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- 1. ELEV8AmeriCorps Seminar November 16, 2012
- 2. What are theMultiplication Basic Facts?All combinations of single digit factors (0 - 9)How many multiplication basic facts are there?
- 3. C-R-A SequenceCONCRETE-REPRESENTATIONAL-ABSTRACTEnables students to understand the concepts of mathematics prior to memorizing facts, algorithms, and operations.
- 4. CONCRETEStudents use three-dimensional objects to solve computational problems.EXAMPLE: 5 times 2After successfully solving multiplication problems at the concrete level, the student proceeds to the representational level.
- 5. REPRESENTATIONALTwo-dimensional drawings are used to solve computational problems.EXAMPLE: 7 times 3 √√√ √√√ √√√ √√√ √√√ √√√ √√√
- 6. 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.
- 7. What Does It Mean to Understand theConcept of Multiplication?Equal groups 3 bags of 5 cookiesArray/area 3 rows with 5 seats in each rowCombinations Outfits made from 3 shirts and 5 pairs of pantsMultiplicative comparison Mike ate 5 cookies. Steve ate 3 times as many cookies as Mike did.
- 8. Thinking StrategiesScaffold to support memorizationInclude properties Zero, One, Commutative, DistributiveInclude patterns and strategies Fives, Nines Skip counting
- 9. Practice StrategiesGamesComputer softwareFlash cardsAnd more . . .
- 10. Assess What FactsStudents 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.
- 11. Thinking StrategiesUsing PropertiesZero PropertyMultiplicative Identity (One)Commutative PropertyDistributive 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” strategyDefinition 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 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! 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 8Use 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 theDistributive 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 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.
- 26. The Common Core Standards“Through skip counting, using area models, andrelating unknown combinations to known ones,students will learn and become fluent with unfamiliarcombinations. 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 StrategiesGames Examples: Circles and Stars The Array Game 24 GameComputer softwareFlash cardsWhat are your most effective practice strategies?
- 28. 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.
- 29. 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!
- 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 InternationalContact us: nbezuk@mail.sdsu.edu moriarty@mail.sdsu.edu

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