MATHEMATICAL REASONING: THE

SOLUTION TO LEARNING THE BASIC MATH

MULTIPLICATION FACTS

Adapted from a presentation by: Sharon MooreSan 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:100 - 19 = 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 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.

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)

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