# Math Coursework-T-Total Investigation

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Introduction

Math Coursework-T-Total Investigation

Part 1

Grid 1

Let n be 23

T number = n

T-total= (n-19)+(n-18)+(n-17)+(n-9)+n

= 5n-63

To get the formula 5n-63, let n be 23( the T number), and calculate the T-total in terms of n, as shown above.

### Part 1 continued

## Grid 2

Let n be 35

T number = n

T-total=(n-19)+(n-18)+(n-17)+(n-9)+n

= 5n-63

I moved the T-shape to other position on the grid and I get the same result, 5n-63, so it proves that the relationship between the T-number and the T-total can be represented by the formula 5n-63.

Part 2

T-shape1

#### T-shape2

##### Grid 1

T-shape 1: Let n be 25

T number = n

T- total= (n-21)+(n-20)+(n-19)+(n-10)+n

=5n-70

### T-shape 2: Let n be 78

###### T number = n

T-total= (n-21)+(n-20)+(n-19)+(n-10)+n

=5n-70

The grid size is 10x10. I move the T-shapes to different position on the grid but I get the same result, 5n-70.

T-shape 1

T-shape 2

Grid 2

###### T-shape1: Let n be 15

T number = n

T total= (n-9)+(n-8)+(n-7)+(n-4)+n

= 5n-28

T-shape2: Let n be 10

T number = n

T total= (n-9)+(n-8)+(n-7)+(n-4)+n

= 5n-28

The grid size is 4x4.

Middle

1) When the T number is less than 20 or when the T number is 1,10,19,28,37,46,55,64 and 73.

(Example is shown on the grid)

If the T number is 19, there will be no grid left on the left hand side of the T-shape and so that part will be stuck out. The same happen to all the numbers in the first row of the grid.

2) When the T number is less than 18

(Example is shown on the grid)

If the T number is 16, there is no grid left on the top part and so the top part will be stuck out.

Part 2 continued

T-shape1

T-shape2

Grid 1

T-shape 1: T-shape 2:

T number = 21 T number =35

Vector: (-2)

2

The difference between the two T numbers: 14

Grid size: 6

Hypothesis:

T number grid size(g) Vector Difference of T no.s

21 6 -2(a) 14

- 2(b)

Difference of T numbers= g(a)+b

Example: 6x2+2=14

I found out that you could never get this result if you multiply the grid size with the negative sign. Also, this negative sign doesn’t indicate anything but the action, moving down.

Part 2 continued

Translation 1

Translation 2

Conclusion

L.H.S.= 23+29+35+34+36

= 157

R.H.S.= 5n+42

=5(23)+42

=157

∵L.H.S.=R.H.S.

∴The reverse in the minus sign has worked.

The two examples above show that the formula for mirror image is 5n+7g.

Part 3 continued-Rotation (90° clockwise or anti-clockwise)

T- shape 1

#### T-shape2

T-shape 3

T number of T-shape1 and T-shape2: 21

T total of T-shape1 and T-shape2: 42/112 (difference=70)

T number of T-shape1 and T-shape2: 67

T total of T-shape1 and T-shape2: 272/342 (difference=70)

T number of T-shape1 and T-shape2: 67

T total of T-shape1 and T-shape3: 272/328 (difference=56)

T number of T-shape1 and T-shape2: 21

T total of T-shape1 and T-shape3: 42/98 (difference=56)

The formula for rotation of 90° clockwise: 5n-7g+70

The formula for rotation of 90° anticlockwise: 5n-7g+56

*These formulas only work for a 9x9 grid*

Part 3 continued-formula for rotation for all grid size

Take a 6x6 grid square as example:

The T number of both T shapes: 21

The T total of the T-shape2: 112

The difference between the 2 T-totals: 49

After my analysis on the two grids of different size, I found out that the formula for clockwise rotation is 5n-7g+7(g+1) while the formula for anti-clockwise rotation is 5n-7g+7(g-1).

Part 3 continued-Combination of transformation-Translation+Rotation

Vector(3,-1)

This student written piece of work is one of many that can be found in our GCSE T-Total section.

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