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Using grids of different sizes, try other transformations and combinations of transformations. Also, to investigate relationships between the T-total, the T-numbers, the grids size and the transformations.

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Introduction

MATH COURSEWORK

T- totals

Part III

Aim:

        Using grids of different sizes, try other transformations and combinations of transformations.  Also, to investigate relationships between the T-total, the T-numbers, the grids size and the transformations.

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Rotation:

A T-shape is rotated about the p number at 90 degrees, 180 degrees and 270 degrees.

T = 12 + 13 + 14 + 22 + 31

T = 92

T = 24 + 33 + 42 +32 +31

T = 162

T = 40 + 48 + 49 + 50 + 31

T = 218

T = 20 + 29 + 38 + 30 + 31

T = 148

We already know the formula for a 0º rotation.  Since it does not move, the formula is the same as the one that I have previously been using:

5p – 7g

I found this formula by making a diagram and then, filling it in appropriately (making many small equations for each box).  I then added all the small equations together to make one big, final formula.  I can use this method again to find out the formula for a 90º, 180º and 270º rotation.

There were 5 p’s, 7 g’s, (-1) and (+1).  The (-1) and (+1) cancelled each other out, leaving 5p, and 7g.  7g was subtracted from 5p because the sum of all the g’s was – 7, therefore giving me the final equation: 5p – 7g

90º Rotation:

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Observations:

  • The left box is p, the position of the T-shape.  
  • The three boxes going across are all consecutive (they are one after another), which means if the first box was p, the next box is p + 1
  • Which also means that the box after that is p + 2
  • The top box subtractsg (the grid size) because it is decreasing.  But, subtracting the grid size means that the box is right above.  For example, in this example:

p – g  = 31 – 9

= 22

22 is directly above 31 because each level increases or decreases by the grid size number.  So, since the number I want it two spaces over, I add 2.  The equation for that box is p – g + 2

  • The bottom box adds g (the grid size) because it is increasing on the grid.  But, since every level increased or decreased means you have to add or subtract the grid size number, adding g (which is 9 in this example) would give me the box directly underneath p.  In this example:
...read more.

Middle

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6 x 6 grid:

Using the Formula:

T = 5p + 7

T = 5(22) + 7

T = 110 + 7

T = 117

T = 22 + 23 + 14 + 18 + 30

T = 117

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7 x 7 grid:

Using the Formula:

T = 5p + 7

T = 5(24) + 7

T = 120 + 7

T = 127

T= 24 + 25 + 26 + 19 + 33

T = 127

180º Rotation

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Observations:

  • The top box p, is the position of the T-shape
  • The box right underneath is one level down, which means the grid size number (g) must be added from p (because this grid increases as it goes down).  Therefore, the equation for this box is p + g
  • The box directly underneath that is two levels down from p, which means you multiply the grid size (g) by 2 (because you moved 2 levels), and then add it to p.  The equation for this box is p + 2g
  • Each time you go up a level the number decreases by the gird size, so therefore, each time you go down a level, the number increases by the grid size
  • The three bottom boxes are also two levels down from p, which means you multiply the grid size number (g) by 2, and then add it to p (because it has decreased levels on the grid).  Since these three numbers are consecutive numbers, the box on the left would be one less than the middle (p + 2g – 1), and the box on the right would be one more than the middle box ( p + 2g +1)

Working out the Formula:

Adding up all these small equations, I can come up with a final formula for a 180º rotation about p, to find T if we only know p.

T = p (p + g) + (p +2g) + (p + 2g – 1 ) + (p + 2g + 1)

T = 5p + 7g

*the (+1) and the (-1) cancel each other out

Testing the Formula:

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5 x 5 grid:

Using the formula:

T = 5p + 7g

T = 5(12) + 7(5)

T = 60 + 35

T = 95

T = 12 + 17 + 21 + 22 + 23

T = 95

6 x 6 grid:

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...read more.

Conclusion

p – 1 (it is one less than p because it is 1 box over) and the next one over would be p – 2  (it is 2 less than p because it is 2 boxes over)The top box is one level up from p, which means it decreases.  Subtract the grid number (g) from p because it has increased one level, which means it has decreased (because the grid increases as it goes down).  This equation is p – g – 2 The bottom box is one level down from p, which means that it increases.  Add the grid number to p, which gives you p + g – 2

Working out the Formula:

I can use these small equations and add them together, to find the final formula for a 270º Rotation about p, to find T if we only know p.

T = p + (p – 1 ) + (p – 2) + (p – g – 2) + (p + g – 2)

T = 5p – 7

*the (+g) and (-g) cancel each other out

Testing the Formula:

5 x 5 grid:

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Using the formula:

T = 5p – 7

T = 5(14) – 7  

T = 70 – 7

T = 63

T = 7 + 12 + 17 + 13 + 14

T = 53

6 x 6 grid:

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Using the Formula:

T = 5p – 7

T = 5(22) – 7

T = 110 – 7

T = 103

T = 14 + 20 + 26 + 21 + 22

T = 103

7 x 7 grid:

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Using the Formula:

T = 5p – 7

T = 5(24) – 7

T = 120 – 7

T = 113

T = 15 + 22 + 29 + 23 + 24

T = 113

Formulas for rotation about p:

Formula

0º Rotation

5p – 7g

90º Rotation

T = 5p + 7

180º Rotation

T = 5p + 7g

270º Rotation

T = 5p – 7

...read more.

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