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  • Level: GCSE
  • Subject: Maths
  • Word count: 3379

T Totals. I will try five different T shapes in the nine by nine grid and record the results as I a go along.

Extracts from this document...

Introduction

T TOTALS

9x9 Grid

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I will try five different T shapes in the nine by nine grid and record the results as I a go along. I will move them all along a straight line so my results do not differ too much.

These are my 5 T shape sequences:

  1. 1+2+3+11+20 = 37                              
  2. 2+3+4+12+21 = 42
  3. 3+4+5+13+22 = 47
  4. 4+5+6+14+23 = 52
  5. 5+6+7+15+24 = 57

I will show what the t number and the t totals were in each of the T shapes:

T Number

T Total

20

37

21

42

22

47

23

52

24

57

I noticed that between each t total there is a difference of 5, so therefore I know that it is using the five timetables, This can be shown as 5x.

X = T Number

I then multiplied five by twenty, as it is the 20th term in the sequence. I took thirty-seven away from it to give me my difference between them.

20 x 5 = 100

100 – 37 = 63

I got a difference of 63. This finished of my formula for working out the T total in a 9 by 9 grid. I ended up with the formula T = 5x – 63. T represents the T total.

I will check it once again to check that it does work on all the terms in the sequence.

CHECK:

21 x 5 = 105

105 – 63 = 42

This shows that my formula for working out the T total in a 9 by 9 grid is correct.

After looking at all the numbers in a T shape in a 9 by 9 grid, I found that all the numbers fitted in to this format.

I came up with this general upright format as I found that the number on top of X was usually 9 taken away from X and the box on top of that was usually double that.

...read more.

Middle

10 by 10 Grid
  1. 1+2+3+12+22 = 40    T= 40  X = 22

After testing this T shape in the 10 by 10 grid I found that the general upright format for a T shape in a 10 by 10 grid is:

I came up with this general upright format because I noticed that the number above X was 10 taken away from it and the number above that was double that difference making it 20 below X. The numbers on either side of that were 21 taken away from X and 19 taken away from X.

I will add the numbers to give me a T total as this what I had to do previously.

X + X – 10 + X – 20 + X – 21 + X – 19 = T

After it is simplified it becomes 5X – 70 = T

This gives me a formula for working out the T total in any upright T shape in a 10 by 10 grid. I will check this formula by checking it.

CHECK:

5 x 22 = 110

110 – 70 = 40

This gives me the same T total as the one I tried before. I will check it once more.

CHECK:

                                                              5 x 55 = 275             34 + 35 + 36 + 45 + 55 = 205

                                              275 – 70 = 205

I multiplied it by 55 as it is the T number and according to the formula, the T number is meant to be multiplied by 5.

These tests show that my formula for working out the T total in an upright T shape in a 10 by 10 grid is correct.

I will now do the same method to work out the formula for working out the T total in an upright T shape in a 12 by 12 grid

12 by 12 Grid

1) 1+2+3+1+26 = 46     T = 46    X = 26

After observing different upright T shapes in a 12 by 12 grid I found that the format was this:

...read more.

Conclusion

270 DEGREES

I have come up with the following format for a T shape that has been rotated 270 degrees clockwise and had a vector applied to it:

I came to this T shape because as you have moved the X across the X axis it will have a new value and that will be the value of the vector for the x axis, but then as you apply the second part of the vector you have to take away the movement in the y axis multiplied by the length of the grid to come to your new T number. Whatever occurs to the T number happens to the rest of the T shape also and that is why they all have the same endings.

I will add the terms in the new T shape format to give me a T Total:

X + A – BL + X – 1 + A – BL + X – 2 + A – BL + X – 2 – L + A – BL + X – 2 + L + A – BL = T

This turns into T = 5X + 5A – 5BL – 7.

From applying vectors to the four different T shapes I have come up with the following formulas along with the formulas without vectors applied to them.

ROTATION

BEFORE VECTORS

AFTER VECTORS

0 DEGREES

T = 5X – 7L

T = 5X + 5A – 5BL – 7L

90 DEGREES

T = 5X + 7

T = 5X + 5A – 5BL + 7

180 DEGREES

T = 5X + 7L

T = 5X + 5A – 5BL + 7L

270 DEGREES

T = 5X - 7

T = 5X + 5A – 5BL  - 7

After looking at these formulas, I found that once a vector had been applied they change. In the new formulas 5A – 5BL has been added to them. This is because the formula for applying a vector is X + A – BL. As there are 5 terms in the T shape the formula has to be applied 5 times. This brings me to the formulas for the T total once having added a vector. So, basically the formula stays the same but has 5A – 5BL added to it.

...read more.

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