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Developing Higher Level Thinking and Mathematical Reasoning

Developing Higher Level Thinking and Mathematical Reasoning. Mathematical Reasoning. The process of problem solving involves Making conjectures Recognizing existing patterns Searching for connections to known mathematics Translating the gist of a problem into mathematical representation.

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Developing Higher Level Thinking and Mathematical Reasoning

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  1. Developing Higher Level Thinking and Mathematical Reasoning

  2. Mathematical Reasoning The process of problem solving involves • Making conjectures • Recognizing existing patterns • Searching for connections to known mathematics • Translating the gist of a problem into mathematical representation

  3. Mathematical Reasoning • Putting together different pieces of information • Developing a range of strategies to use • Verifying the correctness of the solution • Applying skills that require and strengthen student’s conceptual and procedural competencies

  4. Connecting to and building on students’ prior knowledge • Multiplication • Division

  5. Multiplication

  6. Multiplication Word Problems John has 4 bags of cookies. In each bag, he has 2 cookies. How many cookies does he have?

  7. Multiplication Word Problems There are 5 rows in a class. Each row has 3 desks. How many desks are in the class?

  8. Multiplication • What does 3 x 2 mean? • Repeated addition 2 + 2 + 2 • Skip Counting by 2’s – 2, 4, 6 • 3 groups of 2

  9. Multiplication • 3 rows of 2 • This is called an “array” or an “area model”

  10. Advantages of Arraysas a Model • Models the language of multiplication 4 groups of 6 or 4 rows of 6 or 6 + 6 + 6 + 6

  11. Advantages of Arraysas a Model • Students can clearly see the difference between (the sides of the array) and the (the area of the array) factors product 7 units 4 units 28 squares

  12. Advantages of Arrays • Commutative Property of Multiplication 4 x 6 = 6 x 4

  13. Advantages of Arrays • Associative Property of Multiplication (4 x 3) x 2 = 4 x (3 x 2)

  14. Advantages of Arrays • Distributive Property 3(5 + 2) = 3 x 5 + 3 x 2

  15. Advantages of Arraysas a Model • They can be used to support students in learning facts by breaking problem into smaller, known problems • For example, 7 x 8 8 8 4 5 3 4 7 7 + = 56 35 21 28 = 56 28 +

  16. Teaching Multiplication Facts 1st group

  17. Group 1 • Repeated addition • Skip counting • Drawing arrays and counting • Connect to prior knowledge Build to automaticity

  18. Multiplication • 3 x 2 • 3 groups of 2 1 3 5 6 4 2

  19. Multiplication • 3 x 2 • 3 groups of 2 6 2 4

  20. Multiplication • 3 x 2 • 3 groups of 2 2 + 2 + 2

  21. Multiplying by 2 Doubles Facts • 3 + 3 • 2 x 3 • 5 + 5 • 2 x 5

  22. Multiplying by 4 Doubling • 2 x 3 (2 groups of 3) • 4 x 3 (4 groups of 3) • 2 x 5 (2 groups of 5) • 4 x 5 (4 groups of 5)

  23. Multiplying by 3 Doubles, then add on • 2 x 3 (2 groups of 3) • 3 x 3 (3 groups of 3) • 2 x 5 (2 groups of 5) • 3 x 5 (3 groups of 5)

  24. Teaching Multiplication Facts Group 1 Group 2

  25. Group 2 • Building on what they already know • Breaking apart areas into smaller known areas • Distributive property Build to automaticity

  26. Breaking Apart 7 4

  27. Teaching Multiplication Facts Group 1 Group 2 Group 3

  28. Group 3 • Commutative property Build to automaticity

  29. Teaching Multiplication Facts Group 1 Group 2 Group 4 Group 3

  30. Group 4 • Building on what they already know • Breaking apart areas into smaller known areas • Distributive property Build to automaticity

  31. Distributive Property • Distributive Property: The Core of Multiplication Teaching Children Mathematics December 2013/January 2014

  32. Reasoning about Multiplication and Division http://fw.to/sQh6P7I

  33. Multiplying Larger Numbers 23 x 4

  34. Using Arrays to Multiply 23 x 4 80 4 rows of 20 = 80 12 4 rows of 3 = 12 92

  35. Using Arrays to Multiply 23 x 4 12 4 rows of 3 = 12 80 4 rows of 20 = 80 92

  36. Multiplying Larger Numbers 34 x 5

  37. Multiplying Larger Numbers • So what happens when the numbers are too large to actually build? 3 70 73 x 8 24 560 8

  38. Multiplying Larger Numbers 257 x 6

  39. Using Arrays to Multiply • Use Base 10 blocks and an area model to solve the following: 21 x 13

  40. Multiplying and Arrays 21 x 13

  41. 31 x 14 =

  42. Partial Products 31 x 14 300 (10  30) 10 (10  1) 120 (4  30) 4 (4  1) 434

  43. Partial Products 31 x 14 4 (4  1) 120 (4  30) 10 (10  1) 300 (10  30) 434

  44. Pictorial Representation 80 + 4 84 x 57 50 + 7 50  4 50  80 4,000 200 7  80 7  4 560 28

  45. Pictorial Representation 30 + 7 37 x 94 90 + 4 90  7 90  30 2,700 630 4  30 4  7 120 28

  46. Pictorial Representation 300 + 40 + 7 347 x 68 60 + 8 2,400 18,000 420 320 2,400 56

  47. Multiplying Fractions

  48. 3 x 2 Three groups of two 0 1 2 3 4 5 6

  49. Multiplying Fractions • Remember our initial understanding of fractions of fractions • Another way to write this is

  50. Multiplying Fractions • What do we normally tell students to be when they multiply a fraction by a whole number?

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