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.
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
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.
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
Editor's Notes
81 basic multiplication facts
Do we want to put the word reasoning on this slide?
Maybe not include flash cards….maybe something about “ keep facts fresh” (Facts That Last); and an idea about not practicing facts unless they can be “reasoned to”
Maybe this slide comes before the one with the graphic?
Write a number sentence for this array Write 2 number sentences, one fo reach array What do you notice about the number sentence for the first array and the 2 number sentences for the other arrays? Use counters or grid paper
In the interest of slide economy, maybe we should only mention strategies for the 10 tough facts?