The document discusses teaching multiplication basic facts through a concrete-representational-abstract (C-R-A) sequence, emphasizing understanding concepts before memorization. It outlines various strategies, properties, and games for effectively teaching and practicing multiplication, highlighting the importance of recognizing patterns and relationships between numbers. Additionally, it presents assessment methods to gauge student knowledge and suggests linking multiplication with division and fact families.

1. ELEV8
AmeriCorps Seminar
November 16, 2012

2. What are the
Multiplication Basic Facts?
All combinations of single digit factors
(0 - 9)
How many multiplication basic facts are there?

3. C-R-A Sequence
CONCRETE-REPRESENTATIONAL-ABSTRACT
Enables students to understand the concepts of
mathematics prior to memorizing facts, algorithms,
and operations.

4. CONCRETE
Students use three-dimensional objects to solve
computational problems.
EXAMPLE: 5 times 2
After successfully solving multiplication problems at
the concrete level, the student proceeds to the
representational level.

5. REPRESENTATIONAL
Two-dimensional drawings are used to solve
computational problems.
EXAMPLE: 7 times 3
√√√
√√√
√√√
√√√
√√√
√√√
√√√

6. ABSTRACT
The student looks at the computation problem and
tries to solve it without using objects or drawings.
The student reads the problem, remembers the
answer, or thinks of a way to compute the answer,
and writes the answer.
No objects or drawings are used unless the student is
unable to solve the problem.

7. 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.

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

9. Practice Strategies
Games
Computer software
Flash cards
And more . . .

10. 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.

11. Thinking Strategies
Using Properties
Zero Property
Multiplicative Identity (One)
Commutative Property
Distributive Property

12. 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

13. 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

14. Twos
The skip counting
strategy helps students
find the multiples of
two.
Facts remaining:
64 - 15 = 49

15. 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

16. 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

17. 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.”

18. 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 6x4
7x3 7x4 7x6
8x3 8x4 8x6 8x7

19. 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.”

20. 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)

21. 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!

22. 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!

23. 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.

24. 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
8 x 7 = (4 x 7) x 2
This strategy is based on the Associative Property.

25. 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.

26. The Common Core 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.”
(Common Core Principles and Standards)

27. Practice Strategies
Games
Examples:
 Circles and Stars
 The Array Game
 24 Game
Computer software
Flash cards
What are your most effective practice strategies?

28. 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.

29. 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!

30. References and Resources
 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, 4-5, 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
Contact us: nbezuk@mail.sdsu.edu moriarty@mail.sdsu.edu