# Maths Coursework:- T-Total

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Introduction

Vicky Evans Maths Coursework:- T-Total Introduction In this investigation I'm exploring the relationship between the T-total and the T-number on a 10 by 10-sized grid within the confinements of a T-shape. T-total = sum of all the numbers inside the T-shape T-number = number at the base of the T-shape Symbols * t = T-number * p = T-total * g = grid size * x = how far moved on the x-axis * y = how far moved on the y-axis Step One To begin the investigation I am going to explore what happens when the t-shape is moved around the grid. T-Shape (1) T-Shape (2) T-Shape (3) * Shaded number equals the t-number T-Shape T-total (p) = 40 p - t = 18 T-number (t) = 22 +12 T-Shape +3 across T-total (p) = 55 p - t = 30 T-number (t) = 25 +12 T-Shape +6 across T-total (p) = 70 p - t = 42 T-number (t) = 28 Looking at the pattern above we can see that on a 10 by 10 grid every square that the T-shape moves across the T-total increases by 5, the T-number increases by 1and the subtraction of t from p increases by 4. Step Two I am now going to investigate further by generalising the factors within the T-shape. This will eventually allow me to create a formula to find out the T-total or T-number when the T-shape is placed anywhere on the grid. * Numerical Example I decided that I would first of all do a numerical T-shape to see how all the numbers related to each other. My aim is to find a way of obtaining the T-total from the T-number. Starting from the T-number we can see that to get the one above you subtract 10, which just so happens to be the grid size. The square 2 places above t is t -20 or t-2g. ...read more.

Middle

As we can see because the shape is rotated about the T-number they both possess the original T-number. To find out the relationship between the T-total I have to add up all the terms within the new T-shape so as to obtain a formula that will work out the T-total. t + ( t + 1 ) + ( t + 2 ) + ( t - ( g - 2 ) ) + ( t + ( g + 2 ) ) t + t + 1 + t + 2 + t - g + 2 + t + g + 2 = p 5t + 7 = p We can see from this calculation that the new formula to find the T-total when the T-shape has been rotated 90� is very similar to the original. To see if this works for every T-shape which has been rotated 90� I will test a numerical example. This is how the new rotation will appear. Once again I will not place the two shapes as they would be but this is express in the diagram above. If we now place the numbers into the formula we should find that the T-total is equal to 242. 5t + 7 = p ( 5 x 47 ) + 7 = 242 We now know that when a T-shape is rotated 90� around any given rotation point that the formula 5t + 7 = p will work to give the new T-total. 180� The next step is to investigate the other two rotations of 180� and 270�. First however I will investigate 180�. Once again the small diagram expresses the position as the first rotation point will be t itself. Once again the two share the same T-number, now I will add the terms together to get the new T-total. t + ( t + g ) ...read more.

Conclusion

t + ( 2x ) + g( 2y ) = new t This formula is to find the new T-total for any rotation of 180� given the original T-number and the point of rotation. To get the new T-total all you have to do is to apply the formulas done previously for rotation and T-total in Rotations 1. In this case: 5( t + ( 2x ) + g( 2y ) ) + 7g = new T-total 270� Diagram Value of x Value of y T-number 1 3 -1 t + 4 + g(-2y) 2 0 2 t + (-2) + g(y) 3 2 0 t + 2 + g( y + 2 ) * All of them begin with the original t so t will be part 1 * If you subtract y from x you get the middle value so part 2 will be ( x - y ) * To get the final term you add x to y and multiply by g so g( y + x) is part three t + ( x - y ) + g( y + x ) = new t This formula is the to find the new t-total for any rotation of 90� given the original T-number and the point of rotation. To get the new T-total all you have to do is to apply the formulas done previously for rotation and T-total in Rotations 1. In this case: 5( t + ( x - y ) + g( y + x ) ) - 7 = new T-total To conclude for Rotation Rotation Formula for new T-Total 90 5( t + (y+x) + g(y-x) ) + 7 180 5( t + 2x + g(2y) ) + 7g 270 5( t + (x-y) + g(y+x) ) - 7 Conclusion In this investigation you can see how everything is linked together around the T-total and T-number. For example translation is involved in rotation and is modified for reflection. ...read more.

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