# T-total Coursework.

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Introduction

Lauren Willis

T-total Coursework

In this piece of coursework im going to investigate the relationship between the T-total and the T-number. I will do this by using different grid sizes and I will translate the T-shape to different positions on the grid. I will also try other transformations and combinations of transformations.

Here im going to try one T-shape on this 9 by 9 grid and see if I can find a relationship between the T-total and the T-number.

T-total=1+2+3+11+20= 37

T-number= 20

T-total=4+5+6+14+23= 52

T-number= 23

T-total=7+8+9+17+26= 67

T-number= 26

T-total=28+29+30+38+47= 172

T-number= 47

T-total=31+32+33+41+50= 187

T-number= 50

T-total=34+35+36+44+53= 202

T-number= 53

T-total=55+56+57+65+74= 307

T-number= 74

T-total=58+59+60+68+77=332

T-number= 77

T-total=61+62+63+71+80= 337

T-number= 80

This was too difficult to see a pattern so I decided to only use the first three rows of the 9 by 9 grid and then I will see if I can see a pattern.

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 |

T-total =1+2+3+11+20 = 37

T-number= 20

T-total= 2+3+4+12+21 = 42

T-number=21

T-total= 3+4+5+13+22= 47

T-number=22

T-total= 4+5+6+14+23= 52

T-number=23

T-total= 5+6+7+15+24=57

T-number=24

T-number | T-total |

20 | 37 |

21 | 42 |

22 | 47 |

23 | 52 |

24 | 57 |

From this I can see that each time the T-number goes up by one the T-total goes up by five. This is because there are five numbers in the T-shape and they are all being moved up each time. I think the number five will be important in this investigation.

I also looked at the relationship between the T-total and the T-number by finding the difference between them.

T-total | T-number | Difference |

37 - | 20 = | 17 |

42 - | 21 = | 21 |

47 - | 22 = | 25 |

52 - | 23 = | 29 |

57 - | 24 = | 33 |

Middle

17

35 -

14 =

21

From this I can see a pattern that each time the T- number goes up by one. The difference between the T- numbers to the T- total goes up by four.

Because five is an important number in this investigation I will multiply the T- number by 5 times table then take away the T- total to see if this shows a pattern.

T-number | T-total | Difference |

(12x5)=65 - | 25= | 35 |

(13x5)=65 - | 30= | 35 |

(14x5)=70 - | 35= | 35 |

It was a five by five grid and 35 is in the five times tables is it is 7x5. So seven is important in this investigation because before I found that the difference was 63 and that was 7x9.

From this I can see my formula will work

5n – T-total = 7x5 (grid size multiplied by seven)

5x12– 60=35

5x13- 60=35

5x14- 60=35

Now I will try a six by six grid to see if my formula works.

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-total= 1+2+3+8+14=28

T-number= 14

T-total= 2+3+4+9+15=33

T-number=15

T-total= 3+4+5+10+16=38

T-number=16

T-total= 4+5+6+11+17=43

T-number=17

Here is a table to show the T-total and the T-number of this six by six grid.

T-number | T-total |

14 | 28 |

15 | 33 |

16 | 38 |

17 | 43 |

I will now find the difference between the T-total and the T-number.

T-total | T-number | Difference |

28 - | 14 = | 14 |

33 - | 15 = | 18 |

38 - | 16 = | 22 |

43 - | 17 = | 26 |

From this I can see that each time the T-number goes up by one the difference goes up by four.

Because five is an important number in this investigation I will multiply the T-number by five then take away the T-total and see if I can see a pattern.

I will try my formula on each of these T-numbers and multiply them by five then minus 63 to see if I get the T-total for each one.

5n-63

5x20-63=37

5x21-63=42

5x22-63=47

5x23-63=52

5x24-63=57

I will now predict what I will get for 25,26 and 27.

For 25 I think the answer will be 62

For 27 I think the answer will be 67

For 28 I think the answer will be 72

I will now try my formula and see if I am correct.

5n-63

5x25-63=62 The T-total for 25= 6+7+8+16+25 = 62

5x26-63=67 The T-total for 26= 7+8+9+17+26 = 67

5x27-63=72 The T-total for 27= 8+9+10+18+27=72

This proves that my formula does work.

I will now try and see if this formula works on the T-numbers on a five by five grid. Here is a table to show the T-totals and the T-numbers.

Vdvrsbg gbtwrrtg

T-number | T-total |

25 | 12 |

30 | 13 |

35 | 14 |

I am going to use the Formula 5n-35. I will use 5 because there is five numbers in the T-shape and I will minus 35 because that was the difference between the T-total and the T-number each time.

Here is a table to show how I found the difference between the T-number and the T-total. I did this by multiplying the T-number by five each time then I would I minus the T-total and I would get the same answer for the difference each time.

T-number | T-total | Difference |

(12x5)=65 - | 25= | 35 |

(13x5)=65 - | 30= | 35 |

(14x5)=70 - | 35= | 35 |

Conclusion

5x2-63=-53

This will not work so I will try adding instead of misusing

5n+63

5x2+63=73, which was our T-total

T-total= 2+11+19+20+21=73. I will now check this on the other T-totals I did.

5n+63

5x3+63=78 which was the T-total 3+12+20+21+22=78

5x4+63=83 which was the T-total 4+13+21+22+23=83

5x4+63=88 which was the T-total 5+14+22+23+23=88

5x5+63+93 which was the T-total 6+15+23+24+25=93

To make sure my formula works I will try this on a 7by7 grid. I predict my formula will be 5n+49

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 |

T-total=2+9+16+15+17=59

T-number=2

T-total=3+10+16+17+18=64

T-number=3

T-total=4+11+17+18+19=69

T-number=4

T-total=5+12+18+19+20=74

T-number=5

T-total=6+13+19+20+21=79

T-number=6

I will now try the formula I predicted which was 5n+49.

5x2+49=59 which was the T-total 2+9+16+15+17=59

5x3+49=64 which was the T-total 3+10+16+17+18=64

5x4+49=69 which was the T-total 4+11+17+18+19=69

5x5+49=74 which was the T-total 5+12+18+19+20=74

5x6+49=79 which was the T-total 6+13+19+20+21=79

For both of these grids I have found that the number you add is the grid number multiplied by 7 so my formula is 5n+the grid number multiplied by seven. So for a…

5by5 grid the formula will be 5n+35

6by6 grid the formula will be 5n+42

7by7 grid the formula will be 5n+49

8by8 grid the formula will be 5n+56

9by9 grid the formula will be 5n+63

I will now try and find a formula for the T-shape on its side.

5by5grid the formula will be 5n+35

6by6 grid the formula will be 5n+42

7by7 grid the formula will be 5n+9

8by8 grid the formula will be 5n+6

9by9 grid the formula will be 5n+3

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

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