# I am going to investigate the relationship between the T-Totals and T-numbers when the T-shape is translated in different sizes of grids

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Introduction

T-Totals

I am going to investigate the relationship between the T-Totals and T-numbers when the

T-shape is translated in different sizes of grids. A good way of showing translations is by using vectors.

To give you an insight of how the grids look I have used 3 different grid sizes which I will be investigating further on. In each column of the grids we see that every time 9,8 or 7 is added to the number and it follows this sequence and the numbers on a row when added contain, (9 by 9 grid) 81,(8 by 8 grid) 64 or(7 by 7 grid), 49 numbers.

row

The T-shape drawn on grids will look like this

This is called the T-Number,

I will refer this as N

When adding all the number together we will get the T-Total

I will refers this as T.

In the next table I have calculated the T and N which gave me the following results:

T | 37 | 42 | 47 | 52 |

N | 20 | 21 | 22 | 23 |

We can see from this information that every time

Middle

N

5 * 23 = 115 115 – 63 = 52

This has proven that my formula is correct.

In the next section I will be translating a different size of grid which will be

8 by 8 grid.

Which would give me the following results

T | 34 | 39 | 44 | 49 |

N | 18 | 19 | 20 | 21 |

I will find the formula as I did before

(N-10) + (N-1) + (N-2) + (N-3) = T

18-10= 8

18- 2=16

18- 1=17

18- 3=15 +

Total = 56

5N-56= T

5*18= 90 90-56=34

This has proven my formula is correct but lets check again by moving the T-shape I called the T-shape with the T-number 16 = A and the T-shape with the T-number 18=B

Vector AB=

5*20= 100 100-56=44

Yes, the formula is correct

As you see both grids that I have investigated are in the 7 times table and I predict that the following 7 by 7 grid will be 7*7= 49. This gives me the following formula

5N-(7*grid size)=T

Which will give me the following results

T | 31 | 36 | 41 | 46 |

N | 16 | 17 | 18 | 19 |

I will use the same method as before to find the formula.

(N-9) + (N-1) + (N-2) + (N-3) = T

16-9= 7

16-1=15

16-2=14

16-3=13 +

Total = 49

This

Conclusion

As you see above every time 100+10 is added to the T-total when you using this formula you can also see that T-number 21, 22, 23, 24 of the formula 10N-63=T is different as it does not follow the pattern of the formula as you see t-number 21 of the f 5N-63=T is does not get 142 but 147 because it skips one place as we are using the formula 10N-63 = T.

In conclusion I have found out different ways to translate and solve T-shapes in different positions with different grid sizes. I have solved the T-shapes by using a formula which slightly changes in different circumstances the formula is 5N-(7*G)=T which has linked the relationship between the T-Total, T-Number and the Grid size.

Sewita Nazari 10N

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

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