Transcript Document

Mathematical Reasoning:
The Solution to Learning the
Basic Facts
Gail Moriarty
San Diego State University
San Diego City Schools
CMC-SS
What are the
Multiplication Basic Facts?
• All combinations of single digit factors
(0 - 9)
• How many multiplication basic facts are
there?
Three-Step Approach
to Learning Basic Facts
• Understand the Concept of multiplication
• Learn and use Thinking Strategies
• Memorize facts by using a variety of daily
Practice Strategies
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
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Games
Computer software
Flash cards
And more . . .
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
class as a whole is.
• Focus on what students do know through a
lesson that analyzes the multiplication chart.
• Have students keep a self-assessment chart,
shading in the facts they know.
Thinking Strategies
Using Properties
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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”
• CA Standard 3.2.6 NS:
“Understand the special
properties of zero and one in
multiplication.”
• 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”
• CA Standard 3.2.6 NS:
“Understand the special
properties of zero and one in
multiplication.”
• Facts remaining:
81 - 17 = 64
Twos
• The skip counting
strategy helps
students find the
multiples of two.
• 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
 One less than second
factor, then subtract from
9
• Finger strategy
• Facts remaining:
36 - 11 = 25
Squares
• 9 square numbers (plus
0)
• Only one factor to
remember
• Can use associations/
connections:
Sea Squares
• 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.
• CA Standard 3.1.5 AF: “Recognize and
use the commutative and associative
properties of multiplication.”
The Commutative Property
Cuts the Job in Half!
•Only 20 facts left that
can’t be “reasoned to” 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!
4x3
6x3
7x3
8x3
6x4
7x4
8x4
7x6
8x6
8x7
Distributive Property
• “Break-apart” strategy: you can separate a
multiplication problem into two parts. For
example, you can break up the first factor
(number of groups or rows) into two 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.
• CA Standard 5.2.3AF: “Know and use the
distributive property in equations and
expressions with variables.”
Distributive Property
• Break up the first factor
(number of groups or
rows) into two parts.
• You can think, “6 rows of 7 is the
same as 5 rows of 7 and
1 more row of 7.”
• 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”
• 6 x 4= (5 x 4) + (1 x 4)
• 6 x 7= (5 x 7) + (1 x 7)
• 6 x 8 = (5 x 8) + (1 x 8)
These are 4 of the 10 tough facts!
More Distributive Strategies
• Use the “Facts of Five” to find Fours:
• 4 x 6 = (5 x 6) - (1 x 6)
• You can think“4 groups of 6 = 5 groups of
6 minus 1 group of 6”.
• 4 x 7 = (5 x 7) - (1 x 7)
• 4 x 8 = (5 x 8) - (1 x 8)
Three more of the tough facts!
Breaking Apart the Sevens
Use the “Facts of Five” to find Sevens:
• 7 x 3 = (5 x 3) + (2 x 3)
You can think “7 x 3 means 5 groups of 3
and 2 more groups of 3”
• 7 x 4 = (5 x 4) + (2 x 4)
• 7 x 6 = (5 x 6) + (2 x 6)
• 7 x 8 = (5 x 8) + (2 x 8)
• CA MR1.2 Determine when and how to break a problem
into simpler parts.
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.”
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8 x 3 = (4 x 3) x 2
4 x 8 = (2 x 8) x 2
6 x 8 = (3 x 8) x 2
8 x 7 = (4 x 7) x 2
• This strategy is based on the Associative Property.
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.
The NCTM Standards
“Through skip counting, using area models,
and relating unknown combinations to
known ones, students will learn and
become fluent with unfamiliar
combinations. For example, 3 x 4 is the
same as 4 x 3; 6 x 5 is 5 more than 5 x 5; 6
x 8 is double 3 x 8.”
(NCTM Principles and Standards, p. 152)
Practice Strategies
• Games
 Examples:
 Circles and Stars
 The Array Game
 24 Game
• Computer software
• Flash cards
• What are your most effective practice
strategies?
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.
Reasoning Put to Use
Closing Comments
• Timed tests don’t teach!
• Link with division
 Fact families as a concept, not just a
procedure
• Linking reasoning with learning basic facts
accomplishes many objectives at once!
References and Resources
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M. Burns (1991). Math by All Means: Multiplication Grade 3. New Rochelle,
NY: Cuisenaire.
L. Childs & L. Choate (1998). Nimble with Numbers (grades 1-2, 2-3, 3-4, 45, 5-6, 6-7). Palo Alto: Dale Seymour.
J. Hulme (1991). Sea Squares. New York: Hyperion.
L. Leutzinger (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
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