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Math Hot Topic. Multiplication Grades 3-5 September 20, 2012. Schedule Roster Internet Username & Password: k5math Folders & Handouts. Getting Started. Understand concepts of factors and multiples in the CCSSM Develop ways to build multiplication fluency - PowerPoint PPT PresentationTRANSCRIPT
Math Hot Topic
Multiplication
Grades 3-5September 20, 2012
2
Getting Started
• Schedule• Roster• Internet
Username & Password: k5math
• Folders & Handouts
3
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
5
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
6
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
7
Developing “Understanding”
• 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
8
Developing “Understanding”
• 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
9
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
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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
12
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
13
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.
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Questions to Ponder
• Speculate the possible rectangles you will create
• How can you be sure that you found all of possible the rectangles?
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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
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Chart Observations
• List three observations about the chart
17
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
18
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?
19
Multiples
• Defined operationally• Products of a given number• Numbers you say when you skip count
by that number
20
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.
21
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
22
Square Numbers
• A number multiplied by itself(1x1=1, 2x2=4, 3x3=9, 4x4=16…)
• Array forms a square
23
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?
24
Prime and Composite Numbers
• Prime A number with exactly two factors (itself
and one)
• Composite A number that has more than two
distinct factors
25
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
26
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)
28
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?
29
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.
30
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
31
Foundations with MultiplicationThird Grade
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
32
Foundations with MultiplicationThird Grade
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
33
Foundations with MultiplicationThird Grade
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.
35
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
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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
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
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Activity 2: The Factor Game
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
Activity 2: The Factor Game
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
Activity 2: The Factor Game
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
A18
Player B 21
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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
Activity 2: The Product Game
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
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Activity 2: The Product Game
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
Activity 2: The Product Game
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: The Product Game
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
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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
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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
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Activity 2: Target 300
Player 1 Player 22 x 20 = 40 3 x 20 = 60
50
Games
• The Factor Game• The Product Game• Pathways• Target 300
• Zang!
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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
57
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
58
Activity Reflection
• How is this approach to learning about factors and multiples different from the traditional approach?
59
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
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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.
62
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?
63
Problem Solving
• What is the smallest number that has 15 factors?
Activity 4: Mastery of Multiplication
Understanding Multiplication and Developing Multiplication
Fluency
65
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
67
Real World Contexts
What other contexts can you think of?
68
Multiplication
5 x 12 or 12 x 5?Justify your answer.
69
Types of Multiplication Problems
• Equal-sized Groups• Area and Array• Multiplicative Comparison• Counting Interpretation
Combination (ordered pair)
4.1 Interpretations of Multiplication
70
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
71
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
72
Array
• 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
6 66
6
3 × 6 = 18 feet
74
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
C1F
1
C1F
2
C1F
3
C1F
4
C1F
5
C1F
6
C2F
1
C2F
2
C2F
3
C2F
4
C2F
5
C2F
6
C3F
1
C3F
2
C3F
3
C3F
4
C3F
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C3F
6
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
76
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.
77
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.
78
Aligning with the StandardsFourth Grade
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.
79
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CAUTION
Avoid the Key Word Strategy!
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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?
82
Key Words
• Are misleading• All problems do not have key words • Does not promote reasoning, sense making,
or perseverance
https://mathreasoninginventory.com/Home/AssessmentsOverview
83
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.
84
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
facts
85
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
86
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
88Multiplication Strategies
• 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 = ?
89Multiplication Strategies
• Take Half4 x 10 = 404 x 5 = 20
8 x 10 = 808 x 5 = 40
24 x 10 = 24024 x 5 = 120
90Multiplication Strategies
• 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
91Multiplication Strategies
• 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
92
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
93
What Do the Experts Say?
• 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
94
What Do the Experts Say?
• 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
95
What Do the Experts Say?
• 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
96
What Do the Experts Say?
• 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
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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.
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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.
100
Aligning with the StandardsFourth Grade
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.
101
Exploring Multiplication Strategies
Solve 28 × 45 three different ways using strategies you might expect students to use
102
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 = ?
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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
104
Development Of Multiplication
• Direct modeling strategies
• Complete number strategies
• Decomposition number strategies
105
Direct Modeling Strategies
• Counting
• Skip Counting
• Coordinated Skip Counting
17 x 6
106
Complete-Number Strategies
• Approach numbers as units of units
• Repeated addition
• Frequent use of doubling
107
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
1414
1414
14
1414
1414
14
1414
1414
14
1414
1414
14
1414
1414
14
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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
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Decomposition Strategies
110
Decomposition Strategies
What place value understandings are demonstrated by these examples?
111
Decomposition Strategies
112
Decomposition Strategies
113
Decomposition Strategies
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
114
Decomposition Strategies
3 x 48 = 6 x 246 x 24 = 144
25 x 12 = 50 x 650 x 6 = 300
115
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
116
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=
144
117
3
7
Blue 3 × 5 = 15Yellow 3 × 2 = 6Total 15 + 6 = 21
Rectangles and Multiplication
25
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
7
10 2
119
15 x 6 = ?
Blue: 10 × 6 = 60Yellow: 5 × 6 = 30 Total 60 + 30 = 90
10
6
5
120Representing the Distributive Property
An array/area interpretation to show 10 x 13 10 × (10 + 3) = (10 × 10) + (10 × 3)
= 100 + 30 = 130
10
10 3
121
Open Array Model
60 3
5 300 15
Partial Products:
300 + 15 315
63 x 5
122
Representing the Distributive Property
100
20 6
30
13
10
3
10
2
12
An array/area interpretation to show 12 x 13 12 × 13 = (10 + 2) × (10 + 3)
= (10 × 10) + (10 × 3) + (2 × 10) + (2 × 3)
123
Open Array Model
60 3
5060 x 50 =
300050 x 3 =
150
760 x 7 =
4207 x 3 =
21
Partial Products:
3000 150 420 + 21 3591
63 x 57 = ?
124
Illuminations Lesson
Multiply and Conquerhttp://illuminations.nctm.org/
125
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?
126
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.
127
Open Array Model
60 3
50 3000 150
7 420 21
How does this model
connectto the
standard algorithm?
63 x 57 = ?
128
Open Array Model
60 3
50 3000 150
7 420 21
63X 57 44131503591
3150
441
63 x 57 = ?
129
Video
My Kids Can: Making Math Accessible to All Learners
by TERC
Fifth Grade ~ Solving Multiplication Problems
130
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?
131
In Closing…
• Teacher Resources• Literature Books• Book Choice• Evaluations