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Introduction Introduction:

In this investigation I will establish the link between the T-Total and the T-Number. The aim of my coursework is to find any pattern or link between T-Total and T-Number. By obtaining a formula, I will be able to find the T-Total if I am giving the T-Number and vice versa; without having to draw number grids. Furthermore, by obtaining various formulae, I will then be able to deduce a general formula for all grid tables.

After investigating the relationship between the two, I will then use grids of different sizes and translate the T-Shape to different positions; to see whether the pattern changes or stays the same.

In this coursework I will be using the terms T-Total and T-Number frequently, so we need to know what they mean.

 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

We have a grid nine by four with the numbers starting from 1 to 36. There is a shape in the grid called T-Shape. This is highlighted in the colour yellow.

This is a T-Shape drawn on a nine by four number grid.

The total of the numbers inside the T-Shape is 1+2+2+11+20=37.

This is called the T-Total.

The number at the bottom of the T-Shape is called the T-Number.

The T-Number for this T-shape is 20.

Method:

I will carry my investigation out in the following steps:

1. Firstly, I will draw number grids of four, seven and six; plotting 3 adjacent T-Shape in each grid. My results with then be tabulated, this make it easier to compare the results and find a formula for each number grid.
2. The formulae obtained from the grids, I will then be used to deduce a general rule which will apply to all the different grid numbers.

Middle

Five 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
 T-Number (n) T-Total First Difference Second Difference Formula 12 (1+2+3+7+12)= 25 5 25-(5*12)= -35 5n-35 13 (2+3+4+8+13)= 30 5 30-(5*13)= -35 5n-35 14 (3+4+5+9+14)= 35 5 35-(5*14)= -35 5n-35

Table of Results (Five Number Grid)

The above results confirm that my prediction was correct because I obtained the same formula by using the 5n- 7g rule.

Rotation

To investigate further I will also find out whether the 5n- 7g rule applies when it comes to rotation. The yellow highlighted T-Shape below is rotated through 90°, 180° and 270° clockwise in a seven, six and five number grid.

Seven 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

We can work out the formulae for each rotated shape by replacing the number with algebraic terms. This has been done below:

Rotation Through 90°Rotation Through 180°  Rotation Through 270° The above results have been tabulated which makes it easier to see a relationship or pattern.

 Angle (degrees Clockwise) T-Number (n) T-Total Formula Original T-Shape 18 41 5n- 49 Rotated Through 90° 18 97 5n+ 7 Rotated Through 180° 18 139 5n+ 49 Rotated Through 270° 18 83 5n- 7

As we can see from the tabulated result there is a general pattern. We already know the formula for the original T-Shape; 5n- 49, similarly when the shape is rotated through 180°, the formula change, form 5n- 49 to 5n+ 49. This is because the T-shape is in the opposite position, it is a reflection and the signs are reversed with the reversed position.

Another pattern which can be seen form the results above is the rotation through 90° and 270°. The formulae are roughly the same; apart from the opposite sign. The formula for a T-Shape rotated through 90° is 5n+ 7 while the formula for a T-Shape rotated through 270° is 5n- 7.

The most obvious link is that, of the length of grid and when the shape is rotated through 90° and 270°. The length of the gird is 7; similarly when the T- Shape is rotated through 90° and 270°, the formulae are 5n+ 7 and 5n- 7 respectively.

Six 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

Conclusion

270°

5n-7

The next stage of my coursework was to find a relationship between the T-Number, T-Total and translation. Establishing a pattern for translation was very hard because the pattern wasn’t straightforward and harder to spot. Due to this, I resulted to using algebraic terms based on the T-Number. Firstly, I worked out the formula for any horizontal translation, using the column vector (a) and then for any vertical translation using the column vector (b). After working out formulae for both, I used algebraic terms to get a formula for translation; adding all the algebraic expressions together, I arrived with the general formula of 5N+5a-5bg-7g.

Finally, I finished the coursework by establishing the link between T-Number, T-Total and double transformation. When carrying out the double transformation, I chose to translate the original T-Shape before rotating it. This was because it was easier to stop the pattern once the shape had been translated, rather than rotated. By looking at the algebraic terms for translation and rotation; I realized that the formula for double transformation was slightly different than that of translation. This meant that I only needed to alter the formula of translation slightly. After working out formulae for each double transformation, I repeated my step on different number grids to check and make sure the formulae obtained were correct. The formulae I obtained are tabulated below:

 Rotation (degrees) Direction Translated Formula 90° Clockwise Yes 5N+5a-5bg+3 180° Clockwise Yes 5N+5a-5bg+7g 270° Clockwise Yes 5N-5a-5bg-3

Overall, I think that my coursework went very well because I achieved what I set out to do. The aim of the coursework was to find a link between T-Number and T-Total; I managed to do this and further investigations. If I could improve my coursework, I would write it out because it would be clearer to understand and much easier to draw tables and shapes.

Title Page                                                        Page Number

Introduction……………………………………………………………2-4

Main Part………………………………………………………………5-8

Rotation………………………………………………………………9-12

Translation………………………………………………………..13-18

Double Transformation………………………………………..19-24

Conclusion………………………………………………………..25-26

This student written piece of work is one of many that can be found in our GCSE Height and Weight of Pupils and other Mayfield High School investigations section.

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