mathematical reasoning: the solution to learning the basic math multiplication facts adapted from a...
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MATHEMATICAL REASONING: THE
SOLUTION TO LEARNING THE BASIC MATH
MULTIPLICATION FACTS
Adapted from a presentation by: Sharon MooreSan Diego State University
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Three-Step Approach to Learning Basic Multiplication Facts Understand the Concept of Multiplication
Learn and use Thinking Strategies
Memorize facts by using a variety of daily Practice Strategies
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Why Thinking Strategies?
To reach all students Efficiency Long term vs. short term goals
Understanding requires reasoning, not just memorization
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What are the Multiplication Basic Facts?
All combinations of single digit factors (0 – 9)
How many multiplication facts are there?
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What Does It Mean to Understand the Concept of Multiplication?
Equal groups– 3 bags of 5 cookies
Array/area– 3 rows with 5 seats in each row
Combinations– Outfits made from 3 shirts and 5 pairs
of pants Multiplicative Comparison
– Mike ate 5 cookies. Steve ate 3 times as many cookies as Mike did.
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Thinking Strategies
Scaffold to support memorization
Include properties Zero, One, Commutative, Distributive
Include patterns and strategies. Fives, Nines Skip counting
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Practice Strategies
Games Computer software Flash cards And more….
Is practice enough?
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Assess What Facts Students Know
Give students a page of basic facts problems “Just do the ones that are easy for you.”
Examine the results to get a sense of where the students are.
Focus on what students do know through a lesson that analyzes the multiplication
chart. Have students keep a self-assessment
chart, shading in the fact they know.
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Thinking Strategies Using Properties
Zero Property
Multiplicative Identity (One)
Commutative Property
Distributive Property
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Zeros
Zero Property: Multiplying any number
by zero is equal to zero.
“0 groups of __” or “__groups of 0”
Facts remaining:100 - 19 = 81
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Ones
Identity Element: Multiplying any number by one is equal to that number.
“1 groups of__” or “__ groups of 1”
Facts remaining:81 – 17 = 64
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Twos
The skip counting strategy helps students find the multiples of two.
Addition doubles Facts remaining: 64 – 15 = 49
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Fives
The skip counting strategy also helps students find the multiples of five.
Help students realize what they already know.
Facts remaining: 49 – 13 = 36
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Nines
Patterns in Nines facts
Sum of digits in product
Patterns in ones and tens place of product
Facts Remaining: 36 – 11 = 25
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Squares
9 square numbers (plus 0)
Only one factor to remember
Can use associations/connections
Facts remaining: 25 – 5 = 20
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Commutative Property
“Turn around” strategy
Definition of Commutative Property: numbers can be multiplied in any order and get the same result.
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The Commutative Property Cuts the Job in Half!
Only 20 fact left that can’t be reasoned by using 0’s, 1’s, 2’s, 5’s, 9’s and squares.
After “commuting” or “turning around” the factors, only 10 tough facts remain!
4 x 3 6 x 3 6 x 4 7 x 3 7 x 4 7 x 6 8 x 3 8 x 4 8 x 6 8 x 7
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Distributive Property
“Break-apart” strategy: you can separate a multiplication problem into two parts.
Example: Break up the first factor (number of groups or rows) into parts.
7 x 8 = (5 x 8) + (2 x 8) 7 groups of 8 = 5 groups of 8
plus 2 groups of 8 Use known facts to get to unknown facts
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1 x 7
6 x 7
7 x 5
6 X 7 = ( 5 x 7) = ( 1 x 7 )
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Thinking Strategies Based on the Distributive Property Use the “Facts of Five” to find Sixes:
6 x 3 = (5 x 3) + (1 x 3) You can think, “6 x 3 means 5 groups
of 3 and 1 more group of 3” Find Fours:
4 x 6 = (5 x 6) - (1 x 6) Find Sevens:
7 x 3= (5 x 3) + (2 x 3)
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Halving then Doubling
If one factor is even, break it in half, multiply it, then double it:
• 4 x 3 = (2 x 3) x 2
You can think “To find 4 groups of 3, find 2 groups of 3 and double it.”
• 8 x 3 = (4 x 3) x 2• 4 x 8 = (2 x 8) x 2• 6 x 8 = (3 x 8) x 2
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Arrays: Models for Developing Multiplication Fact Strategies
3 x 6 = 18
6 x 3 = 18
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The Array Game
Materials: Grid paper, colored pencils, dice
Object: Fill the grid with arrays generated by rolling dice. Score by adding the products.
Multi-level: Adjust the rules for generating factors and how the grid is to be filled to increase complexity.
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The Array GameLevel One Object: Be first to fill your own board Materials: 2 “Game Boards” (grid
paper), 1 die Factors:
Factor one – number on die Factor two – limited choice (1-6), (0-
9) Label, say, and lightly shade each
array with your own color.
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The Array GameLevel Two
Object: Capture the largest area by making arrays, largest sum of products
wins. Materials: One grid paper game board for
two students to share. Factors:
Factor one: # on one of the dice (choice) Factor two: sum or difference of # on dice Ex: 4, 6 could be (4x2), (4x10), (6x2),
(6x10)
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The CA Reasoning Standards
1.1 Analyze problems by identifying relationships, distinguishing relevant from irrelevant information, sequencing and prioritizing information, and observing patterns.
1.2 Determine when and how to break a problem into simpler parts.
2.2 Apply strategies and results from simpler problems to more complex problems.
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References and Resources
M. Burns (1991) Math by all Means: Multiplication Grade 3. New Rochelle, NY: Cuisenaire.
L. Childs & Choate (1998) Nimble with Numbers (grades 1-2, 2-3, 3-4, 5-6, 6-7). Palo Alto: Dale Seymour.
J. Hulme (1991) . Sea Squares: New York: Hyperion. L, Keytzubger (1999), Facts that Last. Chicago:
Creative Publications Tang, G. (2002) The Best of Times. New York:
Scholastic Publications. Wickett & Burns (2001) Lessons for Extending
Multiplication. Sausalito, CA Math Solutions Publications.
24 Game: Suntex International