The relationship between the T-number and the T-total is:
Whenever the T-number increases by 1, the T-total increases by 5. Therefore the difference is five, which means that the T-total has a constant difference of 5.
Analysis of Results: To find the formula for the 9x9 grid I took the first T-number, 20 and multiplied it by the difference which is 5. This totalled 100. To get the first T-total, I subtracted 37 from 100 to give me 63. When put into a formula it will look like this:
5n – 63.
I will substitute 3 terms in for n:
The first term:
(5 x 20) – 63
100 – 63
= 37
The seventh term:
(5 x 26) – 63
130 – 63
= 67
The fifteenth term:
(5 x 36) – 63
180 – 63
= 117
10x10 grid:
1+2+3+12+22 = 40
2+3+4+13+23 = 45
3+4+5+14+24 = 50
4+5+6+15+25 = 55
5+6+7+16+26 = 60
6+7+8+17+27 = 65
7+8+9+18+28 = 70
8+9+10+19+29 = 75
Table of Results:
Analysis of Results: In order to find the formula for the T-shape on a 10x10 grid I took the first T-number, 22 and multiplied it by 5(the difference). This totalled 110. To get the T-total I subtracted 70 from 110 which equalled 40. When put into a formula it looks like this:
5n – 70
I will substitute three numbers in for n:
The first term:
(5 x 22) – 70
110 – 70
= 40
The seventh term:
(5 x 28) – 70
140 – 70
= 70
The eighth term:
(5 x 29) – 70
145 – 70
= 75
Step 2:
Now I will investigate the relationship between the T-number and the T- total using a 10x10 grid and rotating the T- shape 90o clockwise.
Method: To find the T-total I have to add all the numbers in the T-shape. For example:
3+13+23+12+11 = 62
4+14+24+13+12 = 67
5+15+25+14+13 = 72
6+16+26+15+14 = 77
7+17+27+16+15 = 82
8+18+28+17+16 = 87
Table of Results:
Analysis of Results: To find the formula for the 10x10 grid with the T-shape rotated 90o clockwise I took the first T-number, 11 and multiplied it by the difference, which is 5. The total was 62. To get the T-total I added 7 to 55, which equalled 62. When put in a formula it looks like this:
5n + 7
I will substitute three numbers in for n:
The first term:
(5 x 11 ) + 7
55 + 7
= 62
The third term:
(5 x 13) + 7
65 + 7
= 72
The sixth term:
(5 x 16) + 7
80 + 7
= 87
Step 3: Now I will investigate the relationship between the T-number and the T-total using a 10x10 grid and rotating the T-shape 180o.
Method: In order to find the T-total I have to add all the numbers in the T-shape. E.g.
23+22+21+12+2 = 80
24+23+22+13+3 = 85
25+24+23+14+4 = 90
26+25+24+15+5 = 95
27+26+25+16+6 = 100
28+27+26+17+7 = 105
29+28+27+18+8 = 110
30+29+28+19+9 = 115
33+32+31+22+12 = 130
34+33+32+23+13 = 135
35+34+33+24+14 = 140
36+35+34+25+15 = 145
37+36+35+26+16 = 150
38+37+36+27+17 = 155
39+38+37+28+18 = 160
40+39+38+29+19 = 165
Table of Results:
Analysis of Results: To find the formula for the 10x10 grid with the T-shape rotated 180o, I took the first T-number, 2 and multiplied it by the difference, which is 5. This totalled 10. To get the T-total I added 70 to 10, which gave me 80.
When put in a formula it looks like this:
5n+70
I will substitute 3 numbers in for n:
The first term:
(5 x 2) + 70
10 +70
= 80
The seventh term:
(5 x 8) + 70
40 + 70
= 110
The fifteenth term:
(5 x 16) + 70
80 + 70
= 150
Conclusion: Using the 10x10 grid I have found a pattern in the formulae.
Upright T: 5n – 70
Rotated 90o: 5n + 7
Rotated 180o: 5n + 70
I noticed that all the formulae started with 5n. However for the T-shape rotated 180 o the function sign was opposite to the upright T-shape i.e. 5n – 70 and 5n + 70. Based on this I predict that a T-shape rotated 270o will have the opposite function sign to the T-shape rotated 90o ie. 5n – 7
To test my prediction, I will look at three T-shapes:
1+11+21+12+13 = 58
2+12+22+13+14 = 63
3+13+23+14+15 = 68
(5 x 13) – 7 = 58
(5 x 14) – 7 = 63
(5 x 15) – 7 = 68
From these results I can see that my hypothesis correct.
To conclude my investigation I have shown that the difference is always 5 for a T-total in any grid size. Also when the T-shape is rotated 180o (reflected) the function signs are opposite.