Mathematical reasoning

ELEV8AmeriCorps SeminarNovember 16, 2012

What are the Multiplication Basic Facts?All combinations of single digit factors

(0 - 9)How many multiplication basic facts are

there?

C-R-A SequenceCONCRETE-REPRESENTATIONAL-

ABSTRACT

Enables students to understand the concepts of mathematics prior to memorizing facts, algorithms, and operations.

CONCRETEStudents 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.

REPRESENTATIONALTwo-dimensional drawings are used to solve

computational problems.EXAMPLE: 7 times 3 √√√ √√√ √√√ √√√ √√√ √√√ √√√

ABSTRACTThe 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.

What Does It Mean to Understand the Concept of Multiplication?Equal groups

3 bags of 5 cookies

Array/area3 rows with 5 seats in each row

CombinationsOutfits made from 3 shirts and 5 pairs of pants

Multiplicative comparisonMike ate 5 cookies. Steve ate 3 times as many

cookies as Mike did.

Thinking StrategiesScaffold to support memorizationInclude properties

Zero, One, Commutative, DistributiveInclude patterns and strategies

Fives, NinesSkip counting

Practice StrategiesGamesComputer softwareFlash cardsAnd more . . .

Assess What Facts Students KnowGive 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 PropertiesZero PropertyMultiplicative Identity (One)Commutative PropertyDistributive 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.”


Twos

The skip counting strategy helps students find the multiples of two.


Fives

The skip counting strategy also helps students find the multiples of five.

Help students realize what they already know.


Nines

Patterns in Nines factsSum of digits in

productPatterns in ones and

tens place of productOne less than second

factor, then subtract from 9

Finger strategyFacts remaining:

36 - 11 = 25

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 PropertyCuts 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! 4 x 36 x 3 6 x 47 x 3 7 x 4 7 x 68 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. 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 and1 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 DoublingIf 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.

The CA Reasoning Standards1.1 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 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)

Practice StrategiesGames

Examples: Circles and Stars The Array Game 24 Game

Computer softwareFlash cardsWhat are your most effective practice

strategies?

The Array GameMaterials: Grid paper, Colored pencils, DiceObject: 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.

Closing CommentsTimed tests don’t teach!Link with division

Fact families as a concept, not just a procedureLinking reasoning with learning basic facts

accomplishes many objectives at once!

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

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Mathematical reasoning

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