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What the 'L' - L shape investigation.

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Introduction

Alexander Ford 11L WHAT THE 'L' Mathematics GCSE Coursework L Shape Investigation The diagram below shows a standard L shape, which is drawn on a 9 by 9 number grid. 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 The total of the numbers inside the L-Shape is 1 + 10 + 19 + 20 + 21 = 71. This is called the L-Sum. The number in the bottom left hand corner of the L-Shape is called the L-Number. The L-Number for this L-Shape is 19. This investigation is in three parts. PART 1 Is to investigate the relationship between the L-Sum and the L-Number. PART 2. Using grids of different sizes is to translate the L-Shape to different positions. Investigate the relationship between the L-Sum, the L-Numbers and the grid size. PART 3. Investigate what happens if you use L-Shapes of different sizes. Try other transformations and combinations of transformations. Investigate relationships between the L-Sum, the L-Number, the grid size and the transformations. I will use the following key to illustrate the algebraic terms that I will use during my investigation: KEY L L-Number C Number of cells in L-Shape G Grid Size ? The Sum Of X Number Of Cells Above The L-Number Y Number Of Cells Across From The L-Number To start my investigation I will use the 9 by 9 grid section above I calculated the sum of the first seven L-Shapes; shown below: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 ...read more.

Middle

Thus giving me 5L.The results show that the 1st difference is constant and therefore, the formula must consist of 5L. My results shown in table format are as follows: Number in sequence 5L Difference Sum of L-Shape (L-Sum) 1 55 -12 43 2 60 -12 48 3 65 -12 53 4 80 -12 68 5 85 -12 73 6 90 -12 78 From the table of results I can see that the difference between 5L and the L-Sum is a constant -12. Therefore, we have 5L -12 present in the formula. I will use this formula to verify its correctness and to look further for additional differences. Number In Sequence Formula Formula Equation Results L-Sum 1 5L - 12 ( 5 x 11) - 12 43 43 2 5L - 12 ( 5 x 12) - 12 48 48 3 5L - 12 ( 5 x 13) - 12 53 53 4 5L - 12 ( 5 x 16) - 12 68 68 5 5L - 12 ( 5 x 17) - 12 73 73 6 5L - 12 ( 5 x 18) - 12 78 78 From these answers we can determine that the formula 5L is correct in a 5 cell standard L-Shape in a 5 by 5 grid. Therefore, the algebraic formula to calculate an L-Sum given the L-Number in a standard L-Shape in a 5 by 5 grid is: 5L -12 To prove my formula I am going to utilise the L-Shape, replacing the numbers with letters to prove the formula using any number in a 5 by 5 grid. L-10 L-5 L L+1 L+2 *NB: Note that in this L-Shape numbers along the horizontal will increase and numbers on the vertical will decrease. Also note that the number vertically up from the L-Number is referred to as L-5, the second is L-10. Note that 5 is the grid square and 10 is double the grid size and two rows up. ...read more.

Conclusion

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 These are the four rotations you would get if you rotated a standard L-Shape in a 9 by 9 grid. The four L-Shapes below are my results: - 23 32 41 42 43 Rotating the L-Shape through 90� 41 42 43 50 59 23 32 39 40 41 39 40 41 50 59 Rotating the L-Shape through 180� Rotating the L-Shape through 270� I will now tabulate my results to make it easier to read. Arm 0� 90� 180� 270� X - + + - Y + + - - With the above information I can predict that if the rotation of the L-Shape is changed then the + & - in the equation will have to change. Therefore, the equation to find the L-Sum given just the grid size and the L-Number, in any size grid, in any rotation, with any transition of the L-Shape, with any size arms is: - Now, when using this formula you have to choose a rotation look at the signs needed for that rotation and replace the � signs with the correct symbol. Letter Value Rotation Angle + & - Equation Answer C 7 0� X= - Y= + 365 CORRECT L 59 G 9 X 3 Y 3 Letter Value Rotation Angle + & - Equation Answer C 7 180� X= + Y= - 461 CORRECT L 59 G 9 X 3 Y 3 Thus, I have proved that my formula works for any rotation. For an even further extension I have created an excel spreadsheet that allows my to calculate the L-Sum given just C, L, G, X, Y and the rotation angle using the above formula. A screen dump below shows what the package looks-like and what formulas I have used. The spread sheet is also included on the disc attached. ...read more.

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