math hot topic multiplication grades 3-5 september 20, 2012

Math Hot Topic

Multiplication

Grades 3-5September 20, 2012

Getting Started

• Schedule• Roster• Internet

Username & Password: k5math

• Folders & Handouts

Goals of this Workshop

• Understand concepts of factors and multiples in the CCSSM

• Develop ways to build multiplication fluency

• Understand concepts used in multiplying whole numbers and how these connect to alternate algorithms

www.corestandards.org

CCSS Critical Areas of Study

Instructional time should focus on:• Grade 3: Developing understanding of

multiplication and division and strategies for multiplication within 100

• Grade 4: Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends

• Grade 5: Developing understanding of the multiplication of fractions and of division of fractions in limited cases

Developing “Understanding”

• "Understand" is used in these standards to mean that students can explain the concept with mathematical reasoning, including concrete illustrations, mathematical representations, and example applications.

Presentation by James Williams, NCCTM Leadership Seminar, 10-26-11

• Students who understand a concept can use it to make sense of and explain

quantitative situations incorporate it into their own arguments

and use it to evaluate the arguments of others

bring it to bear on the solutions to problems

make connections between it and related concepts

• Applying Common Core’s definition of “Understanding” forces us to move beyond algorithms, mnemonics and well rehearsed procedures

Presentation by James Williams, NCCTM Leadership Seminar, 10-26-11

Multiplication

3 x 2 = 6• Describe this multiplication equation.• What does it mean?• Create a word problem to match the

equation.• Draw two different models to represent the

equation.

Activity 1: Factors & Multiples

Making Rectangles

• Using 12 square-inch tiles, make as many different rectangles as you can

• How many different rectangles can you make?

• Draw the rectangles on grid paper and cut them out

Making Rectangles

• Work together to make all possible rectangles for your assigned numbers

• Draw the rectangles on grid paper and cut them out

• Write the corresponding multiplication equation on the rectangle and attach the rectangles to the class chart

Making Rectangles

• Group 1: 1, 10, 18, 23• Group 2: 2, 7, 9, 11, 12• Group 3: 3, 13, 17, 24• Group 4: 4, 8, 16, 20• Group 5: 5, 15, 22, 25• Group 6: 6, 14, 19, 21

*When you finish making your rectangles, check the rectangles of the next group.

Questions to Ponder

• Speculate the possible rectangles you will create

• How can you be sure that you found all of possible the rectangles?

Rectangles

1 2 3 4 5 6 7 8 9 10 11 12 13

14 15 16 17 18 19 20 21 22 23 24 25

Chart Observations

• List three observations about the chart

Task Reflection

• How did you work on this task with your group?

• How did you ensure that you found all of the rectangles for your assigned numbers?

• Describe your observations from the chart

What Do You Think?

• Which numbers have rectangles with 2 rows? Which rectangles have a side with two squares on them? Write the numbers from smallest to largest.

• Which numbers have rectangles with 3 rows?

• Which numbers have rectangles with 4 rows?

Multiples

• Defined operationally• Products of a given number• Numbers you say when you skip count

by that number

Multiples

• Multiples of 44, 8, 12, 16, 20, 24, 28…

• Multiples of 88, 16, 24, 32, 40, 48, 56…

How are the multiples of 4 and 8 related? Why?

Would 68 be a multiple of 2? How do you know?

Explain the importance of multiples.

Are They Square?

• Which numbers have rectangles that are squares?

• What would be the next number after 25 that would create a square? Explain how you know

Square Numbers

• A number multiplied by itself(1x1=1, 2x2=4, 3x3=9, 4x4=16…)

• Array forms a square

What Are Your Ideas?

• Which numbers have only one rectangle? List them from smallest to largest

• What is the smallest number that has two different rectangles? Three different rectangles? Four different rectangles?

Prime and Composite Numbers

• Prime A number with exactly two factors (itself

and one)

• Composite A number that has more than two

distinct factors

Number Sort

• Sort the following numbers into three categories

• Numbers may fit into multiple categoriesPrime Numbers

Composite Numbers

Square Numbers

24 8 49 17 33 2 12 1 25 4

9 36 27 16 3 11 21 40 29

Think about . . .

• Is 1 a prime or composite number? Neither!

A prime number has exactly two factors (itself & 1). The number 1 has only one factor (1 x 1).

Composite: Cannot be written as a product of 2 distinct factors

• What about the number 0? Neither!

0 has an infinite number of divisors/factors Cannot be written as a product of 2 distinct factors

(neither of which is itself)

Factorize

http://illuminations.nctm.org/

Factorize Game: Questions to Explore

Consider these questions as you play the game: • Why do you think the length and width of the

rectangles represent the factors of your numbers? • Which number has the most factorizations? Which

has the fewest? Why do you think this is? • What kinds of numbers have only one factorization?

What do the rectangles for these factorizations have in common?

• If you double a number, what happens to the number of factorizations Do you notice a pattern?

Foundations with MultiplicationSecond Grade

Operations & Algebraic Thinking (2.OA.3 & 2.OA.4)Work with equal groups of objects to gain

foundations for multiplication.3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Foundations with MultiplicationThird Grade

Operations and Algebraic Thinking (3.OA)Represent and solve problems involving

multiplication and division.• Interpret products of whole numbers• Use multiplication within 100 to solve word

problems in situations involving equal groups, arrays, and measurement quantities

• Determine the unknown in a multiplication equation

Operations and Algebraic Thinking (3.OA)Understand properties of multiplication and the

relationship between multiplication and division.

• Apply properties of operations (commutative, associative, distributive)

Multiply and Divide within 100• Fluently multiply within 100 and know all

products of two one-digit numbers from memory

Number and Operations in Base Ten (3.NBT)

Use place value understanding and properties of operations to perform multi-digit arithmetic.

• Multiply one-digit numbers by multiples of 10 in the range 10-90 using strategies based on place value and properties

Measurement and Data (3.MD)Geometric measurement: understand

concepts of area and relate area to multiplication and to addition.

• Relate area to the operation of multiplication• Multiply side lengths to find areas of rectangles• Represent products as rectangular areas• Use area models to represent the distributive

property

34Standards for Mathematical Content

4. Operations and Algebraic Thinking (4.OA.4)Gain familiarity with factors and multiples.

4. Find all factor pairs for a whole number in the range of 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range of 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range of 1-100 is prime or composite.

Investigations Correlation

Third Grade: Unit 5 Equal Groups

Investigation 3 Arrays Session 3.2 (square & prime numbers)

Fourth Grade: Unit 1 Factors, Multiples, and Arrays

Investigation 1 Representing Multiplication with Arrays• Session 1.3 (square, prime, & composite numbers)

Investigation 3 Finding Factors

Fifth Grade: Unit 1 Number Puzzles and Multiple Towers

Investigation 1 Finding Factors and Prime Factors• Session 1.2 (square, prime, & composite numbers)

Activity 2: Games

Games to Reinforce Multiplication Fluency

Activity 2: The Factor Game

Directions:• Each player uses a different colored chip• Player A selects a number and places a chip on it• Player B places chips on all of the

proper factors of the selected number.

Proper factors - all the factors of the number, except the number itself. The proper factors for 12 are 1, 2, 3, 4, and 6.

• Player B selects a new number and Player A places a chip on the factors that are not already covered

1 2 3 4 5

6 7 8 910

Directions:• Alternate between players until there are no

factors left for the remaining numbers• If a player selects a number that has no

factors left, the player loses a turn and does not get the points for the number selected

• To determine the winner, add the numbers that are covered. The player with the greater total is the winner

1 2 3 4 56 7 8 9 10

11 12 13 14 1516 17 18 19 2021 22 23 24 2526 27 28 29 30

1 2 3 4 56 7 8 9 10

11 12 13 14 1516 17 18 19 2021 22 23 24 2526 27 28 29 30

Score:Player

Player B 21

Activity 2: The Product Game

Directions:• The object of the game is to get 4 in a row• To begin, Player 1 puts a paper clip on a number

in the factor list of numbers 1-9 along the bottom of the game board

• Player 2 then puts the other paper clip on any number in the factor list. The product of the two marked numbers is determined, and that product is covered with a blue marker

• Player 1 moves either one of the paper clips on the factor list to another number, and the new product is covered with a red marker

1 2 3 4 5 6 7 8 9

1 2 3 4 5 67 8 9 10 12 14

15 16 18 20 21 2425 27 28 30 32 3536 40 42 45 48 4954 56 63 64 72 81

Directions:• Players take turns moving paper clips and

marking each product with a red or blue marker, depending on which player made the product

• If a product is already covered, the player does not get a square for that turn

• Play continues until one player wins (getting four squares in a row vertically, horizontally or diagonally), or until all squares have been colored

1 2 3 4 5 6 7 8 9

1 2 3 4 5 67 8 9 10 12 14

15 16 18 20 21 2425 27 28 30 32 3536 40 42 45 48 4954 56 63 64 72 81

1 2 3 4 5 6 7 8 9

1 2 3 4 5 67 8 9 10 12 14

15 16 18 20 21 2425 27 28 30 32 3536 40 42 45 48 4954 56 63 64 72 81

Activity 2: Pathways

Directions:• Player 1 places a paperclip on two numbers

and marks the product with her color marker• Player 2 moves one paperclip to another

number and marks the product with a marker

• Winner: First player to complete a continuous pathway in the same color across the board (a pathway may include boxes that share a common side or corner

81 54 63 36 7228 18 32 81 2448 64 21 16 5612 9 42 49 27

Activity 2: Pathways

3 4 5 6 7 8 9

81 54 63 36 7228 18 32 81 2448 64 21 16 5612 9 42 49 27

Activity 2: Target 300

Directions:• Object: Get closest to 300 after six rolls • Player 1 rolls the die and decides to multiply

the number by 10, 20, 30, 40, or 50• Record the multiplication sentence• Player 2 follows the same steps• Add each amount to keep a running total• At the end of six turns, compare scores to

see whose total is closest to 300

Activity 2: Target 300

Player 1 Player 22 x 20 = 40 3 x 20 = 60

• The Factor Game• The Product Game• Pathways• Target 300

• Zang!

The Factor Game

The Product Game

Debriefing the Games

• What content are students learning as they play the games? What skills are being reinforced?

• What strategies are students using to play the game?

• Describe actions of teachers that can facilitate learning during games

• Explain the mathematical value of using games during mathematics instruction

The Value of Games

• Develop concepts and practice skills• Provide engaging opportunities for

students to deepen their understanding of numbers and operations

• Encourage strategic mathematical thinking• Provide repeated practice• Cooperation & collaboration with others• Assessment tool

Activity Reflection

• How is this approach to learning about factors and multiples different from the traditional approach?

Multiplication Instruction

“Too often children’s multiplication instruction focused heavily on learning the rules and procedures for performing multiplication calculations at the expense of learning underlying concepts”

~A Collection of Math Lessons by Marilyn Burns (p. 71)

Activity 3: Problem Solving

The Locker Problemand

15 Factors

Problem Solving

In a certain school there are 25 lockers lining a long hallway. All are closed. Suppose 25 students walk down the hall, in file, and the first student opens every locker. The second student comes behind the first and closes every second locker, beginning with locker no. 2. The third student changes the position of every third locker; if it is open, this student closes it; if it is closed, this third student opens it. The fourth student changes the position of every fourth locker, and so on, until the 25th student changes only the position of the 25th locker.

Problem Solving

• After this procession, which lockers are open? Why are they open?

• At the end of the procession, how many times did locker 9 get changed?

• How many times did locker 24 get changed?

• What is different about locker 1?

Problem Solving

• What is the smallest number that has 15 factors?

Activity 4: Mastery of Multiplication

Understanding Multiplication and Developing Multiplication

Fluency

Mastery of Multiplication

Article: Teaching for Mastery of Multiplication by Ann Wallace & Susan Gurganus

• The Case for Requiring Multiplication-Fact Mastery

• Why Do Some Children Fail to Learn the Facts?• How Should Multiplication Facts Be Taught?• How Can We Reach the Hard to Teach?

66Sequence of Multiplication Instruction

• Introduce the concepts through problem situations and linking new concepts to prior knowledge

• Provide concrete experiences and representations prior to symbolic notations

• Teach strategies explicitly• Provide mixed practice

Real World Contexts

What other contexts can you think of?

Multiplication

5 x 12 or 12 x 5?Justify your answer.

Types of Multiplication Problems

• Equal-sized Groups• Area and Array• Multiplicative Comparison• Counting Interpretation

Combination (ordered pair)

4.1 Interpretations of Multiplication

Equal-Group Interpretation

Jean has 3 tomato plants. There are 6 tomatoes in each plant. How many tomatoes are there all together?

4.1 Interpretations of Multiplication

3 × 6 = 18 tomatoes

Array and Area Interpretation

A baker has a fudge pan that measures 3 inches on one side and 6 inches on the other side. If the fudge is cut into square pieces with 1 inch on the side, how many pieces of fudge does the pan hold?

3 × 6 = 18 pieces of fudge

• Rectangular arrangement of things into (horizontal) rows and (vertical) columns

4 x 5 or 5 x 4

What items naturally come in arrays?

73Multiplicative Comparison Interpretation

“n times as many” or “n times as much”

The kangaroo in the zoo is 6 feet tall. The giraffe is 3 times as tall as the kangaroo. How tall is the giraffe?

Giraffe’s height

Kangaroo’s height

3 × 6 = 18 feet

Counting Interpretation

The ice cream store has 3 types of cones and 6 flavors of ice cream. How many different desserts of one cone with one scoop of ice cream can they make?

3 × 6 = 18 kinds of one scoop cones

75Ordered Pair Problems(Combination

Problems)• Determine how many ordered pairs (or combination)

of things can be made

A $5 lunch combo at Ann’s Sandwich Shop includes a sandwich and a bag of chips. Customers may select from 3 different types of chips (potato chips, corn chips, tortilla chips) with each of the 4 types of sandwiches (ham, turkey, roast beef, cheese) that it sells.

ham sandwich & potato chips

Aligning with the StandardsThird Grade

Operations and Algebraic Thinking (3.OA.3)Represent and solve problems involving

multiplication and division.3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Aligning with the StandardsFourth Grade

Operations and Algebraic Thinking (4.OA.1)Use the four operations with whole

numbers to solve problems.1. Interpret a multiplication equation as a comparison, e.g. interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

Operations and Algebraic Thinking (4.OA.2)Use the four operations with whole numbers

to solve problems.

2. Multiply or divide to solve word problems involving multiplicative comparison, e.g. by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

CAUTION

Avoid the Key Word Strategy!

Key Words

• Adrian had 7 marbles in her marble bag. After school, she found some more marbles that she had misplaced and put them in her bag. When she counted all of her marbles there were 15. How many extra marbles did Adrian find and put in her bag?

• Maggie had a large collection of stuffed animals. She gave away 6 of her favorite animals to her little sister, Grace. Maggie still has 15 stuffed animals in her collection. How many did she have before she gave the animals to Grace?

• There are 21 girls in a class. There are 3 times as many girls as boys. How many boys are in the class?

Key Words

• Are misleading• All problems do not have key words • Does not promote reasoning, sense making,

or perseverance

https://mathreasoninginventory.com/Home/AssessmentsOverview

Aligning with the StandardsThird Grade

Operations and Algebraic Thinking (3.OA.7)Multiply and divide within 100.

7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Multiplication Facts

• Fluency is essential to succeed in mathematics Fluency includes efficiency, accuracy, and flexibility

• NOT merely about rote memorization• Children begin fluency with facts by skip

counting and repeated addition• Important to teach relationships among the

Multiplication Facts

• Arrange facts into clusters based on thinking strategies

• Lessons should revolve around special collection of facts to help students see relationships among combinations

• Practice flexible and useful strategies• Drill facts when an efficient strategy is

in place

What Do the Experts Say?

• Children who struggle to commit basic facts to memory often believe that there are “hundreds” to be memorized because they have little or no understanding of the relationships among them.

Fosnot & Dolk, 2001 Young Mathematicians At Work: Constructing Number Sense, Addition & Subtraction

87Multiplication Strategies

• Using the Commutative Property3 x 8 = 8 x 3

• Learning the x2 Combinations8 x 2 = 8 + 8

• Double a Combination You Know2 x 6 = 124 x 6 = (2 x 6) + (2 x 6)4 x 6 = 12 + 124 x 6 = 24

8 x 6 = 24 + 24 Practice: 8 x 6 = 48 8 x 4 = ?

7 x 6 = ?

• Take Half4 x 10 = 404 x 5 = 20

8 x 10 = 808 x 5 = 40

24 x 10 = 24024 x 5 = 120

• Use Combinations that You Know Build Up

7 x 5 = 357 x 6 = (7 x 5) + (1 x 7)7 x 6 = 42

Build Down

10 x 7 = 70 70 – 7 = 63 9 x 7 = 63

• Learning the x12 Combinations6 x 12 = (6 x 10) + (6 x 2)6 x 12 = 60 + 126 x 12 = 72

• Learning the x9 Combinations9 x 8 = ?10 x 8 = 8080 – 8 = 72

Fluency Builders

• Games• Simple Drills

(dice, flash cards, playing cards, fly swatters, beach ball, dominoes, number fans)

• Songs and Videos• Flash Cards at the Door• What’s Your Number

• Timed tests do not measure children’s understanding. An instructional emphasis on memorizing

does not guarantee the needed attention to understanding.

Doesn’t ensure that students will be able to use the facts in problem-solving situations.

Conveys to children that memorizing is the way to mathematical power, rather than learning to think and reason to figure out answers.

Marilyn Buns, 2000, About Teaching Mathematics

• Memorization by drill is NOT faster Kamii (1994)

One class focused on relationships One class focused on drill sheets and

flashcards• 76% correct answers-relationships• 55% correct answers-drill sheets

Fosnot & Dolk, 2001 Young Mathematicians At Work: Constructing Number Sense, Addition & Subtraction

• Timed Tests: Cannot promote reasoned approaches

to fact mastery Will produce few long-lasting results Reward few Punish many Should generally be avoided

Van de Walle, 2006, Teaching Student-Centered Mathematics

• Teachers who use timed tests believe that the tests help children learn basic facts. This makes NO instructional sense. Children who perform well under time

pressure display their skills. Children who have difficulty with skills, or

who work more slowly, run the risk of reinforcing wrong learning under pressure.

Children can become fearful and negative toward their math learning.

Marilyn Buns, 2000, About Teaching Mathematics

Activity 6: Multiplication

Making Sense of Multiplication Algorithms

Fourth Grade: Critical Area

• Developing understanding and fluency with multi-digit multiplication Depending on the numbers and the context,

they select and accurately apply appropriate methods to estimate or mentally calculate products. They develop fluency with efficient procedures for multiplying whole numbers; understand and explain why the procedures work based on place value and properties of operations; and use them to solve problems.

Fourth Grade: Critical Area

• Developing understanding and fluency with multi-digit multiplication Students apply their understanding of models

of multiplication (equal-sized groups, arrays, area models), place value, and properties of operations, in particular the distributive property, as they develop, discuss, and use efficient, accurate, and generalizable methods to compute products of multi-digit whole numbers.

Number & Operations in Base Ten (4.NBT.5)Use place value understanding and properties

of operations to perform multi-digit arithmetic.5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculations by using equations, rectangular arrays, and/or area models.

Exploring Multiplication Strategies

Solve 28 × 45 three different ways using strategies you might expect students to use

Sorting Multiplication Strategies

Analyze the student work samples in order to find commonalities and differences in the multiplication strategies. Sort the student work into groups. Problems:

99 x 5 = ?27 x 8 = ?25 x 14 = ?63 x 38 = ?

Analyzing Multiplication Strategies

• Analyze the student work samplesHow did students solve the

multiplication problems?What common strategies did you

discover?Describe students’ understandings and

misconceptions

Development Of Multiplication

• Direct modeling strategies

• Complete number strategies

• Decomposition number strategies

Direct Modeling Strategies

• Counting

• Skip Counting

• Coordinated Skip Counting

17 x 6

Complete-Number Strategies

• Approach numbers as units of units

• Repeated addition

• Frequent use of doubling

Complete-Number Strategies

14 x 25 5 groups of 14 = 70 (5 x 14 = 70)5 groups of 70 = 350 70 + 70 + 70 + 70 + 70 = 350

Decomposition Strategies

• Decompose numbers in ways that reflect an understanding of base-ten concepts and/or equal-sized groups

• Use friendly numbers and overlap mental math strategies

• Employ operation properties including the distributive property

What place value understandings are demonstrated by these examples?

If there are 17 bags of M&Ms with 70 in each bag… I can do 20 times 70 instead. That’s 1400. I need to take 210 away because I went over by three 70s.

17 x 70 20 x 70 = 14001400 – 210 = 1,190

3 x 48 = 6 x 246 x 24 = 144

25 x 12 = 50 x 650 x 6 = 300

Commutative Property & Associative Property

2 × 73 × 5 6 × 9002 × 73 × 5= 73 × 2 × 5= 73 × 10= 730

6 × 900= 6 × (9 × 100)= (6 × 9) × 100= 54 × 100= 5400

Distributive Property

For all real numbers A, B, and C:A × (B + C) = (A × B) + (A × C)

12 x 12 =12 × (10 + 2)=

(12 × 10) + (12 × 2) =120 + 24=

Blue 3 × 5 = 15Yellow 3 × 2 = 6Total 15 + 6 = 21

Rectangles and Multiplication

118Using the Distributive Property

How can the distributive property of multiplication be used to make the following calculation easier to do mentally?

7 × 12

4.4 Distributive Property

= 7 × (10 + 2)= (7 × 10) + (7 × 2)= 70 + 14= 84

15 x 6 = ?

Blue: 10 × 6 = 60Yellow: 5 × 6 = 30 Total 60 + 30 = 90

120Representing the Distributive Property

An array/area interpretation to show 10 x 13 10 × (10 + 3) = (10 × 10) + (10 × 3)

= 100 + 30 = 130

Open Array Model

5 300 15

Partial Products:

300 + 15 315

63 x 5

Representing the Distributive Property

An array/area interpretation to show 12 x 13 12 × 13 = (10 + 2) × (10 + 3)

= (10 × 10) + (10 × 3) + (2 × 10) + (2 × 3)

Open Array Model

5060 x 50 =

300050 x 3 =

760 x 7 =

4207 x 3 =

Partial Products:

3000 150 420 + 21 3591

63 x 57 = ?

Illuminations Lesson

Multiply and Conquerhttp://illuminations.nctm.org/

Modeling Strategies

• Direct Modeling

• Complete Number

• Decomposition

In groups, solve 29 × 12 using each of the three strategies

How can teachers facilitate the use of such strategies in the classroom?

Aligning with the StandardsFifth Grade

Number & Operations in Base Ten (5.NBT.5)

Perform operations with multi-digit whole numbers and with decimals to hundredths.5. Fluently multiply multi-digit whole numbers using the standard algorithm.

Open Array Model

50 3000 150

7 420 21

How does this model

connectto the

standard algorithm?

63 x 57 = ?

Open Array Model

50 3000 150

7 420 21

63X 57 44131503591

63 x 57 = ?

My Kids Can: Making Math Accessible to All Learners

by TERC

Fifth Grade ~ Solving Multiplication Problems

Thinking About Instruction

• What is the value of teaching alternative algorithms for multiplication?

• What are the challenges to using multiple methods?

• How can we support other teachers in learning and teaching different multiplication strategies?

• How can we help parents understand and value alternative approaches to computation?

In Closing…

• Teacher Resources• Literature Books• Book Choice• Evaluations

math hot topic multiplication grades 3-5 september 20, 2012

different rectangles

possible rectangles

multiplication of fractions

multidigit multiplication

james williams

related concepts

ncctm leadership seminar

division of fractions

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