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# Maths- T-Totals

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Introduction

Contents

• Aim and Method
• Grid 1: 9 by 9
• Across the Grid
• Gown the grid Diagonally
• Down the grid
• Across the Grid:
• Grid 2: 5 by 5
• Grid 3: 6 by 6
• Grid 4: 7 by 7
• Grid 5: 10 by 10
• Grid 6: 11 by 11
• Extension: Transformations Grid 7: 9 by 9 (Going across)
• T at 180 degrees
• T at 90 degrees
• T at 270 degrees
• Conclusion

Aim:

I will investigate the relationship between the T-total (all the numbers in the T added up) and T-number (the number at the end of the T), using grids of different sizes to translate the t-shape to different positions within the grid. I will try different combinations of transformations later within the coursework.

Method:

Part 1

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

Part 2

Use grids of different sizes. Translate the T-shape to different positions. Again investigate the relationship within the different sizes of the grids.

Part 3

Use grids of different sizes again. Try other transformations and combinations of transformations. Investigate relationships between the T-Total, and T-Numbers on the grid of different sizes.

Grid 1: 9 by 9 – Across (      )

In the T below, I noticed that the number at the end of the T (the T-Number) was 20. I will now go across this grid writing down the T-Number on one side and the T-Total in the other aswell showing my working by putting all the numbers within that T.I will also write the trend of the T-Number and T-Total in the table and express I in the nth term after I have acquired results from various points in the grid.

 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 T-Total Trend of         T-Number Trend of       T-Total 20 1+2+3+11+20= 37 +1 +5 21 2+3+4+12+21= 42 +1 +5 22 3+4+5+13+22= 47 +1 +5 23 4+5+6+14+23= 52 +1 +5 24 5+6+7+15+25= 57 +1 +5 25 6+7+8+16+25= 62 +1 +5 26 7+8+9+17+26= 67 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 47 28+29+30+38+47= 172 +1 +5 48 29+30+31+39+48= 177 +1 +5 49 30+31+32+40+49= 182 +1 +5 50 31+32+33+41+50= 187 +1 +5 51 32+33+34+42+51= 192 +1 +5 52 33+34+35+43+52= 197 +1 +5 53 34+35+36+43+53= 202 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 74 55+56+57+65+74= 307 +1 +5 75 56+57+58+66+75= 312 +1 +5 76 57+58+59+67+76= 317 +1 +5 77 58+59+60+68+77= 322 +1 +5 78 59+60+61+69+78= 327 +1 +5 79 60+61+62+70+79= 332 +1 +5 80 61+62+63+71+80= 337 +1 +5

Formula Expressed in the nth term

Middle

16

3+4+5+10+16= 38

+1

+5

17

4+5+6+11+17= 43

+1

+5

 T-Number T-Total Trend of         T-Number Trend of       T-Total 26 13+14+15+20+26= 88 +1 +5 27 14+15+16+21+27= 93 +1 +5 28 15+16+17+22+28= 98 +1 +5 29 16+17+18+23+29= 103 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 32 19+20+21+26+32= 118 +1 +5 33 20+21+22+27+33= 123 +1 +5 34 21+22+23+28+34= 128 +1 +5 35 22+23+24+29+35= 133 +1 +5

Formula Expressed in the nth term

As the tables show every time the T-Number increases by 1 the T-Total increases by 5. I will express in the nth term by using these common differences and start on the basis of ‘if n=1’ but change accordingly if appropriate.

At the T-Number, the common difference is 1.

 n= 1 2 3 4 T= 14 15 16 17

For example take 14 and 15 subtract them (15 -14) and you get 1.

Therefore the formula of this is 1n ± C = T-Number/ n ± C = 14, 14 -1=13.

Therefore nth term= n+13=14

At the T-Total, the common difference is 5.

 n= 1 2 3 4 T= 28 33 38 43

For example take 28 and 33 subtract them (33-28) and you get 5. Therefore the formula of this is 5n ± C = T-Total/ 5n ± C = 28, (if n=14, the T- Number) 14 x 5= 70 – 28=42

Therefore nth term=5n-42=28.

Across- Grid 4: 7 by 7

 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-Number T-Total Trend of         T-Number Trend of       T-Total 16 1+2+3+9+16=31 +1 +5 17 2+3+4+10+17= 36 +1 +5 18 3+4+5+10+18= 41 +1 +5 19 4+5+6+11+19= 46 +1 +5 20 5+6+7+13+20= 51 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 30 15+16+17+23+30= 101 +1 +5 31 16+17+18+24+31= 106 +1 +5 32 17+18+19+25+32= 111 +1 +5 33 18+19+20+26+33= 116 +1 +5 34 19+20+21+27+34= 122 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 44 29+30+31+37+44= 32 +1 +5 45 30+31+32+38+45= 37 +1 +5 46 32+33+34+39+46= 42 +1 +5 47 34+35+36+40+47= 47 +1 +5 48 35+36+37+41+48= 52 +1 +5

Formula Expressed in the nth term

As the tables show every time the T-Number increases by 1, the T-Total increases by 5. I will express in the nth term by using these common differences and start on the basis of ‘if n=1’ but change accordingly if appropriate.

 n= 1 2 3 4 T= 16 17 18 19

At the T-Number, the common difference is 1.

For example take 16 and 17 subtract them (17 -16) and you get 1.

Therefore the formula of this is 1n ± C = T-Number/ n ± C = 16, 16 -1=15.

Therefore nth term= n+15=16.

At the T-Total, the common difference is 5.

 n= 1 2 3 4 T= 31 36 41 46

For example take 21 and 36 subtract them (31-36) and you get 5. Therefore the formula of this is 5n ± C = T-Total/ 5n ± C = 31, (if n=16 (T-Number) ) 16 x 5= 80,  80 -16= 64

Therefore nth term= 5n-64=22

Across - Grid 5: 10 by 10

 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 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
 T-Number T-Total Trend of         T-Number Trend of       T-Total 22 1+2+3+12+22= 40 +1 +5 23 2+3+4+13+23= 45 +1 +5 24 3+4+5+14+24= 50 +1 +5 25 4+5+6+15+25= 55 +1 +5 26 5+6+7+16+26= 60 +1 +5 27 6+7+8+17+27= 65 +1 +5 28 7+8+9+18+28= 70 +1 +5 29 8+9+10+19+29= 75 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 52 31+32+33+42+52= 130 +1 +5 53 32+33+34+43+53= 135 +1 +5 54 33+34+35+44+54= 140 +1 +5 55 34+35+36+45+55= 145 +1 +5 56 35+36+37+46+56= 150 +1 +5 57 36+37+38+47+57= 155 +1 +5 58 37+38+39+48+58= 160 +1 +5 59 38+39+40+49+59= 165 +1 +5
 T-Number T-Total Trend of         T-Number Trend of       T-Total 92 71+72+73+82+92= 130 +1 +5 93 72+73+74+83+93= 135 +1 +5 94 73+74+75+84+94= 140 +1 +5 95 74+75+76+85+95= 145 +1 +5 96 75+76+77+86+96= 150 +1 +5 97 76+77+78+87+97= 155 +1 +5 98 77+78+79+88+98= 160 +1 +5 99 78+79+80+89+99= 165 +1 +5

Formula Expressed in the nth term

As the tables show every time the T-Number increases by 1, the T-Total increases by 5. I will express in the nth term by using these common differences and start on the basis of ‘if n=1’ but change accordingly if appropriate.

At the T-Number, the common difference is 1.

 n= 1 2 3 4 T= 22 23 24 25

For example take 22 and 23 subtract them (22 -23) and you get 1.

Therefore the formula of this is 1n ± C = T-Number/ n ± C = 22, 22 - 1=21

Therefore nth term= n+21=22

At the T-Total, the common difference is 5.

 n= 1 2 3 4 T= 40 45 50 55

Conclusion

• For a 9 by 9 grid – T-number= n+19=20, T-Total=5n-63=37
• For a 5 by 5 grid – T-number= n+16=17, T-Total=5n-10=75
• For a 6 by 6 grid – T-number= n+13=14, T-Total=5n-42=28
• For a 7 by 7 grid – T-number= n+15=16, T-Total=5n-64=22
• For a 10 by 10 grid – T-number= n+21=22, T-Total=5n-70=40
• For a 11 by 11 grid – T-number= n+23=24, T-Total=5n-79=41
•  For a 180 degree transformation at a 9 by 9 grid - T-number=n+9=10, T-Total=5n+7=57
• For a 90 degree transformation at a 9 by 9 grid - T-number= n+1=2, T-Total=5n+63=73
• For a 270 degree transformation at a 9 by 9 grid – T-number=n+11=12, T-Total= 5n-7=53

As you can see this supports my prediction showing all T-Numbers include 1n/n and T-totals include 5n.

From my results, I can also predict that T-Shapes going down a grid would have a nth term including 9n for T-Numbers and 45n for T-totals. This is shown in my results by: the nth term for T-Numbers going down a 9 by 9 grid is 9n+11=20 and the nth term for a T-Total going down a 9 by 9 grid is 45n-8=37.

Also going diagonally down a grid I can predict that T-Shapes have a nth term including 10n for T-Numbers and 50n for T-Totals. This is shown by in my results by: the nth term for T-Numbers going diagonally down a 9 by 9 grid is 10n+10=20 and the nth term for a T-Total going down a 9 by 9 grid is 50n -13=37.

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