x 7 =
420
Partial Products
30 x 3 =
90
3x7=
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).
 Product: A number that is the result of multiplication.
 Quotient: The result of a division problem.
Example: dividend ÷ divisor = quotient
 Remainder: The part of the dividend that is left after all possible equal sized groups
are created.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 10 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
CLASSROOM ROUTINES
The importance of continuing the established classroom routines cannot be overstated. Daily
routines must include such obvious activities as estimating, analyzing data, describing patterns,
and answering daily questions. 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
allowing children plenty of time 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. These routines
contribute to a rich, hands-on, standards based classroom and will support students’
performances on the tasks in this unit and throughout the school year.
STRATEGIES FOR TEACHING AND LEARNING
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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 timely teacher
feedback, peer feedback, and metacognition which includes self-assessment and
reflection.
Students need to write in mathematics class to explain their thinking, to talk about how
they perceive topics, and to justify their work to others.
EVIDENCE OF LEARNING
By the conclusion of this unit, students should be able to demonstrate the following
competencies:
 use mental math to multiply and divide
 be fluent with the multiplication facts up to 10 X 10
 use estimation to determine reasonableness of products and quotients computed
 be able to read, interpret, solve, and compose simple word problems dealing with
multiplication and division
 understand how to use inverse operations to verify accuracy of computation
 be able to write and solve expressions using symbols in place of numbers
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 11 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
TASKS
The following tasks represent the level of depth, rigor, and complexity expected of all third
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
Array Challenge
Multiplication Chart Mastery
Stuck on Multiplication
The Magic Money Machine
Armadillo Stories
Change It Around!
Seating Arrangements
Family Reunion
Multiplication with Base-Ten
Blocks
Array-nging our Fact Families
Division Patterns
Stuck on Division
Making Cents of Division
Sharing Pumpkin Seeds
Culminating Task:
Ice Cream Scoops
Task Type
Grouping Strategy
Learning Task
Partner/Small Group Task
Learning Task
Individual/Small Group Task
Learning Task
Individual/Small Group Task
Learning Task
Individual/Partner Task
Performance Task
Individual/Partner Task
Learning Task
Individual/Small Group Task
Performance Task
Individual/Partner Task
Performance Task
Individual/Partner Task
Learning Task
Individual/Partner Task
Learning Task
Individual/Partner Task
Learning Task
Individual/Partner Task
Learning Task
Individual/Partner Task
Performance Task
Individual/Partner Task
Learning Task
Individual/Partner Task
Performance Task
Individual Task
Skills
Multiplication facts
Multiplication chart patterns
Multiplication concepts
Multiplication using an input-output
machine
Writing multiplication story problems
Commutative property of multiplication
Arrays and multiplication facts
Multiplication and division patterns
One-digit by 2-digit multiplication
Models for multiplication and division
Division patterns
Division concepts
Division word problems
Division word problems
Three-digit dividend, one-digit divisor
Multiplication and division
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 12 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Array Challenge
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of multiplication
of whole numbers and develop the ability to apply it in problem solving.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
e. Apply the identity, commutative, and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
M3P4. 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

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How are multiplication and addition alike?
How are multiplication and addition different?
What are strategies for learning multiplication
facts?
How can we practice multiplication facts in a
meaningful way that will help us remember
them?
How can we connect multiplication facts with
their array models?
How is the commutative property of
multiplication evident in an array model?
MATERIALS
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“Shaded Array Cards” copied on card stock
and cut out
“Array Challenge” game directions and recording sheet
GROUPING
Partner/Small Group
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 13 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students work in small groups to play a game in which array cards are used to
represent area models for multiplication facts. Students have opportunities to display their cards
and respond with the multiplication fact(s) that apply to the array.
Comments
The Shaded Array Cards provide an excellent opportunity for students to make visual
connections between multiplication facts and the corresponding area models. Students are able to
relate the commutative property of multiplication to the model quickly because it represents a
fact and its related fact. For example, the area model for 6 x 7 is the same as 7 x 6 with a
different orientation. Also, familiarity with array models for multiplication facts builds number
sense as students understand that a smaller array represents a smaller product of two facts.
6 rows of 7
or
6 x 7 = 42
7 rows of 6
or
7 x 6 = 42
Background Knowledge
Students should have been introduced to area models for multiplication and understand that
the dimensions of the array represent the two factors and the area represents the product.
Task Directions
Have students follow the directions below:
1. Place the Array Cards face down in a stack.
2. For each round, each player should draw one card from the stack and, using the
commutative property, record both multiplication facts that apply to the card. (If the
array is a square, there will be only one multiplication fact for the array.)
3. At the end of each round, the player with the largest product collects the cards from
the other players.
4. Play continues until all cards have been played.
NOTE: The rules can be changed so that the player with the smallest product collects all the
cards.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 14 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Questions/Prompts for Formative Student Assessment
 How can you use your Array Card to show the commutative property for multiplication?
 How does the size of the array change as the factors get larger? Smaller?
 How are the dimensions of the array and the number of shaded squares related?
 How does an array model show repeated addition?
Questions for Teacher Reflection
 Are students dependent upon counting squares to determine the product in an array
model?
 How fluent are my students with their multiplication facts?
 Are there certain facts that give students more difficulty than others and what are they?
 How can I modify this game so that it challenges students as they learn the facts from this
game?
DIFFERENTIATION
Extension
 Make additional Array Cards that model higher levels of multiplication facts.
 Play Double Challenge where students draw two cards at a time and add the products.
 Have students use the Array Cards to explain the division facts that are related to a given
array and write the corresponding fact family for multiplication and division.
Intervention
 Make Array Cards with lower level multiplication facts, or with other math facts and
concepts that students need to review.
 Use this game in small group instruction to informally assess a student’s level of
multiplication fact mastery and to pinpoint specific areas to target instruction.
TECHNOLOGY CONNECTION
http://www.multiplication.com/ Practice games for multiplication facts as well as teacher
resource pages with instructional ideas on how to introduce multiplication.
Note: This site contains advertising.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 15 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name ___________________________________________ Date ________________________
Array Challenge
Game Directions
Array Challenge is a game for 2 – 4 players.
Materials:
One deck of Array Challenge cards
Array Challenge recording sheet
Directions:
1. Place the Array Cards face down in a stack.
2. For each round, each player should draw one card from the stack and, using the
commutative property, describe both multiplication facts that apply to the card. (If the
array is a square, there will be only one multiplication fact for the array.)
3. At the end of each round, the player with the largest product collects the cards from
the other players.
4. Play continues until all cards have been played.
NOTE: The rules can be changed so that the player with the smallest product collects all the
cards.
Record the multiplication facts for your array cards in the table on the back of this sheet.
Example: If you drew a 6 x 7 array card, two number sentences can be written.
6 rows of 7
or
6 x 7 = 42
7 rows of 6
or
7 x 6 = 42
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 16 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name
_____________________________ Date _____________________________
Array Challenge
Recording Sheet
Record the number sentences for each array card in the table below.
Round
Number Sentence
Number Sentence
Highest Product?
Example
6 x 7 = 42
7 x 6 = 42
 or 
1
2
3
4
5
6
7
8
9
10
11
12
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 17 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Shaded Array Cards
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 18 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 19 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 20 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 21 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 22 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Multiplication Chart Mastery
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply it in
problem solving.
a. Describe the relationship between addition and multiplication, i.e., multiplication is
defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
e. Apply the identity, commutative and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3N4. Students will understand the meaning of division and develop the ability to apply it
in problem solving.
a. Understand the relationship between division and multiplication and between division
and subtraction.
M3P1. 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.
M3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
M3P3. 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.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. 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.
M3P5. Students will represent mathematics in multiple ways.
d. Create and use representations to organize, record, and communicate mathematical ideas.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 23 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS



What patterns of multiplication can we
discover by studying a times table
chart?
How can we determine numbers that
are missing on a times table chart by
knowing multiplication patterns?
What role can arithmetic properties
play in helping us understand number
patterns?
MATERIALS


“Multiplication Chart Mastery”
recording sheet (2 pages)
Manipulatives, if applicable
GROUPING
Individual/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will explain and describe the patterns they find in the multiplication
chart.
Comments
As students discover and verbalize patterns in the multiplication chart, they find more
strategies with which to remember multiplication and division facts. The more familiar
students become with patterns and predicting successive numbers in patterns, the better
prepared they will be for future grade levels.
This task would work well as a math conference interview. Consider using it as an
assessment during the year, adding, deleting, or changing questions as well as parts of the
chart to uncover students’ thinking and learning. Be sure to make manipulatives available to
students who may need them.
Background Knowledge
When learning about multiplication, students need a wide variety of experiences and
opportunities to explore and discover patterns on their own. Students need a good understanding
of how to read the rows and columns on a multiplication chart and how to find products using the
chart as a tool. Students should also have an understanding of the commutative property.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 24 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Task Directions
Have students answer the questions on the “Multiplication Chart Mastery” recording sheet.
Be sure to give students an opportunity to discuss their answers with peers and the teacher.
Questions/Prompts for Formative Student Assessment
 What patterns do you notice in the 9 column?
 If you think of 8 x 4 as 8 x 2 doubled, what is the product of 8 x 4? Will this strategy
always work? How did you know?
 How could a similar strategy be used to find the products for the eights facts?
 Where are examples of the commutative property on the multiplication chart?
Questions for Teacher Reflection
 Which patterns seemed easier for students to see? Which seemed harder for them?
 Were students making natural connections to division patterns?
 Could students verbalize how the commutative property is evident on the chart?
DIFFERENTIATION
Extension
Have students fill in a multiplication chart and purposely put six wrong items. Trade
with a partner and try to be the first to identify the incorrect numbers on the chart and
make corrections.
Intervention
Have students compare the multiplication chart in this table with a completed chart. Elicit
ideas from them about ways the charts are similar and different. Help students develop
strategies for determining what numbers should go in the blank squares by looking at the
completed chart.
TECHNOLOGY CONNECTION


Rectangle Multiplication Students visualize the multiplication of two numbers as an
area.
http://nlvm.usu.edu/en/nav/vlibrary.html Rectangle Multiplication is one of the many
applets from NLVM that allows students to experiment with multiplication using
manipulatives.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 25 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________________ Date _________________________
Multiplication Chart Mastery
1. Michael filled in this chart to practice his multiplication facts. Which facts does he seem
to know best? ___________________________________________________________
How do you know? _______________________________________________________
_______________________________________________________________________
2. Michael has all his nines facts correct, even though he has not memorized them. Explain
one strategy he might have used to fill in his nines on the chart.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 26 of 100
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Third Grade Mathematics  Unit 2  1st Edition
3. Michael is missing some threes and fours facts. Fill them in for him and explain how
you would teach him to find these answers. ____________________________________
_______________________________________________________________________
_______________________________________________________________________
4. How could Michael use the fours facts to help him find the eights facts? Fill those in for
him and explain your strategy. ______________________________________________
_______________________________________________________________________
5. Michael has done a great job filling in all the numbers on the diagonal. What do you
notice about these numbers? ________________________________________________
_______________________________________________________________________
_______________________________________________________________________
6. Do you see any other patterns on the multiplication chart? Describe at least one.
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
7. Explain how the commutative property helps you fill in facts on the multiplication chart.
Give an example. ________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
_______________________________________________________________________
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 27 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARING TASK: Stuck on Multiplication
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply it in problem solving.
a. Describe the relationship between addition and multiplication, i.e., multiplication is
defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit
number.
e. Apply the identity, commutative, and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. 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.
M3P3. 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.
M3P4. 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.
M3P5. 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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 28 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS




How can we model multiplication?
How are multiplication and addition
related?
How can we write a mathematical
sentence to represent a multiplication
model we have made?
Is there more than one way of multiplying
to get the same product?
MATERIALS


12 interlocking blocks per student
“Stuck on Multiplication” recording sheet
GROUPING
Individual/Small Group Task
Comments
The grouping of this activity should not be independent or partner unless students clearly
understand how to write number sentences and how to independently follow written directions.
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will experiment with a set of 24 interlocking blocks to determine the
addition and multiplication patterns for 24.
Comments
It may be necessary to model this task with your students. This will help them understand the
steps and clearly see the connection between the concrete visualization and the number sentence.
Be sure to use this demonstration time as a way to help students make connections between the
language of mathematics and the visual as well as symbolic representations. For example, if you
demonstrate breaking the strip of 12 cubes into two pieces, you may want to explain it this way:
When I break this strip of 12 in two pieces, I can count the pieces and the number of
blocks in each piece. I can show this as an addition sentence. I can say I have a set of six and
I add another set of six to make 12. I can write this as 6 + 6 = 12. I can also show this as a
multiplication sentence. I have two pieces and 6 blocks in each piece so I can say I have two
sets of six blocks. I have one set of six blocks in one hand (hold blocks in left hand) and
another set in the other hand (hold second set in right hand). I can say that I have two sets of
six. When I write it as a number sentence I can say that I have two times. I write it this way: 2
x 6 = 12.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 29 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Next, ask students if there is another way they can break apart 12 into equal groups. The idea is
to make sure students form sets with the same number of blocks in each set. Solutions include:
one set of twelve blocks, two sets of six blocks, three sets of four blocks, four sets of three
blocks, six sets of two blocks, and twelve sets of one block.
Background Knowledge
Students should have had prior experiences using connecting cubes as counting tools. If they
have not, you may want to give them time to explore with them.
Task Directions
Students will follow the directions below from the “Stuck on Multiplication” task directions
sheet and record their work on the “Stuck on Multiplication” recording sheet.
1. Begin with 24 connecting cubes.
2. Think of a way you can make sets with the same number of blocks in each set and use
all 24 blocks. Make a model of your idea with the cubes.
3. In the chart provided, draw a diagram of your model and write an addition
number sentence that describes the model you made. Then write the related
multiplication sentence.
4. Now, show different ways to make equal sets of cubes. Be sure to use all 24
cubes each time.
5. Compare your answers with your classmates. Did everyone have the same
answers? How can you tell whose solutions are correct?
Questions/Prompts for Formative Student Assessment
 How many sets did you make from your strip of 24 blocks? Count the sets for me.
 How many blocks are in each set?
 How can you write this in a number sentence so others will understand your model?
 How can we show this as both a multiplication number sentence and an addition
number sentence?
 Do you have the same number of blocks in each set?
Questions for Teacher Reflection
 Can students explain thoroughly their thinking about making sets of numbers?
 What confusions or misconceptions do students have about the process of writing number
sentences?
 Did students use a pattern and/or their understanding of the multiplication facts to find all
possible groups or did students use trial and error?
DIFFERENTIATION
Extension
Instead of 24, students could be asked to find groups for a larger number of blocks so that
there are more possible number sentences (i.e. 36 or 48).
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 30 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Intervention
Using this task as a direct instruction strategy in small groups will provide support for
students who struggle with these concepts and will enable them to develop the ability to
describe their thinking.
TECHNOLOGY CONNECTION
http://www.funbrain.com/math/index.html One of many game sites designed to support
student understanding of multiplication. Note: This site contains advertising.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 31 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name ________________________________________ Date ___________________________
Stuck on Multiplication
Task Directions
1. Begin with 24 connecting cubes.
2. Think of a way you can make sets with the same number of blocks in each set and use all
24 blocks. Make a model of your idea with the cubes.
3. In the chart provided, draw a diagram of your model and write an addition number
sentence that describes the model you made. Then write the related multiplication
sentence.
4. Now, show different ways to make equal sets of cubes. Be sure to use all 24 cubes
each time.
5. Compare your answers with your classmates. Did everyone have the same answers?
How can you tell whose solutions are correct?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 32 of 100
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Third Grade Mathematics  Unit 2  1st Edition
Name _____________________________________ Date ______________________
Stuck on Multiplication
Recording Sheet
Diagram
Addition
Sentence
Multiplication
Sentence
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 33 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: The Magic Money Machine
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of multiplication of whole
numbers and develop the ability to apply it in problem solving.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
d. Understand the effect on the product when multiplying by multiples of 10.
e. Apply the identity, commutative and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M2A1. Students will use mathematical expressions to represent relationships between
quantities and interpret given expressions.
a. Describe and extend numeric and geometric patterns.
M3P1. Students will solve problems (using appropriate technology).
b. Solve problems that arise in mathematics and in other contexts.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
M3P3. 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.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
c. Recognize and apply mathematics in contexts outside of mathematics.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 34 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS


What patterns can be found when
multiplying numbers?
What pattern is there when we multiply by
ten or a multiple of ten? By one? By zero?
MATERIALS
“The Magic Money Machine” story and
recording sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students become involved in a story about a Magic Money Machine and use an
input-output model to determine what happens when a given quantity is doubled, tripled, and
multiplied by 10.
Comments
An excellent way to introduce this task is to read Two of Everything by Lily Toy Hong, a
Chinese folktale about doubling. Another suggestion is One Potato, Two Potato by Cynthia
DeFelice, a book about a magic pot buried in the O’Grady’s potato patch.
As students are working on the task, they may need access to coins or other manipulatives.
Background Knowledge
Students need a good understanding of the concepts of doubling and tripling. You may want
to ask students to explain these concepts in their own words and provide examples.
There will probably be many students in your class who have never seen a silver dollar, listed
in the table on the recording sheet. Be sure students understand the value of a silver dollar and, if
possible, bring one to show them.
Task Directions
Read aloud the story about the Magic Money Machine:
The Magic Money Machine Story
One sunny day Lucky Luke saw something shiny in the bushes. He discovered a
gold box with a curious slot on the top and a hole on the side. There were three
buttons on the front. One was red, one was blue, and one was yellow.
When he pressed the red button it said, “I double the money you put in.”
Wow! Double your money!
He pushed the blue button and it said, “I triple the money you put in.”
Triple!
Wondering what would happen next, he pushed the yellow button.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 35 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
It said, “The money you put in I multiply by ten.”
Luke had some change in his pocket so he decided to try out the Magic Money
Machine.
The number of coins in Luke’s pocket is listed for you in the table provided.
1. Complete the table to show what happened with each group of coins when he
pressed each button. Show all your work.
2. Make up your own Lucky Luke story problem where he puts in more than
one type of coin at a time. (Example: 2 quarters and 3 dimes)
 Write your story on a separate sheet of paper.
 Be sure to record your problem in the chart and to solve your problem.
Ask students to follow the directions below from the “Magic Money Machine” recording
sheet.
Show the amount of money Luke got from the Magic Money when he pressed each
colored button.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 36 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Questions/Prompts for Formative Student Assessment
 How will you determine how much money Luke will get when he puts nickels into the
machine? Dimes? Quarters? Silver dollars?
 Explain your strategies for finding the total amount the Magic Money Machine will
give Luke.
 Would you like to have a machine like this? Why or why not?
Questions for Teacher Reflection
 Were students able to successfully double, triple, and multiply by 10?
 Were students able to explain their strategies for determining the total amount of money
from the machine?
 Were students able to successfully write and solve the story problem at the end of the
task?
DIFFERENTIATION
Extension
 Have students use bills ($1, $5, $10, $20, $100) instead of coins.
 Have students practice using the distributive property by using combinations of coins
and bills. For example, have them double $1.37 or triple $6.94.
Intervention
 Allow students to use coins as manipulatives. Help them to clearly understand the
difference between counting actual numbers of coins and determining the total value of
the coins.
TECHNOLOGY CONNECTION
http://www.shodor.org/interactivate/activities/WholeNumberCruncher/?version=1.6.0_0
7&browser=MSIE&vendor=Sun_Microsystems_Inc Provides additional practice with
the concept of an input-output machine; enables students to discover for themselves the
patterns among numbers.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 37 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name _____________________________________________ Date ______________________
The Magic Money Machine
One sunny day Lucky Luke saw something shiny in the bushes. He discovered a
gold box with a curious slot on the top and a hole on the side. There were three buttons
on the front. One was red, one was blue, and one was yellow.
When he pressed the red button it said, “I double the money you put in.”
Wow! Double your money!
He pushed the blue button and it said, “I triple the money you put in.”
Triple!
Wondering what would happen next, he pushed the yellow button.
It said, “The money you put in, I multiply by ten.”
Luke had some change in his pocket so he decided to try out the Magic Money Machine.
The number of coins in Luke’s pocket is listed for you in the table provided.
1. Complete the table to show what happened with each group of coins when he
pressed each button. Show all your work.
2. Make up your own Lucky Luke story problem where he puts in more than one type
of coin at a time. (Example: 2 quarters and 3 dimes)
 Write your story on a separate sheet of paper.
 Be sure to record your problem in the chart and to solve your problem.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 38 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name _____________________________________________ Date ______________________
The Magic Money Machine
Show the amount of money Luke got from the Magic Money when he pressed each colored
button.
Coins in
Luke’s
Pocket
Red Button
x 2 (Double)
Blue Button
x 3 (Triple)
Yellow
Button
x 10
Total Amount
3 Nickels
2 Dimes
3 Quarters
1 Silver Dollar
1 Quarter, 2
dimes
Total Amount
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 39 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
PERFORMANCE TASK: Armadillo Stories
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply
it in problem solving.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
d. Understand the effect on the product when multiplying by multiples of 10.
e. Apply the identity, commutative and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3A1. Students will use mathematical expressions to represent relationships between
quantities and interpret given expressions.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the
unknown in a number sentence.
M3P1. 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.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
M3P3. 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.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
c. Recognize and apply mathematics in contexts outside of mathematics.
M3P5. Students will represent mathematics in multiple ways.
c. Use representations to model and interpret physical, social, and mathematical
phenomena.
ESSENTIAL QUESTIONS


What math is involved in the study of Georgia animals?
How can multiplication help us repeatedly add larger numbers?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 40 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
MATERIALS



“Armadillo Stories” recording sheet
Manipulatives, if needed
Research resources such as informational text and/or the
internet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will write and solve word problems and
their accompanying number sentences using a given data set. They
will also include symbol(s) in their number sentences.
Comments
Animals are usually highly motivating subjects for third graders to study. Be sure they note
how science and mathematics are connected as they study Georgian animals and habitats
throughout the school year.
As students solve their multiplication story problems, have them verbalize what each number
in their number sentence represents. In the example on the recording sheet, the number sentence
is 15 x 10 = 150 inches. Be sure students can explain that the 15 represents the length of the
armadillo’s tail in inches and the 10 represents the number of tails.
Background Knowledge
Students need a good understanding of the components of a number sentence, the use of a
symbol to represent what is being found, and how to translate between words and mathematical
symbols.
Task Directions
Students will follow the directions below from the “Armadillo Stories” recording sheet.
Armadillos are native Georgia animals and are they ever strange! Think about these
armadillo facts:
 Armadillos live an average of 12 to 15 years.
 An armadillo can be as long as 59 inches.
 An armadillo’s tail is about 15 inches long.
 An armadillo can jump nearly 5 feet straight into the air.
 The largest armadillos weigh 120 pounds.
 An armadillo mother has 4 identical armadillo babies every time she gives
birth.
These armadillo facts can be used to write multiplication stories.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 41 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Example:
If the tails of 10 average armadillos were placed end to end, how long
would they be?
One armadillo tail is 15 inches long.
There are 10 armadillos.
My number sentence is: 15 x 10 =
inches.
The tails of ten armadillos put together would equal 150 inches.
Example:
Four armadillos weigh 480 pounds. How much does one armadillo
weigh?
My number sentence is: 4 x
= 480 pounds
4 x 120 = 480 pounds
Each armadillo weighs 120 pounds.
Write and solve three more multiplication stories about armadillos or another
interesting Georgian animal.
Questions/Prompts for Formative Student Assessment
 What data did you use for your word problem?
 How did you decide what to include in your number sentences?
 Is there more than one correct way to write your number sentence? How do you
know?
 How did you use a symbol in your number sentence? What does it represent?
 What does each part of the multiplication sentence represent in your story?
 How does multiplication help us represent ideas about the sizes of armadillos?
Questions for Teacher Reflection
 How correct and how sophisticated are the word problems and accompanying number
sentences? Are there common errors or misconceptions?
 Are students correctly incorporating the use of symbols in their number sentences?
 Can students accurately explain what each part of their multiplication sentences
represents?
DIFFERENTIATION
Extension
 Have students discuss and make a list of the ways that measurements are used in
science. Have them construct a chart to show both the English and the metric (when
applicable) measures of length and width, time, speed, and temperature.
 Encourage students to experiment with writing two step word problems.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 42 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Intervention
 Have students model their word problems (using different numbers) on the sample
problem given or a problem that the teacher demonstrates.
 For kinesthetic learners, allow them to use math magnets or other manipulatives to set up
their math sentences on a surface that is easily manipulated prior to recording the number
sentence.
TECHNOLOGY CONNECTION
http://dromus.nhm.uga.edu/~GMNH/gawildlife/index.php A useful website for students to
use to look up additional information on animals and/or habitats in Georgia.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 43 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name ___________________________________________ Date ________________________
Armadillo Stories
Armadillos are native Georgia animals and are they ever strange!
Think about these armadillo facts:






Armadillos live an average of 12 to 15 years.
An armadillo can be as long as 59 inches.
An armadillo’s tail is about 15 inches long.
An armadillo can jump nearly 5 feet straight into the air.
The largest armadillos weigh 120 pounds.
An armadillo mother has 4 identical armadillo babies every time she
gives birth.
These armadillo facts can be used to write multiplication stories.
Example:
If the tails of 10 average armadillos were placed end to end, how long would
they be?
One armadillo tail is 15 inches long.
There are 10 armadillos.
My number sentence is: 15 x 10 = inches.
The tails of ten armadillos put together would equal 150 inches.
Example:
Four armadillos weigh 480 pounds. How much does one armadillo weigh?
My number sentence is: 4 x
= 480 pounds
4 x 120 = 480 pounds
Each armadillo weighs 120 pounds.
Write and solve three more multiplication stories about armadillos or another interesting
Georgian animal.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 44 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Change It Around!
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply it in
problem solving.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3-digit numbers by a 1- digit
number.
d. Understand the effect on the product when multiplying by multiples of 10.
e. Apply the identity, commutative and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3A1. Students will use mathematical expressions to represent relationships between
quantities and interpret given expressions.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the
unknown in a number sentence.
M3P1. 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.
M3P2. Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
M3P3. 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.
M3P4. 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.
M3P5. Students will represent mathematics in multiple ways.
b. Create and use representations to organize, record, and communicate mathematical ideas.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 45 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS


How does the order of the digits in a multiplication
problem affect the product?
How does understanding the commutative property
help us multiply?
MATERIALS
“Change It Around!” recording sheet
GROUPING
Individual/Small Group Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will write multiplication word problems that change the order of the
factors to compare and contrast the number sentences and to determine the products are the
same.
Comments
You may want to make sure students remember that number sentences are different from
expressions. An expression has no relational symbol (e.g. equal sign =, inequality sign <, >)
while number sentences have a relational symbol (e.g. an equal sign with quantities of equal
value on both sides).
Example:
Expressions
Number Sentences
2x8
Equality
9-1
2 x 8 = 16
3x8
Inequalities
4+3
2x8<3x8
9-1>4+3
3 x 8  40
Background Knowledge
Students need a good understanding of how to write number stories and of the meaning of the
commutative property of multiplication. They also need to understand that the two factors in a
multiplication problem can be read as:
□ groups, with  in each group or □ groups of .
For example, 4 x 6 = 24 can be read as “4 groups of 6 equals 24.” Or, when the factors are
changed to 6 x 4 = 24, the multiplication sentence can be read as 6 groups of 4 equals 24.”
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 46 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Task Directions
Students will follow the directions below from the “Change It Around!” recording sheet.
1. Write a story problem where 13 is the number of groups and 7 is the number of
items in each group. Include a number sentence that uses a symbol like a triangle
or a square as the unknown. Use words, numbers, and/or pictures to show how
you found the total number of items. Write your answer in a multiplication
number sentence.
2. Write a story problem where 7 is the number of groups and 13 is the number of
items in each group. Include a number sentence that uses a symbol like a triangle
or a square as the unknown. Use words, numbers, and/or pictures to show how
you found the total number of items. Write your answer in a multiplication
number sentence.
3. On the back of this sheet write about how these two stories are connected. Explain
why the stories are different but the products are the same.
Questions/Prompts for Formative Student Assessment
 Does the order in which the factors appear in a multiplication sentence change the
product?
 Can you make a picture or use a manipulative to demonstrate your word problem?
 Would breaking 13 into 10 + 3 help you solve the problem?
 Did you use words/numbers/pictures to explain your thinking?
Questions for Teacher Reflection
 Do word problems reflect a clear understanding of the difference between the multiplier
(the number of groups) and the multiplicand (the number of items in each group)?
 Can students model and explain why the order doesn’t affect the product?
DIFFERENTIATION
Extension
Have students use larger multipliers (up to 99) as the basis for their number stories.
Intervention
Give students an example with smaller numbers and illustrate or have them act it out.
Point out the connection with division.
Example:
 If three is the number of students and five is the number of dollars each
student has, the product of 3 x 5 is $15.00.
 If five is the number of students and three is the number of dollars each
student has, the product of 5 x 3 is still $15.00.
 Have students explain whether they would rather be a student in the first
example or second example. (Students in the first example have $5.00
because there are fewer students.)
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 47 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
TECHNOLOGY CONNECTION
http://www.naturalmath.com/mult/mult5.html A series of web pages describing patterns and
techniques (starting with the commutative property) that can be used to learn the basic multiplication
facts
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 48 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name _____________________________________ Date __________________
Change It Around!
13 x 7
1. Write a story problem where 13 is the number of groups and 7 is the number of items in
each group.
 Include a number sentence that uses a symbol like a triangle or a square as the
unknown.
 Use words, numbers, and/or pictures to show how you found the total number of
items.
 Write your answer in a multiplication number sentence.
2. Write a story problem where 7 is the number of groups and 13 is the number of items in
each group.
 Include a number sentence that uses a symbol like a triangle or a square as the
unknown.
 Use words, numbers, and/or pictures to show how you found the total number of
items.
 Write your answer in a multiplication number sentence.
3. On the back of this sheet write about how these two stories are connected. Explain why
the stories are different but the products are the same.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 49 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
PERFORMANCE TASK: Seating Arrangements
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply
it in problem solving.
a. Describe the relationship between addition and multiplication, i.e., multiplication is
defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3-digit numbers by a 1-digit
number.
e. Apply the identity, commutative, and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3P1. 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.
M3P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
M3P3. 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.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
ESSENTIAL QUESTIONS



How is multiplication like repeated addition?
How many different ways can you arrange 24 chairs?
How does drawing an array help us think about different ways to decompose a number?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 50 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
MATERIALS



“Seating Arrangements” recording sheet
Grid paper, if needed
Manipulatives, if needed
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will solve a word problem requiring them
to make arrays using the number 24.
Comments
You may want to provide grid paper or have students draw the arrays on plain copy paper.
Students should develop the following arrays: 1 x 24, 2 x 12, 3 x 8, 4 x 6, 24 x 1, 12 x 2, 8 x 3,
and 6 x 4. As students examine both the 4 x 6 array and the 6 x 4 array, for instance, help them
understand that while both arrays have the same area and are congruent, their orientation can
make a difference. For example, when arranging chairs in a room, the shape of the room could
dictate whether there are 6 rows of 4 chairs or 4 rows of 6 chairs.
Background Knowledge
Students should know the meaning of an array and how to write number sentences from a
pictorial or visual display.
Task Directions
Students will follow the directions below from the “Seating Arrangements” recording sheet.
Your class is going to have a special presentation and your teacher has asked you to
figure out a good way to place 24 chairs in your room for seating. There is only one
requirement. All the chairs must be placed in an array.
1. Draw pictures to show all the ways you can arrange the chairs in an array.
2. Label and write matching number sentences for each array.
3. Choose your favorite arrangement and explain why you think it would be the
best arrangement so that every student could see the presentation.
Questions/Prompts for Formative Student Assessment
 Explain how you built each array.
 With 24 blocks, can you have an array with 7 in each row? Why or why not?
 Is there a way to determine the measurements of an array for 24 without building it
with blocks or drawing a diagram?
 How many different solutions do you think there are to this problem? Is there a way
to check to see if you have found all possible solutions?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 51 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Questions for Teacher Reflection
 Where did students experience difficulty? Were they able to resolve it on their own? Is
there evidence of student understanding?
 Did students develop strategies for understanding the math, or were they completely
reliant on their diagrams? What strategies would help students transfer their
understanding to the symbolic level?
DIFFERENTIATION
Extension
 Using 24, or another appropriate number, have students multiply to find the number
of chairs needed for 2, 3, 4, 5, and 6 third grade classrooms that use twenty-four
chairs each. Ask students to develop a strategy to solve the problem. Then allow
students to share their strategies.
 Replace 24 chairs with 30, 36 or 72 for students who can work with larger numbers.
Intervention
 Replace 24 with a smaller number such as 12, 18 or 20.
 Model this task or a similar one in a small group setting.
TECHNOLOGY CONNECTION
http://illuminations.nctm.org/LessonDetail.aspx?id=U109 Numerous ideas for introducing
multiplication, including the array model.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 52 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name ________________________________________ Date ___________________________
Seating Arrangement
Your class is going to have a special presentation and your teacher
has asked you to figure out a good way to place 24 chairs in your room
for seating. There is only one requirement. All the chairs must be placed
in an array.
1. Draw pictures to show all the ways you can arrange the chairs in an array.
2. Label and write matching number sentences for each array.
3. Choose your favorite arrangement and explain why you think it would be the best
arrangement so that every student can see the presentation.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 53 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
PERFORMANCE TASK: Family Reunion
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply it in
problem solving.
a. Describe the relationship between addition and multiplication, i.e., multiplication is
defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
e. Apply the identity, commutative and associative properties of multiplication and verify the
results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3N4. Students will understand the meaning of division and develop the ability to apply it
in problem solving.
a. Understand the relationship between division and multiplication and between division and
subtraction.
f. Solve problems requiring division.
M3P1. Students will solve problems (using appropriate technology).
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
M3P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
M3P3. 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.
M3P4. 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.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
ESSENTIAL QUESTIONS


How can multiplication and division be used to solve real world problems?
How can we use patterns to solve problems?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 54 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
MATERIALS


“Family Reunion” recording sheet
Pattern Blocks
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will use models of tables to decide how
many tables must be used to seat a given number of guests.
Comments
 Emphasize the connection between multiplication and division in these tasks.
 Students should first be allowed time to experiment with their pattern blocks. They will
need time to find the correct number of each block type needed to solve the task.
 Students should represent their solutions using pictures, words, and number sentences.
 If using pentagonal tables, students need to be sure every guest can be seated. With a
remainder of one, an extra table is required so that there is sufficient seating for all of the
guests.
Background Knowledge
Students need a good understanding of how to manipulate pattern blocks in order to solve
tasks. Students should have had prior experiences with the manipulatives; they should be aware
of how to use the blocks as a tool for problem solving.
Task Directions
Students will follow the directions below from the “Family Reunion” recording sheet.
1. Help set up tables for your upcoming family reunion. Thirty-six relatives need a
place at a table to sit and enjoy their food and drinks. You may use the following
table styles:
 Square tables that seat one person to a side for a total of four people at a square
table.
 Circular bistro tables that seat exactly three people.
 Hexagonal tables that seat one person to a side for a total of 6 people.
 Rectangular tables that seat twelve people.
 Pentagonal tables that seat one person to a side for a total of five people.
2. Of which table would you need the most? Show how you figured out how many
of those tables you would need.
3. Of which table would you need the least? Show how you know.
4. Choose two types of tables and draw your method for seating all 36 relatives for
the family reunion. Write a number sentence to describe what you’ve drawn.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 55 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
5. Suppose the only tables you had were pentagonal. Explain how you would seat
all of your relatives.
Questions/Prompts for Formative Student Assessment
 What combinations of blocks have you tried so far?
 How will you know when you find the right combination?
 Do you think there is more than one right solution for this task? Why do you think
so? Do you have a way of finding out?
 How many _____ (square, circular, hexagonal, rectangular, or pentagonal) tables do
you need? How do you know?
Questions for Teacher Reflection
 What various methods did students use for solving this task?
 Did I allow students to describe and explain their methods to each other?
 How did I support students who were experiencing frustration with the task?
 How did I support students in extending their thinking?
DIFFERENTIATION
Extension
 Use square tables that seat one person to a side, but this time push the tables together
end to end and find out how many relatives can be seated. Continue adding tables
this same way until you have enough tables to seat everyone. Enter the information
in a table and describe any patterns you see. How many square tables pushed end to
end would it take?
# Tables
# People
Seated
1
4
2
6
3
8
4
10
5
12
Sketch
Number Pattern
(1 x 4) - 0 = 4
(1 x 2) + 2 = 4
(2 x 4) – 2 = 6
(2 x 2) + 2 = 6
(3 x 4) – 4 = 8
(3 x 2) + 2 = 8
(4 x 4) – 6 = 10
(4 x 2) + 2 = 10
(4 x 5) – 8 = 12
(5 x 2) + 2 = 12
Two possible number patterns
are shown.
The first is the number of seats
for the tables, minus the sides lost
when tables are pushed together.
The second pattern is the
number of seats along the top and
bottom plus the seat at each end.
.
.
.
17
36
(17 x 2) + 2 = 36
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 56 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition




Choose another pattern block shape and see if the same pattern holds as you push the
tables together.
Experiment to see if it will take more or less tables if a hole is left in the center or if all
tables touch another table on all sides except the side where the guests will sit.
Use a different number of relatives or allow students to make up additional types of
tables (octagonal, rhomboidal, triangular, or trapezoidal).
Rather than two types of tables, let students use three types that still yield seating for
36 people.
Intervention
 Use a smaller number of relatives, such as 12 or 20.
 Guided practice that simulates the task, done ahead of time, will enable students to
develop problem solving strategies, particularly if the teacher models the strategies
students are developing.
TECHNOLOGY CONNECTION

http://www.arcytech.org/java/patterns/patterns_d.shtml Allows students to work with
pattern blocks in an interactive applet and easily print their work.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 57 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name _______________________________________ Date ____________________________
Family Reunion
1. Help set up tables for your upcoming family reunion. Thirty-six
relatives need a place at a table to sit and enjoy their food and drinks.
You may use the following table styles:
Square tables that seat one person to a side
for a total of four people at a square table
Circular bistro tables that seat
exactly three people
Hexagonal tables that seat one person
to a side for a total of 6 people
Rectangular tables that seat twelve people
Pentagonal tables that seat one person to a
side for a total of five people
2. Of which table would you need the most? Show how you figured out how many of those
tables you would need.
3. Of which table would you need the least? Show how you know.
4. Choose two types of tables and draw your method for seating all 36 relatives for the
family reunion. Write a number sentence to describe what you’ve drawn.
5. Suppose the only tables you had were pentagonal ones that only seat five people per
table. Explain how you would seat all of your relatives.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 58 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Multiplication with Base-Ten Blocks
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of multiplication of
whole numbers and develop the ability to apply it in problem solving.
a. Describe the relationship between addition and multiplication, i.e.,
multiplication is defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3-digit numbers by a 1- digit
number.
d. Understand the effect on the product when multiplying by multiples of 10.
e. Apply the identity, commutative, and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3P2. Students will reason and evaluate mathematical arguments.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
M3P3. 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.
M3P4. 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.
M3P5. 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 can base-ten blocks help us understand how to multiply a two-digit number?
How does understanding the distributive property help us multiply large numbers?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
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Third Grade Mathematics  Unit 2  1st Edition
MATERIALS


Base-ten manipulatives for each student
“Multiplication with Base-Ten Blocks” recording sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will model multiplication of 2-digit
numbers using base-ten blocks to create partial products.
Comments
Students need to know more than one way to denote multiplication. The “x” may
become confusing for some students when they begin using variables, so they should also
recognize that a dot and parentheses are also symbols indicating multiplication.
Students need many experiences with arrays and base-ten blocks to be successful with
this task.
Detailed examples follow below. Two colors are used to emphasize the placement of the
base-ten blocks.
2 • 13 means there are two groups of 13. Using the base-ten blocks, ask students to
build two rows of thirteen.
Have students make one row of 13 with one rod and three units joined together.
10
+ 3
= 13
Repeat.
Place the two rows of thirteen into an array. The diagram below shows 2 x 13 as two
groups of 13 combined: two rods joined together, making two rows of ten, and six units
joined together, forming two rows of three.
Students should see how to visually group the two rods to make twenty and the two
rows of three units to make six, totaling 26.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 60 of 100
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Third Grade Mathematics  Unit 2  1st Edition
In the next example, 5(15) is five groups of fifteen. Have students build one row of
fifteen with one rod and five units joined together.
Repeat four more times until they have five rows of 15. Join them together to form an array of
five groups of fifteen.
Some students will quickly discover they can multiply the tens first, 5 x 10 = 50,
because the rods in the model are easy to see as groups of ten. Then they may see the
units as an array, 5 x 5 = 25. Finally, they can add the two partial products, 50 + 25, to
reach the total of 75.
As students practice while you
model these examples, they often
become quickly adept with this
method. After sufficient practice with
actual base 10 blocks, have them
draw and label the arrays. Some will
begin to do partial calculations in
their heads and add them to get the totals much more quickly than they would with the
traditional algorithm. This joining together of arrays clearly models the distributive property
of multiplication.
Another way to think about the array is to describe it in terms of its dimensions of length
and width. For example, the same array can be shown as follows:
The 5 and 15 are shown as
dimensions of the array, and can be
described as “5 by 15.” The area of
the array is visibly shown as 50 + 25,
or 75. This method of building arrays
using dimensions reinforces the idea
of the product shown as an area
model and the dimensions as factors
in the multiplication problem.
As students become more comfortable with this model, some will be able to move to
using basic sketches to illustrate the model shown above. Rather than using grid paper or
drawing each row, their sketches may evolve to look like the sketch shown below:
10
5
+
5 x 10 = 50
5
5 x 5 = 25
50 + 25 = 75
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 61 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Background Knowledge
Students need multiple experiences with base-ten blocks and how to represent ones, tens, and
hundreds with them. Students should also understand how to trade pieces for equal values. For
example, ten rods (of 10) can be traded for one flat (100).
Students need to have a good understanding of basic multiplication facts. They should
also understand the various ways that multiplication number sentences can be written using
an x, a dot, or parenthesis.
Task Directions
Students will follow the directions below from the “Multiplication with Base-Ten Blocks”
recording sheet.
Model each expression with a drawing of base 10 blocks. Show how you use the model
to find the product. Label the dimensions of each array. Write number sentences to help
explain your drawings.
Comments
Students need the opportunity to work with manipulatives on their own or with a partner in
order to develop the understanding of 2-digit multiplication. From the manipulatives, students
will be able to move to pictorial representations of the blocks, then more abstract representations
of the blocks (see the sketch above), and finally to abstract representation of multiplication using
numbers. It is important to remember that this progression begins with concrete representations
using manipulatives.
Questions/Prompts for Formative Student Assessment
 How did you know which pieces and how many to use for your array model?
 What partial products did you create?
 How does the arrangement of the base-ten pieces help you see partial products?
 What are the dimensions of your array?
 What product/area does your model represent?
Questions for Teacher Reflection
 Are students’ models accurate and reflective of the multiplication expressions assigned?
 Are students able to describe the dimensions and the areas of the arrays?
 Can students describe the partial products within the array?
 Are students able to regroup the partial products accurately into one whole product?
DIFFERENTIATION
Extension
 Give students a base-ten block array or a drawing of an array and have them
determine the product and its factors.
 Have students decide on a number, build it with base 10 blocks, and then trade seats
with a neighbor to determine the factors and find the product.
 Have students use an array to write/solve division problems.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 62 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Intervention
 Begin with much smaller arrays, such as 2 x 3, 3 x 4, and 2 x 6. Have students describe
the dimensions and area of each array. Then connect dimensions and area to the actual
multiplication sentence.
 Use grid paper and allow students to place the base-ten blocks onto the grid paper first
and then to count the grid squares as part of their calculations.
 If necessary, allow students to use a times table chart or other cueing device if full
mastery of the basic multiplication facts has not yet been attained.
TECHNOLOGY CONNECTION
http://nlvm.usu.edu/en/nav/frames_asid_192_g_2_t_1.html?from=category_g_2_t_1.html
Base-ten model using virtual grid paper. Click on the “Common” button to allow the use of
numbers larger than 10 and remember to keep one dimension less than 10.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 63 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________ Date _______________________
Multiplication with Base-Ten Blocks
Model each expression with a drawing of base 10 blocks. Show how you use the
model to find the product. Label the dimensions of each array. Write number
sentences to help explain your drawings.
4 x 14
12 • 7
5(15)
(13)(6)
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 64 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Array-nging Our Fact Families
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of
multiplication of whole numbers and develop the ability to apply it in
problem solving.
a. Describe the relationship between addition and multiplication, i.e. multiplication is
defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
e. Apply the identity, commutative, and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3N4. Students will understand the meaning of division and develop the ability to apply it
in problem solving.
a. Understand the relationship between division and multiplication and between division
and subtraction.
b. Recognize that division may be two situations: the first is determining how many equal
parts of a given size or amount may be taken away from the whole as in repeated
subtraction, and the second is determining the size of the parts when the whole is
separated into a given number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
e. Divide a 2 and 3-digit number by a 1-digit divisor.
M3P1. 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.
M3P3. 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.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. 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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 65 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
M3P5. 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 are multiplication and division related?
How can the same array represent both
multiplication and division?
MATERIALS



Grid paper
Colored pencils or markers
“Array-nging Our Fact Families” recording sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will make models on grid paper of arrays that show both
multiplication and division number sentences.
Comments
This task makes important connections between multiplication and division. Students will
become familiar with division as the inverse operation of multiplication as they learn that the
numbers in a multiplication sentence can also be used in a related division sentence.
An excellent way to introduce this concept of an area model for division is to give students
12 blocks that represent the total area of an array. Have them arrange the blocks in an array and
identify the dimensions of their array, noting different arrays are possible for 12. Then ask if
there is a way they can make a division sentence with the dividend represented by the total area
of the array. For example, a student may make a 4 x 3 array. The dividend (area of 12) can be
divided by 4 or 3, both factors of 12. Both dimensions are utilized, one as the divisor and the
other as the quotient.
Background Knowledge
Before working on this task, students need many experiences building arrays and
recording their attributes. Give students many opportunities to write story problems related
to the arrays they build. Students should understand the vocabulary in the task, such as “by”
when describing the dimensions of a 4 by 3 array. If possible, provide grid paper for
students to record their arrays.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 66 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Task Directions
Students will follow the directions below from the “Array-nging Our Fact Families”
recording sheet.
1. Draw the following arrays:
6 by 3
4 by 8
2 by 7
2. Use the example to complete the following for each array:
 Label the dimensions and total area.
 Write a multiplication sentence and tell the factors and the product.
 Write a division sentence and indicate the divisor, dividend, and quotient.
3. Select one of your arrays and write two story problems that can be modeled with
the array, one for multiplication and one for division.
Questions/Prompts for Formative Student Assessment
 How can you describe your array?
 How does the array show both multiplication and division?
 What does the word “by” mean in the directions (i.e. 6 by 3)?
 What is the difference between a factor and a product? With what operation would
you use these words?
 Explain the meaning of the divisor, dividend, and quotient in a division sentence?
Questions for Teacher Reflection
 Are student models accurate? What misconceptions are present and how should they be
addressed?
 Can students effectively explain how both multiplication and division are represented by
the same array?
 Are students regularly using specific math vocabulary words appropriately?
DIFFERENTIATION
Extension
 Have students build an array of their choice and have a partner describe the
dimensions and area of the array and all related vocabulary relating to both
multiplication and division.
 Have students build arrays for multiplication and division that involve larger
numbers. Limit the dimensions to a three-digit number times a one-digit number.
Intervention
 If students are not ready to transition to grid paper without the use of the base-ten blocks,
allow the use of these manipulatives to guide student work.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 67 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
TECHNOLOGY CONNECTION
http://www.eduplace.com/math/mw/background/3/08/te_3_08_overview.html Provides
background information on the relationship between multiplication and division.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 68 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________ Date ______________________
Array-nging Our Fact Families
This 3 by 5 array has a
total of 15 square units.
3 x 5 = 15.
Fifteen divided by three equals
five.
15 ÷ 3 = 5
15 is the dividend.
3 is the divisor.
5 is the quotient.
3 and 5 are factors.
15 is the product.
1. Draw the following arrays listed in the table below.
2. Following the example above, complete the following for each array:
 Label the dimensions and total area.
 Write a multiplication sentence and label the factors and the product.
 Write a division sentence and label the divisor, dividend, and quotient.
6 by 3
4 by 8
2 by 7
3. Select one of your arrays. On the back of this paper, write two story problems that can be
modeled with the array, one for multiplication and one for division.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 69 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Division Patterns
STANDARDS ADDRESSED
M3N4. Students will understand the meaning of division and
develop the ability to apply it in problem solving.
a. Understand the relationship between division and multiplication and between
division and subtraction.
b. Recognize that division may be two situations: the first is determining how many
equal parts of a given size or amount may be taken away from the whole as in
repeated subtraction, and the second is determining the size of the parts when
the whole is separated into a given number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
e. Divide a 2 and 3-digit number by a 1-digit divisor.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3A1. Students will use mathematical expressions to represent relationships between
quantities and interpret given expressions.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the
unknown in a number sentence.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
M3P2. 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.
M3P3. 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.
d. Use the language of mathematics to express mathematical ideas precisely.
M3P4. 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.
M3P5. Students will represent mathematics in multiple ways.
d. Create and use representations to organize, record, and communicate mathematical ideas.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 70 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS




How do the parts of a division problem relate to
each other?
What is the relationship between the divisor and
the quotient?
What happens to the quotient when the dividend
increases or decreases?
What do the parts of a division problem
represent?
MATERIALS
“Division Patterns” recording sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will analyze patterns in division.
Comments
You may want to demonstrate how to use the times table chart to determine the answers to
basic division problems if students have not yet learned the division facts. Memorization of
division facts is not required until fourth grade. Teaching the algorithm for long division is not
required at this point, it will be addressed later in this unit.
You may want to open or close this task by reading and discussing, the events in The
Doorbell Rang by Pat Hutchins or similar book. The Doorbell Rang is a story about dividing a
batch of cookies by a varying number of children. Focus on how the number of cookies each
child gets changes as the number of children increases.
Background Knowledge
Students need a good understanding of the terms divisor, dividend, and quotient. They should
use these words correctly orally and in writing.
Task Directions
Students will follow the directions below from the “Division Patterns” recording sheet.
There are three parts to every division problem: the dividend, the divisor,
and the quotient. Look at the division problem below to understand what these
terms mean:
28
÷
4
=
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 71 of 100
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Third Grade Mathematics  Unit 2  1st Edition
28
4
is the dividend, the total amount before we divide.
is the divisor, the number of groups we will make or the number of
items in each group.
is the quotient, the number of items in each group.
1. Complete the chart.
2. What do you notice about the dividend numbers as you go
from the top of the chart to the bottom of the chart?
3. What do you notice about the divisor numbers as you go
from the top of the chart to the bottom of the chart?
4. What do you notice about the quotient numbers as you go
from the top of the chart to the bottom of the chart?
5. Describe the pattern that shows the relationship between the
dividend, divisor, and quotient.
Questions/Prompts for Formative Student Assessment
 What is the same about all of the division problems?
 What is different about all of the division problems?
 What do you notice about the quotients of the division problems?
 Can you describe a pattern you see in this task?
Questions for Teacher Reflection
 What math reasoning skills were students able to express when explaining the patterns
they discovered?
 Are students able to make logical predictions about the relative size of dividends,
divisors, and quotients that are not on the chart?
DIFFERENTIATION
Extension
Have students experiment with keeping a different part of the division problem constant
such as the quotient or dividend and make predictions about the outcomes. Have students
record their results and describe their conclusions.
Intervention
Use base-ten manipulative pieces or grid paper as necessary for students who may need
to model each division problem.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 72 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
TECHNOLOGY CONNECTION



http://mcq.wrdsb.on.ca/Admin/Documents/WORC/PDFs/LESSON%20PrimaryMath.pdf
http://www.lessonplanspage.com/MathLAMultiplicationDivisionUsingTheDoorbell
Rang23.htm Both websites above provide teacher resources for the book The
Doorbell Rang by Pat Hutchins.
http://www.softschools.com/math/games/division_practice.jsp Division practice; the
student or teacher can determine the parameters for the divisor, dividend, and
number of problems
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 73 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name ________________________________________ Date ___________________________
Division Patterns
There are three parts to every division problem: the dividend, the divisor, and
the quotient. Look at the division problem below to understand what these
terms mean:
28
÷
4
=
28
is the dividend, the total amount before we divide.
4
is the divisor, the number of groups we will make or the number of
items in each group.
is the quotient, the number of items in each group.
1. Complete the following chart:
Dividend
Divisor
Quotient
4
4
1
8
4
12
4
16
4
20
÷
4
24
4
28
4
32
4
36
4
40
4
=
2. What do you notice about the dividend numbers as you go from the top of the chart to the
bottom of the chart?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 74 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
3. What do you notice about the divisor numbers as you go from the top of the chart to the
bottom of the chart?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
4. What do you notice about the quotient numbers as you go from the top of the chart to the
bottom of the chart?
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
5. Describe the pattern that shows the relationship between the dividend, divisor, and quotient.
___________________________________________________________________________
___________________________________________________________________________
___________________________________________________________________________
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 75 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Stuck on Division
STANDARDS ADDRESSED
M3N4. Students will understand the meaning of division and develop
the ability to apply it in problem solving.
a. Understand the relationship between division and multiplication and between division
and subtraction.
b. Recognize that division may be two situations: the first is determining how many equal
parts of a given size or amount may be taken away from the whole as in repeated
subtraction, and the second is determining the size of the parts when the whole is
separated into a given number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3P1. Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M3P2. 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.
M3P3. 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.
M3P4. 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.
M3P5. 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.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 76 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS





How can we model division?
How are multiplication and division
related?
How are subtraction and division related?
How can we write a mathematical sentence
to represent division models we have
made?
Is there more than one way to divide a
number to get the same quotient?
MATERIALS



12 connecting cubes per student
“Stuck on Division” task sheet
“Stuck on Division” recording sheet
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will experiment with a set of 12 connecting cubes to determine the
division patterns when the dividend is 12.
Comments
You may choose to open this task by reading, discussing, and modeling the events in Divide
and Ride by Stuart J. Murphy. Divide and Ride is a story about dividing a group of children to
ride amusement park rides. Another suitable book about division is One Hundred Hungry Ants
by Elinor J. Pinczes. Focus on the different ways division can be described (separating into equal
groups, repeated subtraction, and inverse of multiplication.)
The three ways of looking at division are closely related and may be difficult for students to
verbalize initially as they make connections between concrete models and their corresponding
number sentences. Therefore, students need multiple experiences using a given number of cubes
to model repeated subtraction, form equal groups, and explain how these two activities are alike
and different. They also need to understand the inverse relationship of multiplication and
division. Help students make connections to the language of mathematics and between visual and
symbolic representations.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 77 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Background Knowledge
Students should clearly understand how to write number sentences and how to follow written
directions before working independently.
One possible solution is shown below:
Task Directions
Students will follow the directions below from the “Stuck on Division” task sheet.
Use 12 connecting cubes to complete this task.
1. Begin with 12 cubes and remove the same number of cubes over and over again until
there are none left. Remember, you must remove the same number each time. Make
a model of your idea with the cubes.
2. Use the first row of the “Stuck on Division” recording sheet to
a. write about what you did
Georgia Department of Education
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Third Grade Mathematics  Unit 2  1st Edition
3.
4.
5.
6.
7.
b. draw a diagram of your model
c. write a subtraction number sentence that describes your model
Find a way to separate your cubes into equal groups. How can you show the
dividend, divisor, and quotient with your cubes?
Use the second row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write a division number sentence
Now think of a multiplication fact whose product is twelve. Can you make groups of
cubes that prove that division is the opposite of multiplication?
Use the third row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write the fact family for your diagram
Compare your answers with your friends. Did everyone have the same answers?
How can you tell whose solutions are correct?
Questions/Prompts for Formative Student Assessment
 Can you explain more than one way to think about dividing a number?
 How can you write your model in a number sentence so others will understand your
model?
 How can we show your model as both a division number sentence and a subtraction
number sentence?
Questions for Teacher Reflection
 Can students effectively explain their thinking about division?
 Are students making connections between subtraction and division?
 Are students making connections between multiplication and division?
 What confusions or misconceptions do students have about the process of writing number
sentences?
DIFFERENTIATION
Extension
Have students to complete the chart with 13 blocks. Ask students to include leftovers in
their explanations, diagrams, and number sentences.
Intervention
Direct instruction in small groups can provide support for students who struggle with
these concepts and can enable them to develop the ability to describe their thinking.
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Kathy Cox, State Superintendent of Schools
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Third Grade Mathematics  Unit 2  1st Edition
TECHNOLOGY CONNECTION



http://mcq.wrdsb.on.ca/Admin/Documents/WORC/PDFs/LESSON%20PrimaryMath.
pdf
http://www.lessonplanspage.com/MathLAMultiplicationDivisionUsingTheDoorbellR
ang23.htm Both websites above provide teacher resources for the book The Doorbell
Rang by Pat Hutchins.
http://www.stuartjmurphy.com/activities/activity_ideas.php Stuart Murphy website
with activity suggestions for Divide and Ride. (Click on level 3 and then click on the
title of the book.)
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Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 80 of 100
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Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________________ Date _________________________
Stuck on Division
Task Sheet
Use 12 connecting cubes to complete this task.
1. Begin with 12 cubes and remove the same number of cubes over and over again until
there are none left. Remember, you must remove the same number each time. Make
a model of your idea with the cubes.
2. Use the first row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your model
c. write a subtraction number sentence that describes your model
3. Find a way to separate your cubes into equal groups. How can you show the
dividend, divisor, and quotient with your cubes?
4. Use the second row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write a division number sentence
5. Now think of a multiplication fact whose product is twelve. Can you make groups of
cubes that prove that division is the opposite of multiplication?
6. Use the third row of the “Stuck on Division” recording sheet to
a. write about what you did
b. draw a diagram of your cube groups
c. write the fact family for your diagram
7. Compare your answers with your friends. Did everyone have the same answers?
How can you tell whose solutions are correct?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 81 of 100
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Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________________ Date _________________________
Stuck on Division
Recording Sheet
Division is…
Diagram
Number Sentence
Repeated subtraction
Separating a whole into equal
groups
The opposite of multiplication
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Third Grade Mathematics  Unit 2  1st Edition
PERFORMANCE TASK: Making Cents of Division
STANDARDS ADDRESSED
M3N4. Students will understand the meaning of division and develop
the ability to apply it in problem solving.
a. Understand the relationship between division and multiplication and between
division and subtraction.
b. Recognize that division may be two situations: the first is determining how many
equal parts of a given size or amount may be taken away from the whole as in
repeated subtraction, and the second is determining the size of the parts when
the whole is separated into a given number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
d. Explain the meaning of a remainder in division in different circumstances.
e. Divide a 2 and 3-digit number by a 1-digit divisor.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3P1. Students will solve problems (using appropriate technology).
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.
M3P2. Students will reason and evaluate mathematical arguments.
b. Recognize reasoning and proof as fundamental aspects of mathematics.
c. Make and investigate mathematical conjectures.
M3P3. 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.
M3P4. Students will make connections among mathematical ideas and to other disciplines.
c. Recognize and apply mathematics in contexts outside of mathematics.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 83 of 100
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Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS



How can we divide larger numbers?
What is the meaning of a remainder?
Does a remainder mean the same thing in every
division problem?
MATERIALS


“Making Cents of Division” recording sheet
Play Money
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will determine the fair amount of money each student should receive
when dividing an amount of money among groups of three, four, and nine students.
Comments
Students will have opportunities in this task to work with dividing numbers that have
remainders. There will be no remainder in the group of three students. The remainder can
be fairly divided among the group of four students.
However, in the group of nine students, each will receive $3.33 and there will be $0.03
left over. In this case, three cents cannot be divided fairly among nine children.
Note the difference in dividing $30.00 into four equal groups and dividing the number
30 by four. Four is not a factor of 30 and 30 is not evenly divisible by four. Yet, $30.00 can
be divided evenly among four children. Have students discuss why this is true.
Background Knowledge
Students may come into this task with different concepts of the meaning of a remainder.
They should know that if there is no remainder, then the number can be “evenly divided.” In this
case, be sure students avoid the misconception that the quotient must be an even number.
Task Directions
Students will follow the directions below from the “Making Cents of Division” recording
sheet.
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Third Grade Mathematics  Unit 2  1st Edition
A group of three students had a car wash to raise money
for a field trip. They received a $20.00 bill for the first car and
a $10.00 bill for the second car.
Use words, pictures, numbers, and your play money to
explain how the three students could divide the money fairly.
What if there were four students in the group that was
washing cars? What about nine students? Use the table below
to show your solutions for each group of students.
Questions/Prompts for Formative Student Assessment
 How will you determine the fair amount for each
student?
 Explain your solution using a number sentence.
 Model your solution using a drawing or play money.
 Was there any group of students that received the exact amount of money with no
remainder?
 Are there times a remainder can still be divided fairly? Explain.
Questions for Teacher Reflection
 Are student explanations of remainders valid and logical?
 What additional strategies might students need to help them solve this task?
 Do students have an understanding that dividing “fairly” can include a fair division of the
remainder as well?
DIFFERENTIATION
Extension
Extend this lesson by having students determine by which numbers 30 can be evenly
divided. After seeing which numbers are factors of thirty, have them choose other divisors
that leave a remainder. Which remainders can be evenly divided by using coins? Which ones
cannot? Have them explain the difference in the quotient and the remainder.
Examples are shown in the table below. Students will discover that $30.00 divided
among 8 children allows each of them to get $3.75 with no money left over. If there were 11
children, each would get $2.72 and there would be 8 cents left over.
Intervention
For students who have difficulty with the concept of a remainder, have them use money
to solve a word problem involving a smaller amount of money.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
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Third Grade Mathematics  Unit 2  1st Edition
TECHNOLOGY CONNECTION
http://www.thinkingblocks.com/ThinkingBlocks_MD/TB_MD_Main.html This website supports
students solving multiplication and division word problems with the use of interactive models.
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August 2009  Page 86 of 100
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Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________________ Date _________________________
Making Cents of Division
A group of three students had a car wash to raise money for a field trip.
They received a $20.00 bill for the first car and a $10.00 bill for the second car.
Use words, pictures and numbers and your play money to explain how the three students
could divide the money fairly.
What if there were four students in the group that was washing cars? What about nine
students? Use the table below to show your solutions for each group of students.
3 students
4 students
9 students
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Third Grade Mathematics  Unit 2  1st Edition
LEARNING TASK: Sharing Pumpkin Seeds
STANDARDS ADDRESSED
M3N4. Students will understand the meaning of division and develop
the ability to apply it in problem solving.
a. Understand the relationship between division and multiplication and
between division and subtraction.
b. Recognize that division may be two situations: the first is
determining how many equal parts of a given size or amount may
be taken away from the whole as in repeated subtraction, and the
second is determining the size of the parts when the whole is separated into a given
number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
d. Explain the meaning of a remainder in division in different circumstances.
e. Divide a 2 and 3-digit number by a 1-digit divisor.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3P1. 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.
M3P3. 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.
M3P4. 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.
M3P5. Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 88 of 100
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Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
ESSENTIAL QUESTIONS



How can we divide larger numbers?
What is the meaning of a remainder?
Does a remainder mean the same thing in every division problem?
MATERIALS


“Sharing Pumpkin Seeds” recording sheet
Base 10 blocks or other materials for counting available for students who wish to use
them
GROUPING
Individual/Partner Task
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this task, students will decide how to share pumpkin seeds fairly with a group of children.
Comments
This task can be paired with the following science standard: S3L1b. Identify features of
green plants that allow them to live and thrive in different regions of Georgia.
There are many children’s books about pumpkins and pumpkin seeds, any one of them
could be used as an introduction to this task. One book that deals directly with the number of
seeds in a pumpkin is How Many Seeds in a Pumpkin? by Margaret McNamara, Illustrated
by G. Brian Karas.
Background Knowledge
This task provides students with an opportunity to develop and discuss strategies for dividing
a two- or three-digit number by a one-digit number. Possible strategies students may use to solve
this type of problem include, using base 10 blocks, using their knowledge of multiplication and
inverse operations, or using repeated subtraction. Third grade is students’ first exposure to larger
number division and it is important to allow students time to make sense of this operation, so that
students will continue to be successful with division in later grades.
Task Directions
Students will solve the two sharing problems on the “Sharing Pumpkin Seeds” recording
sheet.
Problem 1
Ben and his 3 friends toasted 116 pumpkin seeds from their pumpkin. How many seeds
will each child get if they share the pumpkin seeds fairly?
Clearly explain your thinking using words, numbers, and/or pictures.
Students may approach the problem 116 ÷4 in a variety of ways. Some students may
build on their understanding of multiplication as the inverse of division to solve the problem.
Example 1
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Third Grade Mathematics  Unit 2  1st Edition
I know 4 x 25 = 100, 4 x 4 = 16, and 100 + 16 = 116. If I add 25 groups of four and 4
groups of four, I know there are a total of 29 groups of 4 in 116. Therefore, each child
will get 29 pumpkin seeds.
Other students may build on their understanding of division as repeated subtraction.
Example 2
4 x 10 = 40
116 – 40 =76
Each child got 10 pumpkin seeds.
4 x 10 = 40
76 – 40 = 36
Each child got 10 more pumpkin seeds.
4 x 9 = 36
36 – 36 = 0
Each child got 9 pumpkin seeds.
Each child received a total of 10 + 10 + 9 pumpkin seeds or 29 pumpkin
seeds.
Some students may choose to use base 10 blocks to represent the division problem.
Example 3
First I took out blocks equal to 116.
Next I traded the 100 block for 10 ten strips.
Then I started sharing the ten strips among four groups. The ten strips I had left over
I traded for unit blocks.
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MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
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Third Grade Mathematics  Unit 2  1st Edition
Finally, I shared the remaining unit blocks with the four groups. I had none left over.
Each group received a total of 29 blocks. Therefore, each child will get 29 pumpkin
seeds.
Comments
After students have had plenty of time to develop an understanding of division using a
method that makes sense to them, begin to talk with students about an efficient way to
record the various strategies they now use.
The following division examples show how the examples above could be recorded:
Example 1
Example 2
Example 3
In each example, the number of groups is recorded to the right; the total number of items is
being subtracted. (In example 1, if there are 25 groups of 4, there are a total of 100 objects.
Therefore 100 is subtracted from 116.) The total number of groups (quotient) is found and
recorded at the bottom.
An example for the second problem on the “Sharing Pumpkin Seeds”
recording sheet is shown. Here the remainder is circled and written with
the quotient.
Questions/Prompts for Formative Student Assessment
 Do you have enough pumpkin seeds for each child to get 10? 25? 50?
100?
 What is your plan to solve this problem?
 How do you know your answer is correct?
 How does this help you answer the question in the problem?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 91 of 100
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Third Grade Mathematics  Unit 2  1st Edition
Questions for Teacher Reflection
 Are students able to use their understanding of addition, multiplication, and/or
subtraction – as it relates to division – to solve these problems?
 Are students able to explain how the approach they chose to explain division can be
recorded using an algorithm (i.e. the division problems shown above)?
DIFFERENTIATION
Extension
Have students to compare strategies used to solve each problem. Encourage them to look
for similarities and differences in their approaches to the problem and to discuss the
efficiency of each. Ask students to present their findings to the class.
Intervention
Before asking students to solve the problems on the “Sharing Pumpkin Seeds” recording
sheet, be sure students have been able to solve similar problems with two-digit dividends.
TECHNOLOGY CONNECTION
http://mason.gmu.edu/~mmankus/whole/base10/asmdb10.htm#div A site for teachers and
parents provides information on using base 10 blocks to solve division problems with an area
model.
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Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 92 of 100
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Third Grade Mathematics  Unit 2  1st Edition
Name __________________________________________ Date _________________________
Sharing Pumpkin Seeds
Ben and his 3 friends toasted 116 pumpkin seeds from their pumpkin.
How many seeds will each child get if they share the pumpkin seeds fairly?
Clearly explain your thinking using words, numbers, and/or pictures.
Sarah and her 4 friends toasted 188 pumpkin seeds from their pumpkin.
How many seeds will each child get if they share the pumpkin seeds fairly?
Clearly explain your thinking using words, numbers, and/or pictures.
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Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
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Third Grade Mathematics  Unit 2  1st Edition
UNIT TWO CULMINATING TASK
PERFORMANCE TASK: ICE CREAM SCOOPS
STANDARDS ADDRESSED
M3N3. Students will further develop their understanding of multiplication of
whole numbers and develop the ability to apply it in problem solving.
a. Describe the relationship between addition and multiplication, i.e.,
multiplication is defined as repeated addition.
b. Know the multiplication facts with understanding and fluency to 10 x 10.
c. Use arrays and area models to develop understanding of the distributive property and to
determine partial products for multiplication of 2- or 3-digit numbers by a 1- digit
number.
d. Understand the effect on the product when multiplying by multiples of 10.
e. Apply the identity, commutative and associative properties of multiplication and verify
the results.
f. Use mental math and estimation strategies to multiply.
g. Solve problems requiring multiplication.
M3N4. Students will understand the meaning of division and develop the ability to apply it
in problem solving.
a. Understand the relationship between division and multiplication and between division
and subtraction.
b. Recognize that division may be two situations: the first is determining how many equal
parts of a given size or amount may be taken away from the whole as in repeated
subtraction, and the second is determining the size of the parts when the whole is
separated into a given number of equal parts as in a sharing model.
c. Recognize problem-solving situations in which division may be applied and write
corresponding mathematical expressions.
d. Explain the meaning of a remainder in division in different circumstances.
e. Divide a 2 and 3-digit number by a 1-digit divisor.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3A1. Students will use mathematical expressions to represent relationships between
quantities and interpret given expressions.
c. Use a symbol, such as □ and Δ, to represent an unknown and find the value of the
unknown in a number sentence.
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Kathy Cox, State Superintendent of Schools
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Third Grade Mathematics  Unit 2  1st Edition
M3P1. 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.
M3P2. Students will reason and evaluate mathematical arguments.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M3P3. 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.
M3P4. 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.
M3P5. 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 QUESTION
How do estimation, multiplication, and
division help us solve problems in everyday
life?
MATERIALS
“Ice Cream Scoops” recording sheet
GROUPING
Independent Task
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Third Grade Mathematics  Unit 2  1st Edition
TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION
In this culminating task, students will use multiplication and division to show different ways
they can spend $3.00 on different flavors of ice cream. In the process, they will double, triple, or
quadruple the price for a single scoop of ice cream.
Comments
You may want to introduce this task by having students think of a number, then double it,
triple it, and quadruple it. Discuss what mathematical operations students used to determine their
answers. Then discuss prices of a single scoop of ice cream and ask what the price would be if
they ordered a double-scoop, triple scoop, or double-double scoop (4 scoops). Talk about the
operations used to use to find the answers and how they can use inverse operations to be sure that
their work is correct. Students should recognize what multiplication is indicated by doubles,
triples, and quadruples.
As students work on this task, those with good mental math skills may be able to quickly
determine the prices of various ice cream scoops. Students who laboriously multiply may need
more practice with mental math skills. You may want to encourage students to use their
estimation skills while working on this task.
While this task is intended to serve as a summative assessment, it also may be used for
teaching and learning. If used as an assessment, it is important that all elements of the task be
addressed throughout the unit so that students understand what is expected of them. Also, if
using a rubric, students should be given a copy of the rubric as part of the teacher introduction of
the assessment, so that they are aware of the expected rigor and quality for their work. A sample
rubric is provided below.
Background knowledge
As students begin to work on this task, they need to understand the meaning of the
terms single, double, triple and double-double scoops of ice cream. The term “doubledouble” is another way of saying “quadruple” and you may want to ask students to explain
why this is true.
Task Directions
Have students follow the directions on the “Ice Cream Scoops” Recording Sheet.
The Super Delicious Ice Cream Shop has the very best ice cream in town.
They sell their ice cream in double scoops, triple scoops, or double-double (that’s
four) scoops. The top selling ice creams are listed on the sign below. You have
$3.00 to spend. Don’t worry about tax.
Use words, pictures, and numbers to show all your work as you answer the
questions below. Think about using estimation to help you consider your choices.
Be sure to show your estimation work.
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MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
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Third Grade Mathematics  Unit 2  1st Edition
Ice Cream Flavors and Prices for a Single Scoop
Varoom Vanilla
Cha-cha Chocolate
Cheery Cherry
Rockin’ Rocky Road
Striped Strawberry
Kid’s Delight
$0.67
$1.33
$1.04
$1.12
$0.89
$0.98
1. With $3.00, which flavor can you buy, triple Varoom Vanilla, or triple Cheery
Rockin’
Rocky
Cherry? Would
you have
anyRoad
money left? $1.12
2. To spend most of your money, should you buy a double, triple, or double-double
Stripled
Strawberry
scoop of Rockin’
Rocky
Road? How much $0.89
money would you have left?
3. Which ice cream flavor can you buy if you order a double-double scoop?
Delight
4. On a differentKid’s
day, you
and 5 of your friends$0.98
decide to combine your money. You
have $11.76 total. You all want to order the same ice cream in a double scoop.
Which flavors are you able to buy?
5. You have been saving pennies for a whole year! You have saved 425 pennies. If you
and two of your friends share the pennies fairly, how many pennies will each of you
have to buy ice cream?
Questions/Prompts for Formative Student Assessment
 How are you using estimation to help you solve this task?
 What math facts would help you solve this problem?
 Can you use an inverse operation to be sure your solution is correct?
Questions for teacher reflection
 Are students able to explain doubling, tripling and quadrupling and the multiplication
needed to determine the prices of the scoops?
 What misconceptions are present and how will I address them?
 What further opportunities will my students need to reinforce their multiplication and
division skills?
DIFFERENTIATION
Extension
Have students make up their own flavors and prices, use different amounts of money,
and write their own Ice Cream Scoops stories to share with their classmates.
Remediation
While fluency with multiplication facts is required of third graders, it is not required
that all facts will be acquired in the first marking period of the school year. You may
want to allow students to use cueing devices like a times table chart during this
performance assessment as needed.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 97 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
Name ________________________________________ Date ____________________________
Ice Cream Scoops
The Super Delicious Ice Cream Shop has the very best ice cream in town. They
sell their ice cream in double scoops, triple scoops, or double-double (that’s four)
scoops. The top selling ice creams are listed on the sign below. You have $3.00
to spend. Don’t worry about tax.
Use words, pictures, and numbers to show all your work as you answer the
questions below. Think about using estimation to help you consider your choices.
Be sure to show your estimation work.
Ice Cream Flavors and Prices for a Single Scoop
Varoom Vanilla
$0.67
Cha-cha Chocolate
$1.33
Cheery Cherry
$1.04
Rockin’ Rocky Road
$1.12
Striped Strawberry
$0.89
Kid’s Delight
$0.98
1. With $3.00, which flavor can you buy, triple Varoom Vanilla, or triple Cheery
Cherry? Would you have any money left?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 98 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
2. To spend most of your money, should you
buy a double, triple, or double-double
scoop of Rockin’ Rocky Road? How
much money would you have left?
5. On a different day, you and 5 of your
friends decide to combine your money. You
have $11.76 total. You all want to order
the same ice cream in a double scoop.
Which flavors are you able to buy?
3. Which ice cream flavor can you buy if
you order a double-double scoop?
4. You have been saving pennies for a whole
year! You have saved 425 pennies. If you
and two of your friends share the pennies
fairly, how many pennies will each of you
have to buy ice cream?
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 99 of 100
Copyright 2009 © All Rights Reserved
Georgia Performance Standards Framework
Third Grade Mathematics  Unit 2  1st Edition
3rd Grade Unit 2 Performance Assessment Rubric
Standard ↓
M3N3. Students will further
develop their understanding of
multiplication of whole numbers
and develop the ability to apply it
in problem solving.
f. Use mental math and estimation
strategies to multiply.
g. Solve problems requiring
multiplication.
Exceeding
Meeting
– Multiplication work
– Multiplication
shows use of
calculations are
diagrams, words,
correct
and/or other suitable
– Evidence of
representations for
estimation is shown
demonstrating mastery
– Evidence of estimation
is shown with
explanations
Not Yet Meeting
– Multiplication
calculations are
– incorrect or omitted
– No evidence of
estimation
– Work shows all
– Division number
– Division number
division sentences
sentence
sentence does not
correctly
corresponds to the
correspond to
– Thorough explanation
question asked in
question
of
remainders
is
given
word
problem.
–
No mention is made
c. Recognize problem-solving situations
–
Explanation
of
all
the
–
Response
indicates
of remainder
in which division may be applied and
possible
solutions
is
the
presence
or
lack
–
Solution to division
write corresponding mathematical
given
with
reasons
for
of
a
remainder
and
problem is incorrect
expressions.
which
solution
is
the
what
this
indicates
d. Explain the meaning of a remainder in
best
– Solution to division
division in different circumstances.
problem is correct
e. Divide a 2 and 3-digit number by a 1-
M3N4. Students will understand
the meaning of division and
develop the ability to apply it in
problem solving.
digit divisor.
f. Solve problems requiring division.
g. Use mental math strategies to divide.
M3P3. Students will communicate
mathematically.
b. Communicate their mathematical
thinking coherently and clearly to
peers, teachers, and others.
M3P5. Students will represent
mathematics in multiple ways.
a. Create and use representations to
organize, record, and communicate
mathematical ideas.
– Explanations are
thorough and detailed
and include reasoning
as well as multiple
representations to
support conclusions
– Explanations are
logical and use
specific math
vocabulary to
describe
multiplication or
division process
– All data relevant to the – Work shown is
solutions of both
organized and
multiplication and
logically presented
division problems are – Work shown
accurately recorded in
supports conclusions
an organized fashion
about which ice
cream to buy
– Explanations are
omitted or illogical
– Explanations do not
describe the process
used to derive an
answer to the
question asked
– Work is not shown
– Work shown is
disorganized,
inaccurate, or fails to
communicate
mathematical ideas
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
MATHEMATICS  GRADE 3  UNIT 2: MULTIPLICATION AND DIVISION OF WHOLE NUMBERS
August 2009  Page 100 of 100
Copyright 2009 © All Rights Reserved