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• Level: GCSE
• Subject: Maths
• Word count: 3627

# Find the relationship between the T-total and the T-number.

Extracts from this document...

Introduction

T-Totals

Aim:In this Investigation I will try and find the relationship between the T-total and the T-number. By changing the size of the grid and translating the T-shape to different positions in the grid I will find the pattern and rule for the T-Total and T-number.

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The Total of the Numbers inside the T-shape is 1+2+3+11+20= 37

The number at the bottom is know as the T-number                T-Total

The T-number for this T-shape is 20

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

If you take the other Numbers in the T-shape away from the T-number you get a T-shape like this.

 T-19 T-18 T-17 T-9 T

## = 5T-63

2) Testing

For T-Total I will use the letter X

For T-Number I will use the letter T

So X = 5T-63

T=20

X=5x20-63

= 100-63

= 37

For this T-Shape the T-Number is 20 and the T-Total is 37

Here is the difference once more in the T-shape:

 T-19 T-18 T-17 T-9 T

This shows the difference T = T-number

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5T + 7 = T-Total

5x70 + 7 = 357

Check

T-Number = 70

T-Total = 70+71+72+63+81 = 357

If we were to put the T-shape diagonally on the grid we find that the same rule applies gain apart from you cannot use the 2nd rule where you multiply the grid size by seven.

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T-number = 33

T-Total = 7+17+26+25+33 = 109

Difference

33-25 = 8

33-7 = 26

33-17 = 16

33-27 = 6

Total = 56

5T+56 = T-total

5x33-56 = 109

the reverse shape should be changed to a plus.

T-Number = 13

T-Total = 19+29+39+21+13 = 121

5T + 56 = T-total

5x13+56 = 121

 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

T-Number = 32

T-total = 32+42+52+60+44 = 230

Formula

5T+70 = T-total

Difference

42-32 = 10

52-32 = 20

60-32 = 28

44-32= 12

TOTAL = 70

5 x 32 + 70 = 230

If you reverse this procedure however the Formula now will be 5T-70

32-42 = -10

32-52 = -20

32-60 = -28

32-44 = -12

TOTAL = -70

5 x 32 -70 = 90

 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
 T-Square T-number T-total Difference 1+2+3+11+20 20 37 A+17 2+3+4+12+21 21 42 A+21 3+4+5+13+22 22 47 A+25 4+5+6+14+23 23 52 A+29 5+6+7+15+24 24 57 A+33

As we can see from this table the difference between the t-number and the T-total goes up up four each time. So I have decided to put the difference in Algebraic terms: -

A = T-number

A + 17 = 37

A + 21 = 42

A + 25 = 47

A + 29 = 52

A + 33 = 57

 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
 T-Square T-number T-total Difference 1+2+3+11+20 20 37 A+17 10+11+12+20+29 29 82 A+53 19+20+21+29+38 38 127 A+89 37+38+39+47+56 56 217 A+161 46+47+48+56+65 65 262 A+197

 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
 T-Square T-number T-total Difference 19+20+21+11+2 2 73 A+71 20+21+22+12+3 3 78 A+75 21+22+23+13+4 4 83 A+79 22+23+24+14+5 5 88 A+83 23+24+25+15+6 6 93 A+87

 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
 T-Square T-number T-total Difference 19+20+21+11+2 2 73 A+71 28+29+30+20+11 11 118 A+107 37+38+39+29+20 20 163 A+143 46+57+48+38+29 29 218 A+199 55+56+57+47+38 38 253 A+215

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Conclusion

2">

19+28+37+29+30

30

143

A+113

28+37+46+38+39

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188

A+149

37+46+55+47+48

48

233

A+185

 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
 T-Square T-number T-total Difference 1+10+19+11+12 12 53 A+41 2+11+20+12+13 13 58 A+45 3+12+21+13+14 14 63 A+49 4+13+22+14+15 15 68 A+53 5+14+23+15+16 16 73 A+57

 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
 T-Square T-number T-total Difference 3+12+21+11+10 10 57 A+47 12+21+30+20+19 19 102 A+83 21+30+39+29+28 28 147 A+119 30+39+48+38+37 37 192 A+155 39+48+57+47+46 46 237 A+191

 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
 T-Square T-number T-total Difference 3+12+21+11+10 10 57 A+47 4+13+22+12+11 11 62 A+51 5+14+23+13+12 12 67 A+55 6+15+24+14+13 13 72 A+59 7+16+25+15+14 14 77 A+63

 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
 T-Square T-number T-total Difference 13+21+29+11+1 1 75 A+74 22+30+38+20+10 10 120 A+110 31+39+47+29+19 19 165 A+146 40+48+56+38+28 28 210 A+182 49+57+65+47+37 37 255 A+218

 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
 T-Square T-number T-total Difference 13+21+29+11+1 1 75 A+74 14+22+30+12+2 2 80 A+78 15+23+31+13+3 3 85 A+82 16+24+32+14+4 4 90 A+86 17+25+33+15+5 5 95 A+90

 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
 T-Square T-number T-total Difference 19+11+3+21+31 31 85 A+54 28+20+12+30+40 40 130 A+90 37+29+21+39+49 49 175 A+126 46+38+30+48+58 58 220 A+162 55+47+39+57+67 67 265 A+198

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