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Developing Higher Level Thinking and Mathematical Reasoning

This resource focuses on developing higher-level thinking and mathematical reasoning skills through problem-solving and recognizing patterns. It provides strategies, examples, and activities for multiplication and multiplication word problems.

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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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