Fairfield Public Schools Elementary School Mathematics A Guide to Multiplication & Division

# Fairfield Public Schools Elementary School Mathematics A Guide to Multiplication & Division

## Fairfield Public Schools Elementary School Mathematics A Guide to Multiplication & Division

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1. Fairfield Public SchoolsElementary School MathematicsA Guide toMultiplication & Division Fairfield Public Schools 2011-2012

2. “Our experience, discussions, and review of the literature have convinced us that school mathematics demands substantial change” – - Adding it Up National Research Council Center for Education Fairfield Public Schools 2011-2012

3. Mathematics in the 21st Century It is no longer sufficient to just learn arithmetic. Today’s students need to be mathematical thinkers and problem solvers. Fairfield Public Schools 2011-2012

4. Consider this… How does the standard algorithm work? The following is a sampling of different ways of thinking about how to solve basic computation problems. Fairfield Public Schools 2011-2012

5. How would you mentally calculate? 4 x 39 = ? Think about how you would solve this problem before you continue. Fairfield Public Schools 2011-2012

6. How would you record your thinking? 3 39 x 4 4 x 9 = 36 carry the 3 4 x 3 = 12 add the 3 = 15 15 6 Actually, it is 4 x 9 = 36 carry the 3 tens 4 x 3 tens = 12 tens add the 3 tens = 15 tens Digit-Oriented Right to Left Fairfield Public Schools 2011-2012

7. Another way to record thinking 3 39 x 4 39 x 4 Right to Left OR 3 6 120 + 120 + 36 156 156 4 x 9 = 36 & write the 3 tens Then 4 x 3 tens = 120 Number-Oriented Left to Right Fairfield Public Schools 2011-2012

8. Another way to record thinking OR re-write the problem horizontally and decompose 39 according to place value. 4 x (30 + 9) Fairfield Public Schools 2011-2012

9. Connecting an alternative strategy to algebraic properties 4 x 39 = ? 4 x 30 + 9 = 120 + 36 = 156 ( ) Distributive Property of multiplication over addition Fairfield Public Schools 2011-2012

10. Connecting an alternative strategy to algebraic properties 4 x 39 = ? 4 x 40 - 1 = 160 - 4 = 156 Distributive Property of multiplication over subtraction ( ) Fairfield Public Schools 2011-2012

11. Try another problem and solve it mentally: 25 x 49 = ? How would you record your thinking? Fairfield Public Schools 2011-2012

12. Did you think… Money? (4 quarters) = \$1.00 (or 100) 8 quarters = 200, 12 quarters = 300, 16 quarters = 400… 48 quarters = 1200 + 1 more quarter is 49 quarters = 1225, Fairfield Public Schools 2011-2012

13. OR did you think? 25 x 49 (20 + 5) x 49 = (20 x 49) + (5 x 49) (2 x 49 x 10) + (5 x (40 + 9)) = (98 x 10) + (200 + 45) = (980 + 200) + 45 = 1225 Fairfield Public Schools 2011-2012

14. 49 x 25 is (50 – 1) x 25 or, 5 x 25 = 125 and 10 x 125 = 1250 (50 – 1) x 25 is one less group of 25 from 1250 (50 x 25) – (1 x 25) = 1225 ORdid you think… Associative Property ( ) 10 x 5 x 25 = Distributive Property Fairfield Public Schools 2011-2012

15. OR did you think… (4) 49 x 25 5 Fairfield Public Schools 2011-2012

16. OR did you think… (4) 49 x 25 245 Fairfield Public Schools 2011-2012

17. OR did you think… (4) 49 x 25 245 0 Fairfield Public Schools 2011-2012

18. OR did you think… (1) (4) 49 x 25 245 80 Fairfield Public Schools 2011-2012

19. OR did you think… (1) (4) 49 x 25 245 + 980 Fairfield Public Schools 2011-2012

20. OR did you think… (1) (4) 49 x 25 245 + 980 1225 Fairfield Public Schools 2011-2012

21. Did you have another approach? Which way is most efficient for you? Fairfield Public Schools 2011-2012

22. Compare your strategy with someone else. Which approach is the “correct” approach? Why does the traditional algorithm in the US work? When might you use alternative strategies for computation? Look at the numbers, then decide… Fairfield Public Schools 2011-2012

23. Division How would you divide 156 ÷ 12 = ? Solve it mentally and think about how you approached the problem. Fairfield Public Schools 2011-2012

24. How would you record your thinking? 1 3 12 goes into 15 once. 6 Put down the 12 and subtract from 15 to get 3. -12 • 36 3 Bring down the 6 to make 36. 12 goes into 36 three times. Digit Oriented Fairfield Public Schools 2011-2012

25. OR did you think… 10 x 12 = 120 - 120 10 120 from 156 leaves 36 36 3 13 3 x 12 = 36 Number Oriented Fairfield Public Schools 2011-2012

26. How do models support thinking? The following examples use the array model to represent thinking. Fairfield Public Schools 2011-2012

27. Area Model Open Array Closed Array 3 rows of 4 or, 3 groups of 4 3 x 4 = 12 4 3 Fairfield Public Schools 2011-2012

28. Algebraic properties are used with whole numbers in the elementary grades. Flexible strategies foster a greater sense of number, equivalence, estimation, and the use of algebraic properties. Fairfield Public Schools 2011-2012

29. Multiplication with Partial Products 3 x (5 + 4) 9 5 4 3 Or (3 x 5)+(3 x 4) The Distributive property of multiplication over addition 15 + 12 = 27 Fairfield Public Schools 2011-2012

30. 10 + 3 13 10 x 10 = 10 + 2 12 100 30 10 x 3 = 2 x 10 = 2 x 3 = 6 20 12 x 13 = (10 + 2) x (10 + 3) = Fairfield Public Schools 2011-2012

31. DivisionPartial Quotients (Partial Factors) 3 + 3 = 6 9 18 9 3 Fairfield Public Schools 2011-2012

32. DivisionMake it friendly to solve mentally 10 = 13 + 3 120 36 12 156 Fairfield Public Schools 2011-2012

33. OR + 1 = 13 12 OR Record it: 12 1 156 144 Fairfield Public Schools 2011-2012

34. OR use your understanding of 5 x 12… + 5 + 3 5 = 13 Which can be recorded as: 36 60 156 60 12 Fairfield Public Schools 2011-2012

35. Differences between invented strategies and the traditional algorithm Invented Traditional Rigid Right to left Digit oriented • Flexible • Left to right • Number-oriented Fairfield Public Schools 2011-2012

36. Advantages of traditional algorithms Better for calculations with large numbers One procedure for all problems within a given operation Fairfield Public Schools 2011-2012

37. Advantages of invented strategies Built on understanding - easier to make sense Builds understanding with estimation Reinforces place value concepts Much easier to perform mentally Avoid system errors Fairfield Public Schools 2011-2012

38. The numbers and problem often lend themselves to different strategies. If you wanted to buy 3 rolls of wrapping paper on sale for \$1.99, how would you calculate your answer? Did you use the standard algorithm or did you use an alternative strategy? Is one strategy more correct than another? Fairfield Public Schools 2011-2012

39. “The single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students.” (Hiebert et al., 1996) It is important that students solve problems not to apply mathematics but to learn new mathematics. When students engage in well chosen problem based tasks and focus on the solution methods, what results is new understanding of the mathematics embedded in the task. (Van de Walle and Lovin, 2006) Fairfield Public Schools 2011-2012

40. Ways you can support your child: • Ask your child “How they solved a problem and how do they know if they are correct?” (prove it!) • Play math games • Include them in everyday activities and highlight the mathematics – e.g. cooking, measuring, shopping, traveling, building… • Delay the “standard algorithm” until your child understands why it works. Fairfield Public Schools 2011-2012