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• Level: GCSE
• Subject: Maths
• Word count: 2630

1) Investigate the relationship between the T-total and the T-number. 2) Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

Extracts from this document...

Introduction

T-Totals

Introduction

If you look at the 9x9 grid with the T-shape, you can see that the total of the numbers added together is 37 because it is1+2+3+11+21 which equals 37.

This is what we call the T-total (37)

And T-number is the number at the bottom of the T-shape which in this case is 20

 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

The tasks I have been set are:

1) Investigate the relationship between the T-total and the T-number.

2) Use grid of different sizes. Translate the T-shape to different position. Investigate relationships between the T-total, the T-numbers and the grid size.

3) Use grids of different sizes again. Try other transformation and combinations of transformations. Investigate relationship between the T-total, the T-numbers, the grid size and the transformations.

Standard T-shapes

 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

If I place the results from 7 T-Shapes into a table then it would look like this.

 T-number 20 21 22 23 24 25 26 T-total 37 42 47 52 57 62 67

This table clearly shows that the numbers go up by 5 every time the T-number goes up by 1.

Now I can use trial and error to try to find the formula, I will then test what I believe to be the correct formula to see if it is.

Because there is a difference of 5 between the T-totals I think that it is a logical place to start the formula.

Middle

24-1=23

24-2=22

24-3=21

24-13=11

TOTAL=77

I can now take an educated guess that this is the number that changes as the grid size alters. I now have to alter the formula to allow the size of the grid to be multiplied or added to another number to reach the difference of 77 I require. So I did this and I found the formula, with g representing the grid number.

5T-(7g)

I shall now test this formula on another grid size to make sure that it is correct.

 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

With my formula I have calculated that the T-total for T-number 22 on a 6 by 6 grid will be 5x22-(7x6) =68

After studying the T-shapes for some time now and studying my formula, I have realize that A always equals T - 2 x The grid size (g) – 1, B equals T -  2g, C = T – 2g + 1, and D = T – g

As the T-Total is A+B+C+D+T I can conclude therefore that the

T-total = T – 2g – 1 + T – 2g + T – 2g + 1 + T – g + T

This can be simplified to 5T – 7g my prior formula.

I have now proven that my formula works on smaller sized grids.

Conclusion

Remainder- this is the column of the T-number. We know this is correct because 9-2= 7 so the middle seven columns are valid and two is in the centre.

So the three steps to validating a T-number are:

1. T/S - The T-Number (T) is divided by the Grid Size (S) , the answer will be a number and a remainder (R=Row, C=Column).

T / S = R .C

2. Row - The row number +1 must be higher then 2 and lower then the grid size +1

(The inequality 2 > (R+1) < S+1)

3. Column - The column number must be lower then one and also lower then the grid size -1

(The inequality 1 > C < S-1)

If the two numbers fit the inequalities then the T-Number will fit inside the grid. I will now test the formula using random figures.

 T-Number Manually Using Formula 2 No No 11 No No 20 Yes Yes 28 No No 63 No No 42 Yes Yes

Conclusion

In conclusion I have worked out many different formulae, including how to work out the T-total from just the T-number, how to work out a rotated T’s T-total from the T-number, working the T-total out by using vectors, checking if all of a T-shape is in a grid, and checked everything I have done. I enjoyed this work as it used a lot of formulae and think I explained my method to the best of my abilities.

By Keir Wyndham-Ayres

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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