# To investigate and discover an equation, numerically and algebraically.

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Introduction

Maths coursework: January 2001. Aim: To investigate and discover an equation, numerically and algebraically. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 Method: We experimented with different grid sizes, including 9 and 7. We found out the total of the numbers inside the t-shape. (Red) T = 1+2+3+11+10 = 37 If I move the T across I can investigate the relationship by comparing the various t-totals. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 T= total G= grid size N= t number (e.g.) 20 N G T Difference 20 9 37 5 21 9 42 5 22 9 47 5 To try and find a formula for the red T, I used the difference, 5 and multiplied it by the N number. ...read more.

Middle

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 N becomes 8: T= 8+9+10+3+17= 47 When N= 9 T= 9+10+11+4+18= 52 When N= 10 T= 10+11+12+5+19= 57 G N T Diff 7 8 47 5 7 9 52 5 7 10 57 5 N=8 Nx5= 40 +7 = 47 N=9 Nx5= 45 +7= 52 N=10 Nx5 = 50 +7= 57. The formula has been found to be the same as on Grid size 9: T=5N+7 Again, if we mirror the image the formula can be predicted to be: T=5N-7. First lets try Grid size 9. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 N=12 G N T Diff 9 12 53 5 9 13 58 5 9 14 63 5 T= 12+11+10+1+19= 53 N= 13 T= 13+12+11+2+20= 58 N= 14 T= 14+13+12+3+21= 63 Predicted Formula: T= 5N-7 When N=12 5xN= 60 60-7= 53 When N= 13 5x13= 65 65-7= 58 When N= 14 5x14= 70 70-7= 63 So we have found that when a T-shape N number is placed upright and facing to the right, the outcome is positive. ...read more.

Conclusion

25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 N= 23 G N T Diff 7 23 73 5 7 24 78 5 7 25 83 5 T= 23+17+11+3+19= 73 N= 24 T= 24+18+12+4+20= 78 N= 25 T= 25+19+13+5+21= 83 When the Grid size was nine and the shape was in the same rotation, the formula was: T= 5N-56 but will it be the same for when G= 7? When N= 23 5N= 115 115 -73 42 This gives us a possibility for a formula: T= 5N-42. 42 is 6x7 or 6G so why can't G be involved in the formula? G cannot be involved because it does not work for grid size nine. 6x9= 54 and the formula for when G= 9 is: T= 5N-56. So, we have found a numerical formula now I have to test it. Formula: T= 5N-42 When N= 24 5N= 120 120-42= 78 When N= 25 5N= 125 125-42= 83 This proves the formula to be successful. Evaluation: From this study I have been able to deduce that for shapes that are mirrored the formulas will be the same but the signs will be opposite. Also G can only be involved when the shape was upright or upside down. G could have been involved more if we were only using grid size seven but then the formula would be specific. It could not have been a general algebraic formula. ...read more.

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