This is a piece about 'T' Numbers and 'T' totals. The 'T' number is the number at the bottom the of shape, the 'T' total is the total of all the numbers within the shape.

I will, from now, use ‘W’ as the width of grid.

The first thing I found was that the number above the ‘T’ number was the ‘T’ minus the width.

The number above that was the ‘T’ number minus 2 times the width.

The numbers either side of the previous added up to 2T-4W

So a possible formula would be:

N= T + (T-W) + (T-2W) + (T-2W-1) + (T-2W+1)

Simplified this is N = 5T – 7W

I will now test this in various sizes of grid:

Test 1

 

N = 9+10+11+16+22

N = 68

N = N = 5T – 7W

N = 5 x 22 - 7 x 6

N = 68

Test 2

N = 25+26+27+36+46

N = 160

N = N = 5T – 7W

N = 5 x 46 - 7 x 10

N = 160

Test 3

N = 162+163+164+178+193

N =  860

N = N = 5T – 7W

N = 5 x 193 - 7 x 15

N =  860

Now I can again assume the I have found a formula to relate the width of grid, ‘T’ total and ‘T’ number.

The formula for T totals on any grid: N = 5T – 7W

Task 3

I will now investigate firstly translations.

This is a translation (3-3)

The first thing I noticed was that there was relationship (21 difference) between corresponding numbers (see diagram)

I will begin by looking at horizontal translations.

X = the horizontal movement

Y = the vertical movement    

When the shape is moved horizontally each number will increase or decrease by the amount moved.

There are 5 squares in each ‘T’, so the total increase in the numbers is the number or squares (5) times the individual increase (X). In this case X=3

Therefore…

5 x X

5 x 3 = 15

I predict that formula for horizontal translations on any grid is:

N = 5T – 7W + 5X

(Where X can be positive for translating right or negative for translating left)

Test 1

N = 25+26+27+36+46

N =  160

N = 5T – 7W + 5X

N = 5 x 42 - 7 x 10 + 5 x 4

N =  160

I am only testing this once because I will test the formula of translations of vertical and horizontal.  

Now I will find a rule for vertical translations.

X = 0

Y = - 3

Note: The negative Y direction is downwards, however this is the direction in which the Y values increase. This means I will have to use a double negative within my equations to make the outcome positive for downwards movements, as it should be.

The number on the row below a certain number is the number above plus the width.

So, 2 rows below it will be the number plus two times the width. And if it is ‘Y’ rows bellow it will be the number plus Y x W.

Because we have 5 squares, the total of all the number will be the original total plus

5YW

I predict that formula for vertical translations on any grid is:

N = 5T – 7W - 5YW  (where Y can be negative for moving down or positive for moving up)

N = 51+52+53+62+72

N = 290

N = 5T – 7W -  5YW

N = 5 x 42 – 7 x 10 + 5 x 3 x 10

N = 290

Now I will combine both vertical and horizontal translation equations to give me an equation for any transition within any size gird.

N = 5T – 7W - 5YW + 5X (where Y and X can both be positive or negative)

Test 1

N = 57+58+59+68+78

N = 320

N = 5T – 7W - 5YW + 5X

N = 5 x 42 – 7 x 10 + 5  x 3 x 10 +5 x 6

N = 320

Test 2

N = 14 + 15 + 16 +25 +35

N = 105

N = 5T – 7W - 5YW + 5X

N = 5 x 78 – 7 x 10 + 5 x –4 x 10 + 5 x -3

N = 105

Test 3

N = 22 + 23 + 24+ 29+ 35

N = 133

N = 5T – 7W - 5YW + 5X

N = 5 x 14 – 7 x 6 + 5 x 3 x 6 + 5 x 3

N = 133

The formula for any translation on any grid is:

N = 5T – 7W - 5YW + 5X (where Y and X can both be positive or negative)

Now I will look at Reflection

The first reflection I will look at is reflection over a horizontal line at the bottom of the ’T’

N1 = 5T – 7W

N2= the total of N1 plus additions for each of the 5 squares

= 5T – 7W   + W (addition for 14)

                 + 3W (addition for 8)

                   + 5W x 3 (additions for 1, 2 and 3 )

So simplified formula for Reflection Top to Bottom.

N = 5T + 12W

Test 1

N = 49 + 58 + 67 + 66 + 68

N = 308

N = 5T + 12W

N = 5 x 40 + 12 x 9

N = 308

Test 2

N = 56 + 66 +76 +75 + 77

N = 350

N = 5T + 12W

N = 5 x 46 + 12 x 10

N = 350

If I were to have a gap of ‘R’ rows in between the reflection the formula would be: N = 5T +12W + 5RW

This is because a gap of one row adds W units to each number. So, R rows adds RW units to each number. But there are 5 numbers, so the total to be added is 5RW

If I now look at reflection in the same axis across the top of the “T” (that is reflecting from bottom to top), I will get the following result

N2=Total for bottom minus changes for each individual number

N2 =5T-7W- (3xW + 3W + 5W)

   =5T – 18W

And if there were a gap of “R” rows between the reflections, the formula would be :

N  = 5T – 18W – 5RW 

Test 1

N = 22 + 31 + 39 + 40 +41

N = 173

N = 5T – 18W

N = 5x67 – 18 x 9

N = 335 – 162

   = 173

Test 2

N = 23 + 32 + 40 + 41 +42

N = 178

N = 5T – 18W – 5RW

N = 5x77 – 18 x 9 – 5 x 1 x 9

N = 335 – 162

   = 178

It is possible to combine two or more combinations

E.g. 2 translations, 2 reflections and combinations of them

Example – A translation to right (x = 3) and down (y = -3), followed by a reflection in a horizontal line at the bottom of a ‘T’ shape.

 

Using

N2 = 5T1 + 12W

N1 = 5T – 7W

• 5YW
• 5X

 

N2 = 5T1 + 12W

N1 = (5T – 7W) + 19W

N2 = N1 + 19 W

     = (5T – 7W – 5YW + 5X) + 19W

     = 5T + 12W – 5YW + 5X

Test 1

N = 60 + 69 + 78 + 77 + 79

N = 363

N = 5T + 12W – 5YW + 5X

N = 105+108 +135+15

N = 363

Test 2

N = 56 + 66 + 76 + 75 + 77

N = 350

N = 5T + 12W – 5YW + 5X

N = 160 +120 + 50 + 20

N = 350

Example – Translation to the left (X = -3) and up (Y = 3) Followed by a reflection in a Horizontal line on the top edge.

Using:

N2  = 5T1 - 18W   

N1 = 5T – 7W

             - 5YW

             + 5X   

N2 = (5T1 – 7W) – 11W

     = N1 – 11W  

     = (5T – 7W – 5YW + 5X) –11W

     = 5T – 18W – 5YW +5X

Test 1

N = 22 + 23 + 24 + 14 + 5

N = 88

N = 5T – 18W – 5YW +5X

N = 5x80 – 18x9 – 5x3x9 +5*-3

N = 400 – 162 – 135 –15

   = 88

Test 2

N = 27 + 37 + 46 + 47 + 48

N = 205

N = 5T – 18W – 5YW +5X

N = 165-180+200+20

   = 205