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Introduction

T-Shapes Coursework We have been asked to investigate the relationship between the T-total and the T-number for our T-shapes coursework. We have been given our first T-shape on a 9 by 9 grid. This T-shape is shown below: The total of the numbers inside the T-shape is 1 + 2 + 3 + 11 + 20 = 37 this is called the T-total. The T-number for this T-shape is 20. T-number I have now moved the shape over one square on the grid. The new total of the numbers inside the shape is 2 + 3 + 4 + 12 + 21 = 42. Looking at this I can see that 5 has been added to the T-total. This is because when moved over, each number increases by one, and because there are 5 squares in the T-shape, the number increases by 5. Knowing this I can now predict that if I move the T-shape over by another square, the new T-total will be 47. I have now moved the T-shape over by another square and the new T-total is 3 + 4 + 5 + 13 + 22 = 47. This is what I originally predicted. T-total of original T-shape. 37 T-total after first translation. 42 T-total after second translation. 47 I have recorded the results of the original T-shape total and the T-total after each translation in the table above. ...read more.

Middle

- - - - - - - - - - - - - - - - - I will now repeat the steps I did for a 9 by 9 and 10 by 10 grid for an 8 by 8 grid. The T-total for this shape on an 8 by 8 grid is 2 + 3 + 4 + 11 + 19 = 39 I have moved the T-shape over by one square and I believe that, just the same as the 9 by 9 and 10 by 10 grid, 5 will be added to the T-total. So the T-total of this shape should equal 44. 3 + 4 + 5 + 12 + 20 = 44 I have moved the T-shape over another square and I expect the new T-total to be 49. 4 + 5 + 6 + 13 + 21 = 49 T-total of original T-shape. 39 T-total after first translation. 44 T-total after second translation. 49 I have recorded the results of the original T-shape total and the T-total after each translation in the table above. I have noticed that after each translation 5 has been added to the T-total. This is because when the T-shape is moved across once on the grid, one is added to each number, and because there are 5 numbers in the T-shape, 5 is added all together. The T-total for this shape is 94. ...read more.

Conclusion

To demonstrate this I will use the T-shape to my left. I will use a 10 by 10 grid. The numbers in the original T-shape add up to 110. By using the algebra I know that after carrying out a reflection through the T-number my T-total would be 250. This is because 5t has been added (180) and 7g has been added (70.) 36 + 46 + 56 + 55 + 57 = 250. My algebra has been tested and proven correct. Finally I am going to reflect my T-shape through a consecutive horizontal line and see how this affects my algebra. T T-g T-2g T-2g+1 T-2g-1 5T-7g T T+g T+2g T+2g+1 T+2g-1 5T+7g T+2g T+2g+1 T+2g-1 T+3g _T+4g_ 5T+13g T+4g T+5g T+6g T+6g+1 T+6g+2 5T + 27g I have noticed that each time the shape is reflected to make a consecutive pattern the number that you multiply the grid by increases by one per square. I will now test the algebra on a size 10 grid to see if my theory works. 1 2 3 12 22 20 To work out if my formula works I will use my T-number 22 (22) to calculate the T-totals. 32 42 41 43 180 41 42 43 52 62 240 62 72 82 81 83 380 Yellow T = 22 5t - 7g 110 - 70 = 40 Pink 5t + 7g 110 + 70 = 180 Black 5t + 13g 110 + 130 = 140 Green 5t + 27g 110 + 270 = 380 ...read more.

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