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• Level: GCSE
• Subject: Maths
• Word count: 1902

# T-shape - investigation

Extracts from this document...

Introduction

Math’s GCSE Course work

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Translate the T shape to different positions on the grid.

1. Investigate the relationship between the T-total and the T-number.
2. 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.
3. 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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 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.

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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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