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The Numeracy Professional Development Project in Secondary Schools Addition and Subtraction Strategies Kevin Hannah National Coordinator, Secondary Numeracy Project 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 Strategy Framework 0 1 2 3 4 5 6 7 8 Emergent One-to-one counting Counting from one with materials Counting from one by imaging Counting on Early Part-Whole Advanced Part-Whole Advanced Multiplicative Advanced Proportional 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. Answers Only Please 1. 2. 3. 4. 5. 6. 46 + 19 61 – 27 14 + ? = 101 78 + 124 9000 – 8985 403 - 98 7. 8. 9. 10. 11. 12. 47 + y = 83 53 - m = 27 2x + 1 = x + 7 2x - 1 = 8 - x 26 + 7 = ? + 12 88 + x = 120 + ? 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 Using Materials 46 + 37 = 83 10 10 10 4 0 10 20 30 40 3 50 46 60 70 80 83 90 100 Using Materials 48 + 33 = 81 10 10 10 2 0 10 20 30 40 48 1 50 60 70 80 90 100 80 90 100 81 29 + 46 = 75 40 1 0 10 20 30 29 40 50 5 60 70 75 Encouraging Imaging 39 + 24 = 63 1 30 40 39 10 10 50 3 60 70 63 Encouraging Imaging 28 + 26 = 54 2 20 28 30 10 10 40 4 50 54 16 + 57 = 73 4 16 20 50 3 70 73 60 Using Number Properties 18 + 44 = 62 From 18: add 2 to get to 20 add 40 to get to 60 add 2 to get to 62 Total: add 44 Using Number Properties 39 + 54 = 93 27 + 25 = 52 46 + 36 = 82 55 + 17 = 72 17 + 47 = 64 Where to from here? A subtraction strategy Other strategies on number line Link to Algebra - solving equations Using Materials- subtraction 81 - 47 = 34 10 10 10 3 0 10 20 30 40 1 50 47 60 70 80 81 90 100 Imaging - subtraction 64 - 37 = 27 3 30 40 37 10 10 50 4 60 70 64 Imaging - subtraction 83 - 26 = 57 4 26 30 50 3 80 83 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 Other strategies & problems 0 10 20 30 40 50 60 70 80 90 100 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. Other subtraction strategies 81 - 47 = 34 6 0 10 20 30 34 1 40 41 10 10 50 10 60 10 70 80 81 90 100 Other subtraction strategies 81 - 47 = 34 10 0 10 20 30 3 31 10 40 34 10 50 10 60 10 70 80 81 90 100 Other subtraction problems 92 - 27 = 65 5 0 10 20 30 10 40 10 10 50 27 92 - 65 = 10 60 70 10 10 80 90 92 100 Other subtraction problems 92 - 27 = 65 5 0 10 20 30 40 50 60 70 65 65+ = 92 10 10 80 2 90 92 100 Other subtraction problems 61 - 18 = 43 10 0 10 20 30 40 43 61 50 8 60 70 61 = 43 + 18 80 90 100 Other subtraction problems 61 - 18 = 43 10 0 10 20 18 10 30 10 40 10 50 3 60 61 18 + 43 = 61 70 80 90 100 Other addition problems 45 10 0 10 20 30 27 + 27 = 72 10 10 10 40 50 60 5 70 72 + 27 = 27 + 80 90 100 Other addition problems 45 + 27 = 72 7 0 10 20 30 40 50 45 45 10 10 60 70 72 = 72 - 27 80 90 100 Other addition problems 57 + 26 = 83 10 0 10 20 30 40 50 60 57 10 70 6 80 83 90 100 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 Solving Equations 47 + = 83 47 47 83 83 Solving Equations 53 - x = 27 53 - 53 - x = 27 - 53 - x = -26 x = -26 ÷ -1 x = 26 Solving Equations 53 - x = 27 53 - x +x = 27 +x 53 = 27 +x 53 - 27 = 27 -27+x 26 =x Solving Equations 53 - = 27 27 53 27 53 Solving Equations 2X + 1 = X + 7 X X X 1 7 Solving Equations 2X + 1 = 7 X X 7 1 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 Solving Equations 2X - 1 = X + 7 X X X 7 1 Solving Equations X - 1 = 2X - 7 7 X X X 1 Solving Equations X - 1 = 2X - 7 7 X X X 7 = X+1 X = 6 1 Solving Equations 2X - 1 = 8 - X X X 8 X 1 Solving Equations 2(X + 1) = 18 X 1 X 18 1 Solving Equations 2(X + 1) = 18 X 1 9 X 1 9 Solving Equations 2(X + 1) = 18 X X 18 1 1 Solving Equations X+3=2 X 3 2 Solving Equations x+4 3 = 10 X x 3 x 3 x 3 4 10 Solving Equations x+4 3 = 10 X 10 10 4 10 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