MATHEMATICAL REASONING: THE SOLUTION TO LEARNING THE BASIC MATH MULTIPLICATION FACTS Adapted from a presentation by: Sharon Moore San Diego State University

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

Why Thinking Strategies?  To reach all students  Efficiency  Long term vs. short term goals  Understanding requires reasoning, not just memorization

What are the Multiplication Basic Facts?  All combinations of single digit factors (0 – 9)  How many multiplication facts are there?

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.

Thinking Strategies  Scaffold to support memorization  Include properties  Zero, One, Commutative, Distributive  Include patterns and strategies.  Fives, Nines  Skip counting

Practice Strategies  Games  Computer software  Flash cards  And more….  Is practice enough?

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.

Thinking Strategies Using Properties  Zero Property  Multiplicative Identity (One)  Commutative Property  Distributive Property

Zeros  Zero Property:  Multiplying any number by zero is equal to zero.  “0 groups of __” or “__groups of 0”  Facts remaining: = 81

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

Twos  The skip counting strategy helps students find the multiples of two.  Addition doubles  Facts remaining: 64 – 15 = 49

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

Nines  Patterns in Nines facts  Sum of digits in product  Patterns in ones and tens place of product  Facts Remaining: 36 – 11 = 25

Squares  9 square numbers (plus 0)  Only one factor to remember  Can use associations/ connections  Facts remaining: 25 – 5 = 20

Commutative Property  “Turn around” strategy  Definition of Commutative Property: numbers can be multiplied in any order and get the same result.

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

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

1 x 7 6 x 7 7 x 5 6 X 7 = ( 5 x 7) = ( 1 x 7 )

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)

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

Arrays: Models for Developing Multiplication Fact Strategies 3 x 6 = 18 6 x 3 = 18

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.

The Array Game Level 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.

The Array Game Level 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)

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.

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