Scientific Reasoning: The Solution to Learning the Basic Facts

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´╗┐Numerical Reasoning: The Solution to Learning the Basic Facts Gail Moriarty San Diego State University CMC-N December 6, 2003

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What are the Multiplication Basic Facts? All mixes of single digit components (0 - 9) what number increase essential certainties are there?

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Three-Step Approach to Learning Basic Facts Understand the Concept of duplication Learn and utilize Thinking Strategies Memorize actualities by utilizing an assortment of day by day Practice Strategies

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What Does It Mean to Understand the Concept of Multiplication? Measure up to gatherings 3 sacks of 5 treats Array/territory 3 columns with 5 situates in every line Combinations Outfits produced using 3 shirts and 5 sets of jeans Multiplicative correlation Mike ate 5 treats. Steve ate 3 times the same number of treats as Mike did.

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Why Thinking Strategies? To achieve ALL understudies Efficiency Long term versus fleeting objectives Understanding requires thinking, not simply remembrance

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Thinking Strategies Scaffold to bolster retention Include properties Zero, One, Commutative, Distributive Include examples and systems Fives, Nines Skip numbering

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Practice Strategies Games Computer programming Flash cards And more . . . Is practice enough?

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Assess What Facts Students Know Give understudies a page of fundamental truths issues "Simply do the ones that are simple for you" Examine the outcomes to get a feeling of where the class in general seems to be. Concentrate on what understudies do know through a lesson that investigates the duplication graph. Have understudies keep a self-appraisal graph, shading in the actualities they know.

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

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Zeros Zero Property: Multiplying any number by zero is equivalent to zero. "0 gatherings of __" or "__ gatherings of 0" CA Standard 3.2.6 NS: "Comprehend the exceptional properties of zero and one in augmentation." Facts staying: 100 - 19 = 81

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Ones Identity Element: Multiplying any number by one is equivalent to that number. "1 gatherings of __" or "__ gatherings of 1" CA Standard 3.2.6 NS: "Comprehend the unique properties of zero and one in increase." Facts remaining: 81 - 17 = 64

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Twos The skip tallying methodology helps understudies discover the products of two. Expansion copies Facts remaining: 64 - 15 = 49

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Fives The skip numbering technique likewise helps understudies discover the products of five. Help understudies acknowledge what they definitely know. Actualities remaining: 49 - 13 = 36

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Nines Patterns in Nines certainties Sum of digits in item Patterns in tens place of item One not as much as second component, then subtract from 9 Finger methodology Facts remaining: 36 - 11 = 25

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Squares 9 square numbers (in addition to 0) Only one variable to recollect Can utilize affiliations/associations: Sea Squares Facts staying: 25-5=20

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Commutative Property "Pivot" technique Definition of Commutative Property: numbers can be increased in any request and get a similar result. CA Standard 3.1.5 AF : "Perceive and utilize the commutative and affiliated properties of increase."

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The Commutative Property Cuts the Job in Half! Just 20 truths left that can't be "contemplated to" by utilizing 0's, 1's, 2's, 5's, 9's and Squares. In the wake of "driving" or "pivoting" the elements, just 10 intense certainties remain! 4 x 3 6 x 3 6 x 4 7 x 3 7 x 4 7 x 6 8 x 3 8 x 4 8 x 6 8 x 7

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Distributive Property "Break-separated" methodology: you can isolate an augmentation issue into two sections. Case: Break up the main component (number of gatherings or lines) into two sections. 7 x 8 = (5 x 8) + (2 x 8) 7 gatherings of 8 = 5 gatherings of 8 in addition to 2 gatherings of 8 Use known actualities to get to obscure certainties . CA Standard 5.2.3AF : "Know and utilize the distributive property in conditions and expressions with factors."

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Distributive Property Break up the principal figure (number of gatherings or columns) into two sections. You can think, " 6 lines of 7 is the same as 5 lines of 7 and 1 more column of 7." 6 x 7 = (5 x 7)+(1 x 7)

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Thinking Strategies Based on the Distributive Property Use the "Truths of Five" to discover Sixes: 6 x 3= (5 x 3) + (1 x 3) You can think, " 6 x 3 implies 5 gatherings of 3 and 1 more gathering of 3 " Use the "Actualities of Five" fo discover Fours: 4 x 6 = (5 x 6) - (1 x 6) Use "Certainties of Five" to discover Sevens: 7 x 3 = (5 x 3) + (2 x 3) CA MR1.2 Determine when and how to break an issue into more straightforward parts.

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Halving then Doubling If one variable is even, soften it up half, duplicate it, then twofold it: 4 x 3 = (2 x 3) x 2 You can think " To discover 4 gatherings of 3, discover 2 gatherings of 3 and twofold 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 This system depends on the Associative Property. See Greg Tang's The Best of Times

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Practice Strategies Flash cards Computer programming Games: The Array Game

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The Array Game Materials: Grid paper, Colored pencils, Dice Object: Fill the network with clusters created by moving shakers. Score by including the items. Multi-level: Adjust the tenets for producing variables and how the network is to be filled to build unpredictability.

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The Array Game Level One Object: Be first to fill your own board Materials: 2 "Diversion Boards" (matrix paper), 1 bite the dust Factors: Factor one - number on bite the dust Factor two - constrained decision (1-6), (0-9) Label, say, and delicately shade every exhibit with your own particular shading

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The Array Game Level Two Object: Capture the biggest zone by making clusters, biggest aggregate of items wins. Materials: One matrix paper amusement board for two understudies to share Factors: Factor one: # on one of the dice (decision) Factor two: aggregate or distinction of # on dice Ex: 4, 6 - (4 x 2), (4 x 10), (6 x 2) or (6 x 10)

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The Array Game Level Three Object: Make neighboring exhibits , score is total of results of adjoining arrays.(Several sets of nearby permitted) Materials: Same as Level Two (One network, two dice, shaded pencils) Factors: Same as Level Two (One decision, one entirety or contrast)

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The NCTM Standards "Through skip checking, utilizing range models, and relating obscure blends to known ones, understudies will learn and get to be familiar with new mixes. For instance, 3 x 4 is the same as 4 x 3; 6 x 5 will be 5 more than 5 x 5; 6 x 8 is twofold 3 x 8." (NCTM Principles and Standards, p. 152)

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The CA Reasoning Standards 1.1 Analyze issues by recognizing connections , recognizing important from immaterial data, sequencing and organizing data, and watching designs . 1.2 Determine when and how to break an issue into less difficult parts . 2.2 Apply techniques and results from less difficult issues to more mind boggling issues.

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Reasoning Put to Use

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Closing Comments Timed tests don't instruct! Interface with division Fact families as an idea, not only a technique Linking prevailing upon learning fundamental realities finishes numerous targets without a moment's delay!

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References and Resources M. Smolders (1991). Math definitely: Multiplication Grade 3 . New Rochelle, NY: Cuisenaire. L. Childs & L. Choate (1998). Deft with Numbers (grades 1-2, 2-3, 3-4, 4-5, 5-6, 6-7). Palo Alto: Dale Seymour. J. Hulme (1991). Ocean Squares . New York: Hyperion. L. Leutzinger (1999). Actualities 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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Contact Information Professional Development Collaborative Website at SDSU: Handout accessible on site