# In this investigation I will be looking at the relationship between the T-total and the T-number with simple drawings and try to identify a general rule.

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Introduction

Maths Coursework

## Introduction

This piece of coursework is about T- totals. I will be looking a T- Shape (see below) drawn on a 9x9 number grid. The number at the bottom of the shape is called the T- number. In this investigation I will be looking at the relationship between the T-total and the T-number with simple drawings and try to identify a general rule.

## Drawings

To help me in my investigation I have produced 3 drawings, the 4th being my prediction in a grid which is 9x9.

(T-n = T- Number T-t = T-Total)

T-n= 21 + 12 + 1 + 2 +3 = 39 (T-t)

T-n= 41 + 32 + 22 + 23 + 24 =142 (T-t)

T-n= 61 + 52 + 42 + 43 + 44 =242 (T-t)

(Prediction)

T-n= 81 + 72 + 62 + 63 + 64 = 337 (T-t)

## Table of Results

## With the T-shapes drawn in the grid above I can now put my results in a table to see if there are any patterns occurring.

Identifying the pattern

As you can see from my table of results that the T-total is increasing by 100 every time you add 20 to the T-number.

Prediction

I predict that the next T-total will be 337 and the T-number will =80.

Middle

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

T - n = 16 + 9 + 1 + 2 + 3 = 31(T-t)

T - n = 24 + 17 + 9 + 10 + 11 = 71(T-t)

T – n = 32 + 25 + 17 + 18 + 19 = 111(T-t)

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 |

T – n = 18+ 10 + 1 + 2 + 3 = 34 (T- n)

T – n = 27 + 19 + + 10 + 11 + 12 = 79 (T- n)

T – n = 36 + 28 + 19 + 20 + 21 = 124 (T- n)

Table of Results

My results for the 7x7 and 8x8 grid will be put into a table, so that I can see if there are any patterns emerging.

7 x 7 Grid 8 x 8 Grid

T - n | T- t |

18 | 34 |

27 | 79 |

36 | 124 |

T- n | T - t |

16 | 31 |

24 | 71 |

32 | 111 |

General Rule

These are the nth terms for Grids 7x 7 and 8 x 8.

7 x 7 8 x 8

n - 15 | n - 14 | n - 13 |

n - 7 | ||

n | ||

N - 17 | N - 16 | N - 15 |

N - 8 | ||

N |

T-total = n + n – 7 + n – 15 + n – 14 + n – 13 T-total = n + n – 8 + n + 17+ n – 16 +n -15

=5n – 49 = 5n – 56

Identifying the pattern

There is a clear pattern on the 7 x 7 grid as the T- n goes up by 8 the T- t increases by 40 every time. On the 8 x 8 grid the T-n goes up by 9 every time and the T-t increases by 45.

I have also noticed that every time the grid size gets smaller the nth term decreases by 7 but the 5n does not change.

9 x 9 = | 5n – 63 |

8 x 8 = | 5n - 56 |

7 x 7 = | 5n - 49 |

Conclusion

Overall Rule

I am going to use all the nth terms from the 3 grids and put them in a table and try to find a rule that will work on any grid size.

n - 17 | n - 16 | n - 15 |

n - 8 | ||

n |

7 x 7 8 x 8 9 x 9

n – 15 | n – 14 | n - 13 |

n - 7 | ||

n | ||

n-19 | n- 18 | n-17 |

n-9 | ||

n |

9 = | n | + | n - 7 | + | n - 15 | + | n - 14 | + | n – 13 |

8 = | n | + | n - 8 | + | n - 17 | + | n - 16 | + | n - 15 |

7 = | n | + | n - 9 | + | n - 19 | + | n - 18 | + | n - 17 |

G = | n | + | n - g | + | n – 2g + 1 | + | n - 2g | + | n – 2g –1 |

n = T - number

G = Grid size

General Rule

I will now solve the rule.

T-total=n + n–g + n–2g +1 + n–2g + n-2g – 1 = 5n – 7g

The rule to find any t – total in any grid size is 5n – 7g.

Testing overall rule

To prove that my rule does work I will test it on 6 x 6 and 10 x 10 grids

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 |

T- n = 32

5n – 7g =(5 x 32) 160 – (7 x 6) 42 = 118 (T-t)

T-t = 32 + 26 + 19 + 20 + 21 = 118

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 |

T-n = 19

5n – 7g = (5 x 19) 95 – (7 x 5) 35 = 60 ( T- t)

T – t = 19 + 14 + 8 + 9 + 10 = 60

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

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