# Mathematics Coursework - T-totals

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Introduction

GCSE MATHEMATICS COURSEWORK T-totals 10/3/'02 Jonathan Briggs INTRODUCTION Introduction The question I have been asked has been set in three parts. The question is about T-shapes on different grids. The bottom number in the T is called the T-number the T-number is the largest number in the T-shape. All the numbers in the T-shape added together are called the T-total. In the whole of the coursework you are not allowed to expand the T-shape but you are allowed to turn it around. In part 1of my coursework I am asked to investigate the relationship between T-totals and T-numbers. At the moment I do not have any notion how the T-total could be calculated in a standard way or of the relationship between the T-number and the other numbers in the T-shape. I did know that they had to be related because they were all numbers with in a sequence. I also knew that the T-total would depend on: ~ * The position of the T-shape on the grid; * The size of the T-shape (for the purpose of this exercise the T-shape is fixed with only five numbers); * The orientation of the T-shape on the grid. These points are fairly obvious as any change to the T-shape will change the numbers within it. From this knowledge I initially experimented practically by putting T-shapes in various positions on a 9 x 9 grid. At this stage (part 1 of this paper) the T-shape was fixed in a normal T position. Part 1 shows my approach and findings from experimenting with a standard T-shape on a 9 x 9 grid In part 2 I did further experiments with grids of different dimensions attempting to repeat my findings from part 1 with these grids and attempting to develop a general formula for grids of all sizes In part 3 I did more experiments, this time with the T-shape in different orientations on the grid; that is on its side in two positions and upside down. ...read more.

Middle

Equation to work out the T total based on expressing numbers in the T-shape relative to the first number (9 � 9 grid) On the diagram above A = 13, B = 14, C = 15, D = 23 and E = 32 A = A B = A + 1 C = A + 2 D = A + 10 E = A + 19 T-total = 5A + 32 After doing this I found there was an alternative way of expressing numbers relative to each other and the grid size as follows If N is the T number the number above N can be expressed as N - G where G is the width of the grid i.e.: nine in this case The number above N - G can be expressed as N - 2G The numbers each side of N - 2G can be expressed as N - 2G - 1 and N - 2G + 1 correspondingly. This gives the formula Equation: ~ Where~ N = T-total X = number of numbers in the T-shape G = grid size T-total = (N) + (N - 9) + (N - 2G) + (N - 2G - 1) + (N - 2G +1) This simplifies to: ~ T-total = 5N - 7G 7 is the relationship of the T-total and the other numbers in the T-shape. PART 3 Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-total, the T-number, the grid size and the transformations Part 3 Constraints 1. As in Part 2 except that it is possible to move the alignment of the T-shape in relation to the 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 ...read more.

Conclusion

Equation Where: ~ G = grid size TN = T-number A = 4 - (dimensions of vector) B = 3| (dimensions of vector) T-total = (TN + 4 + 3G) + (TN + G + 4 + 3G) + (TN + 2G + 4 + 3G) + (TN + 2G + 1 + 4 + 3G) + (TN + 2G - 1 + 4 + 3G) T-total = (TN + A + BG) + (TN + G + A + BG) + (TN + 2G + A + BG) + (TN + 2G + 1 + A + BG) + (TN + 2G - 1 + A + BG) T-total = 5TN + 5A + 5BG + 7G 5 (TN + A +BG + 1 G ) CONCLUSION Conclusion I tested my rules by experimenting with a variety of examples. I found the rule by experimenting with different grid sizes and different numbers inside my various grids. I have found out that there is a definite relationship between the T-number and the T-total. I found that the T-total can be found by using several different formulae, depending on the size of the grid which the T-shape is in and the orientation of the T-shape. I have made a variety of tables to test the formulae and to show the different orientations and the variations between the different sizes of grid. The main equation that I have used for parts one and two is T-total = 5N - 7G. This equation shows the relationship between the T-totals, the T-number and the grid size. I have also used different transformations and orientations. It was also possible to create different equation for different orientations of the T-shape. Then I tried using vectors which was more difficult. There was a formula for the vectors which I could find, I could find a link between vectors size and the extent to which T-number and T-total changed with a transformation. ?? ?? ?? ?? 1 ...read more.

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