Mathematical Reasoning: The Solution to Learning the Basic Facts

1. Mathematical Reasoning: The Solution to Learning the Basic Facts Gail Moriarty San Diego State University CMC-N December 6, 2003

2. What are the Multiplication Basic Facts? • All combinations of single digit factors (0 - 9) • How many multiplication basic facts are there?

3. Three-Step Approach to Learning Basic Facts • Understand the Concept of multiplication • Learn and use Thinking Strategies • Memorize facts by using a variety of daily Practice Strategies

4. 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.

5. Why Thinking Strategies? • To reach ALL students • Efficiency • Long term vs. short term goals • Understanding requires reasoning, not just memorization

6. Thinking Strategies • Scaffold to support memorization • Include properties • Zero, One, Commutative, Distributive • Include patterns and strategies • Fives, Nines • Skip counting

7. Practice Strategies • Games • Computer software • Flash cards • And more . . . • Is practice enough?

8. 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.

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

10. 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

11. 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

12. Twos • The skip counting strategy helps students find the multiples of two. • Addition doubles • Facts remaining: 64 - 15 = 49

13. 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

14. 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

15. Squares • 9 square numbers (plus 0) • Only one factor to remember • Can use associations/ connections: Sea Squares • Facts remaining: 25-5=20

16. 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.”

17. 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 3 6 x 36 x 4 7 x 37 x 47 x 6 8 x 38 x 48 x 68 x 7

18. Distributive Property • “Break-apart” strategy: you can separate a multiplication problem into two parts. • Example: 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.”

19. 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)

20. 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” • Use the “Facts of Five” fo find Fours: 4 x 6 = (5 x 6) - (1 x 6) • Use “Facts of Five” to find Sevens: 7 x 3 = (5 x 3) + (2 x 3) • CA MR1.2Determine when and how to break a problem into simpler parts.

21. 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 This strategy is based on the Associative Property. See Greg Tang’s The Best of Times

22. Practice Strategies • Flash cards • Computer software • Games: • The Array Game

23. 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.

24. The Array GameLevel One • Object: Be first to fill your own board • Materials: 2 “Game Boards” (grid paper), 1 die • Factors: • Factor one - number on die • Factor two - limited choice (1-6), (0-9) • Label, say, and lightly shade each array with your own color

25. The Array GameLevel Two • Object: Capture the largest area by making arrays, largest sum of products wins. • Materials: One grid paper game board for two students to share • Factors: • Factor one: # on one of the dice (choice) • Factor two: sum or difference of # on dice • Ex: 4, 6 - (4 x 2), (4 x 10), (6 x 2) or (6 x 10)

26. The Array GameLevel Three • Object: Make adjacent arrays, score is sum of products of adjacent arrays.(Several sets of adjacent allowed) • Materials: Same as Level Two (One grid, two dice, colored pencils) • Factors: Same as Level Two (One choice, one sum or difference)

27. The NCTM 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.”(NCTM Principles and Standards, p. 152)

28. 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.

29. Reasoning Put to Use

30. 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!

31. 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

32. Contact Information moriarty@mail.sdsu.edu Professional Development Collaborative Website at SDSU: • http://pdc.sdsu.edu Handout available on website