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The Numeracy Professional Development Project in Secondary Schools

Explore a teaching progression for addition and subtraction strategies in secondary schools. Learn how to use materials and develop mental images to solve problems. Discover the link between numeracy and algebra.

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The Numeracy Professional Development Project in Secondary Schools

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  1. The Numeracy Professional Development Project in Secondary Schools Addition and Subtraction Strategies Kevin Hannah National Coordinator, Secondary Numeracy Project

  2. Addition and Subtraction strategies • The Number Strategy Framework • A Model for a Teaching Progression • Some Questions • Using materials • Encouraging Imaging • Towards number properties • Subtraction strategies • Subtraction and addition problems • From Number to Algebra - solving equations

  3. Strategy Framework 0 Emergent 1 One-to-one counting 2 Counting from one with materials 3 Counting from one by imaging 4 Counting on 5 Early Part-Whole 6 Advanced Part-Whole 7 Advanced Multiplicative 8 Advanced Proportional

  4. Objective • To explore links between numeracy and algebra In particular: • To show that images used to help solve number problems also develop understanding for solving complicated linear equations.

  5. 46 + 19 61 – 27 14 + ? = 101 78 + 124 9000 – 8985 403 - 98 7. 47 + y = 83 8. 53 - m = 27 9. 2x + 1 = x + 7 10. 2x - 1 = 8 - x 11. 26 + 7 = ? + 12 12. 88 + x = 120 + ? Answers Only Please

  6. A Teaching Progression Start by: • Using materials, diagrams to illustrate and solve the problem Progress to: • Developing mental images to help solve the problem Extend to: • Working abstractly with the number property

  7. 46 + = 83 46 83 Using Materials 37 10 10 10 4 3

  8. 48 + = 81 29 + = 75 10 10 10 2 1 48 75 81 29 40 5 1 Using Materials 33 46

  9. 39 + = 63 10 10 3 1 30 40 50 60 70 39 63 Encouraging Imaging 24

  10. 28 + = 54 16 + = 73 10 10 3 4 2 73 28 16 54 20 20 30 40 50 70 60 50 4 Encouraging Imaging 26 57

  11. 18 + = 62 Using Number Properties 44 From 18: add 2 to get to 20 add 40 to get to 60 add 2 to get to 62 Total: add 44

  12. 39 + = 93 27 + = 52 46 + = 82 55 + = 72 17 + = 64 54 25 36 17 47 Using Number Properties

  13. Where to from here? • A subtraction strategy • Other strategies on number line • Link to Algebra - solving equations

  14. 47 81 Using Materials- subtraction 81 - 47 = 34 10 10 10 3 1

  15. 10 10 4 3 30 40 50 60 70 37 64 Imaging - subtraction 64 - 37 = 27

  16. 3 26 83 30 80 50 4 Imaging - subtraction 83 - 26 = 57

  17. Using Number Properties 82 - 17 = 65 From 17: add 3 to get to 20 add 60 to get to 80 add 2 to get to 82 Total: add 65

  18. Other strategies & problems • The number line is a versatile tool and image. • It can be used to support and explain a variety of strategies. • It can be used to solve a wide range of problems. • It can prepare students for algebra.

  19. 10 10 10 10 6 1 41 81 34 Other subtraction strategies 81 - 47 = 34

  20. 10 10 10 10 10 3 31 81 34 Other subtraction strategies 81 - 47 = 34

  21. 10 10 10 10 10 10 5 27 92 92 - 65 = Other subtraction problems 92 - = 65 27

  22. 10 10 5 2 65 92 65+ = 92 Other subtraction problems 92 - = 65 27

  23. 10 8 43 61 = 43 + 18 61 Other subtraction problems - 18 = 43 61

  24. 10 10 10 10 3 18 61 18 + 43 = 61 Other subtraction problems - 18 = 43 61

  25. 5 10 10 10 10 27 72 + 27 = 27 + Other addition problems + 27 = 72 45

  26. 7 10 10 45 72 = 72 - 27 45 Other addition problems + 27 = 72 45

  27. 10 10 6 57 83 Other addition problems 57 + 26 = 83

  28. What is Numeracy? • It’s about making sense of numbers. • It’s about problem solving with numbers. • It’s about understanding base 10. • It’s about learning with meaning. • It’s about preparing for algebra

  29. 47 83 47 83 Solving Equations 47 + = 83

  30. Solving Equations 53 - x = 27 53 - 53 - x = 27 - 53 - x = -26 x = -26 ÷ -1 x = 26

  31. Solving Equations 53 - x = 27 53 - x +x = 27 +x 53 = 27 +x 53 - 27 = 27 -27+x 26 = x

  32. 27 53 Solving Equations 53 - = 27 53 27

  33. Solving Equations 2X + 1 = X + 7 X X 1 X 7

  34. X X 1 Solving Equations 2X + 1 = 7 7

  35. A Teaching Progression Start by: • Using materials, diagrams to illustrate and solve the problem Progress to: • Developing mental images to help solve the problem Extend to: • Working abstractly with the property

  36. Solving Equations 2X - 1 = X + 7 X X X 7 1

  37. Solving Equations X - 1 = 2X - 7 7 X X X 1

  38. Solving Equations X - 1 = 2X - 7 7 X X X 1 7 = X + 1 X = 6

  39. Solving Equations 2X - 1 = 8 - X X X X 8 1

  40. Solving Equations 2(X + 1) = 18 X 1 X 1 18

  41. 9 9 Solving Equations 2(X + 1) = 18 X 1 X 1

  42. Solving Equations 2(X + 1) = 18 X X 1 1 18

  43. Solving Equations X + 3 = 2 X 3 2

  44. Solving Equations X 4 10

  45. 10 10 10 Solving Equations X 4

  46. What is Numeracy? • It’s about making sense of numbers. • It’s about problem solving with numbers. • It’s about understanding base 10. • It’s about learning with meaning. • It’s about preparing for algebra

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