French Holidays
4 star(s)
T-shape - investigation
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Introduction
Math’s GCSE Course work
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10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
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37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 |
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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 |
Task
Translate the T shape to different positions on the grid.
- Investigate the relationship between the T-total and the T-number.
- Use different sized grids. Translate the T shape to different position and investigate the relationships between the T-total the T-numbers and the grid size.
- Use grids of different sizes again. Try other transformation and combinations of transformations. Investigate relationships between the T-total, the T-total, the gridsize and the transformations.
Part 1
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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 | 20 | 30 | 40 | 50 | 60 | 70 | 80 |
T-total | 37 | 87 | 137 | 187 | 237 | 287 | 337 |
From my results I would say that every time the T- number goes up by 10 the T-total goes up by 50.
I am now going to try and work out a formula for the relationships between the T-total and the T-number for anywhere on the grid. I am going to do this by working out all of the numbers in relation to N (T-number.)
1 | 2 | 3 | n-19 | n-18 | n-17 | |
11 | = | n-9 | ||||
20 | n |
Now I will add up all of the N’s and all of the numbers to hopefully get a formula.n+n-9+n-18+n-19+n-17= 5n-63.
Middle
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1 | 2 | 3 | n-21 | n-20 | n-19 | |
12 | = | n-10 | ||||
22 | n |
62 | 63 | 64 |
73 | ||
83 |
N=83 83x5-70=345
62+63+64+73+83=345
36 | 37 | 38 |
47 | ||
57 |
N=57 57x5-70=215
57+47+37+36+38=215
My formula does work anywhere on the grid so now I am going to work out the formula for 8 by 8 and see if I can find something that relates them all together.
This is the first T-shape on an 8by 8 grid:
1 | 2 | 3 | n-17 | n-16 | n-15 | |
10 | = | n-8 | ||||
18 | n |
Adding up all the Ns and the numbers I get: 5n-56. So if N is 18 then the formula becomes 18x5-56=34 and 1+2+3+10+18=34. I shall now test the formula on other points in the grid.
41 | 42 | 43 |
50 | ||
58 |
With this shape n is 58: 58x5-56=234 and 58+50+42+41+43=234. I shall try it with one more shape just to make sure.
14 | 15 | 16 |
23 | ||
31 |
N is 31 so the formula is: 31x5-56=99 and 31+23+14+15+16=99. I have proved now that the formula works for this 8by 8 grid. I am now going to put all of my results into a table and see if there is a link for them.
8 by 8 | 9 by 9 | 10 by 10 |
5n-56 | 5n-63 | 5n-70 |
By looking at my results I can see that each time the grid size goes up by 1 the formula changes as the second number goes up by 7. And also the grid size (g) x7 =the second number. I predict that the formula will be 5n-7g. I am going to check it by working it out the same way as I did the other formulae by working out the T-shape in relation with N but this time I’m going to add G (grid size).
1 | 2 | 3 | n-2g-1 | n-2g | n-2g+1 | |
10 | = | n-g | ||||
18 | n |
Adding up all of the Ns and the Gs and the numbers I get: 5n-7g. I am now going to test this formula on the T-shapes I used before. Firstly I will test the 8 by 8 grids.
1 | 2 | 3 |
10 | ||
18 |
N=18 and G=8 so the formula is 5x22-7x8=34 and 18+10+2+1+3=34
41 | 42 | 43 |
50 | ||
58 |
N=58 G=8. 5x58-7x8=234. 58+50+42+41+43=234.
14 | 15 | 16 |
23 | ||
31 |
N=31 g=8. 31x5-7x8=99 31+23+14+15+16=99
I have now proved that the formula works on an 8 by 8 grid now ill try it on 9 by 9.
9 by 9
55 | 56 | 57 |
65 | ||
74 |
N=74 5x74-9x7=307
74+65+56+55+57=307
1 | 2 | 3 |
11 | ||
20 |
5x20-7x9=37 20+11+2+3+1=37
11 | 12 | 13 |
21 | ||
30 |
Conclusion
Normal= 5n-7g Rotated 90o= 5n+7
To see a link I need one more formula so I will work out the formula for a shape rotated by 180o.
34 | |||||
42 | |||||
49 | 50 | 51 | |||
n | |||||
n+g | |||||
n+2g-1 | n+2g | n+2g+1 | |||
This become equal to 5n+7g I am going to test this on 2 shapes from each grid size to check if it works.
First I will try 8 by 8.
34 | ||
42 | ||
49 | 50 | 51 |
34x5+7x8=226 and the sum of T-shape is 34+42+40+49+51=22
22 | ||
30 | ||
37 | 38 | 39 |
22x5+7x8=166 and the T-total is 22+30+38+37+39=166
Now I will try it 9 by 9.
2 | ||
11 | ||
19 | 20 | 21 |
2x5+9x7=73 and 2+11+20+21+19=73
34 | ||
43 | ||
51 | 52 | 53 |
5x34+9x7=233 and 34+43+51+52+53=233 so the formula works on 9 by 9 I will now try it on 10 by 10.
54 | ||
64 | ||
73 | 74 | 75 |
5x54+7x10=340 and 54+64+74+73+75=340
78 | ||
88 | ||
97 | 98 | 99 |
5x78+7x10=460 78+88+98+99+97=460 I have now proved that my formula works on all sized gids so now I shall compare the three formulae I have.
Normal=5n-7g rotated 90o clockwise=5n+7 rotated 180o=5n+7g
Degrees | Formulae | |
0 | 5n-7g | 5n+7(-g) |
90 | 5n+7 | 5n+7(1) |
180 | 5n+7g | 5n+7(g) |
270 | 5n-7 | 5n+7(-1) |
I predict that 270 is 5n+7. I will now check.
1 | n-g-2 | |||||
9 | 10 | 11 | = | n-2 | n-1 | n |
17 | n+g-2 |
5n-7so I was correct.
This student written piece of work is one of many that can be found in our GCSE T-Total section.
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