mathematical reasoning
DESCRIPTION
Basics of Multiplication Facts. How to break down by sequences that kids need to learn for mathematical success.TRANSCRIPT
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ELEV8AmeriCorps SeminarNovember 16, 2012
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What are the Multiplication Basic Facts?All combinations of single digit factors
(0 - 9)How many multiplication basic facts are
there?
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C-R-A SequenceCONCRETE-REPRESENTATIONAL-
ABSTRACT
Enables students to understand the concepts of mathematics prior to memorizing facts, algorithms, and operations.
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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.
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REPRESENTATIONALTwo-dimensional drawings are used to solve
computational problems.EXAMPLE: 7 times 3 √√√ √√√ √√√ √√√ √√√ √√√ √√√
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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.
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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.
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Thinking StrategiesScaffold to support memorizationInclude properties
Zero, One, Commutative, DistributiveInclude patterns and strategies
Fives, NinesSkip counting
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Practice StrategiesGamesComputer softwareFlash cardsAnd more . . .
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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.
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Thinking Strategies Using PropertiesZero PropertyMultiplicative Identity (One)Commutative PropertyDistributive Property
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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
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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
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Twos
The skip counting strategy helps students find the multiples of two.
Facts remaining: 64 - 15 = 49
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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
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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
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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.”
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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
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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.”
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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)
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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!
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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!
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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.
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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.
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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.
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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)
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Practice StrategiesGames
Examples: Circles and Stars The Array Game 24 Game
Computer softwareFlash cardsWhat are your most effective practice
strategies?
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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.
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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!
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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: [email protected] [email protected]