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 San Diego City Schools CMC-SS November 8, 2002

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

6. Practice Strategies • Games • Computer software • Flash cards • And more . . .

7. 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.

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

9. 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

10. 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

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

12. 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

13. 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

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

15. 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.”

16. 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

17. 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.”

18. 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)

19. 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!

20. 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!

21. 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.

22. 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.

23. 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.

24. 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)

25. Practice Strategies • Games • Examples: • Circles and Stars • The Array Game • 24 Game • Computer software • Flash cards • What are your most effective practice strategies?

26. 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.

27. Reasoning Put to Use

28. 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!

29. 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: nbezuk@mail.sdsu.edu moriarty@mail.sdsu.edu