The investiagtion betwwen the relationship of the T-number and T-total

Part 1. The investigation into the relationship between the T number (N) and the T total (Tt).

The T number is said to be the number at the bottom of the shape when it is in this shape. The T number always stays as the same box even if rotated.

The T total is the sum of all the 5 boxes.

T I firstly want to find the relationship between the T number and the T total in a 10 x 10 grid, and then in any grid.

From these results I worked out that the Nth term was 5Tn - 70.

However this only applied a 10 by 10 grid and so if I wanted a formula that applied to any grid then I would have to make all the parts to my formula dependant upon the grid size. For this I needed some new lettering. I needed letters that were dependant on the grid size so I used M to represent the increase in value in one movement down the grid, and R to represent the increase in value in one movement right in the grid.

N-2M-R N-2M N -2M+R This grid allowed me to work out a formula

which would apply to any grid size by adding up the terms. I ended up with 5N – 7M. This is the formula which links the T number to the T total

N-M on any sized grid.

Part 2. The investigation into the effect of transformations on the T total.

I started this investigation with simple translations. I wanted to work out what the T total would be after any transformation in any grid size. I began by looking at the relationship between the T totals after a translation along the x-axis in a positive direction.

These results told me that for every movement right the T total increased by 5, in a 10 by 10 square. This allowed my to deduce the following formula Tt + 5X. X is the number of movements right on the x-axis.

I then looked at the increase in movements down the y-axis.

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I then looked at the increase in movements down the y-axis.

From these results I could see that for on movement down the T total increased by 50, in a 10 by 10 grid. I then deduced the formula Tt + 50Y. Y is the number of movements down the y-axis. However unlike the previous formula this one is limited only to a 10 by 10 grid. I therefore had to add a variable that is dependant on the grid size to it. Any movement down will add 5M to the value so therefore Tt + Y (5M) would be the formula that encompasses any grid size.

However I wanted a formula which would apply to any combinations of translations do I had to combine the to formulae. I got Tt + 5X + Y (5M).

I then moved on to reflections this time I aimed to get a formula for any x-axis reflection and any y-axis one.

I fist tried reflections in the y-axis.

I got these results

The difference was constant at 15. I therefore devised the formula Tt + 15N. However this was only for a reflection in the line touching the extreme right edge. I therefore experimented with changing the distance between the edge and reflection line.

I noticed that for each increase in distance there was an increase in value of 10. Therefore my formula became Tt + 15R + D (10R).

I then moved onto x-axis. Again I started with 0 distance reflections but I also incorporated other distance into my data.

From these results I could see that by increasing the distance by 1 you increase the value by 100. I also saw that at 0 distance the increase was constant at 190. I therefore got the formula Tt + 19M + D (10M).

Having finished reflections in moved on to enlargements.

1 2 3

26 34 42

108 135 162

114

154

405

These are the values of each square in the T shape after first

Enlargement by scale factor 2 and then scale factor 3.

From this I couldn’t see any obvious patterns so I then tried to 675

work out the formula that linked each of the T numbers to the

T total. I came out with; 5Tn – 7M

5Tn – 56M

5Tn – 189M

I then noticed that all the Ms were 7 x the enlargement number3.

From this I was able to produce the formula 5Tn – 7m(e3). However, this formula only worked if you already knew the T number. To do that you would need a grid so the formula was pointless with out an additional part to tell you the T number.

This also failed to give me any leads. Finally I noticed that the number of Ms and Rs was drawn from the height and width of the square. If you multiplied the width by the number of Ms added by one column you got you answer. By putting the number of increasing Ms in one column into a pattern I got my formula.

1 3 6 10 15

2 3 4 5

1 1 1 I then applied the formula for finding quadratic nth terms and got 0.5a2 + 0.5a when I fitted this into my formula I got

5(S (T + (3M + R) – M (0.5A2+0.5A) + R (0.5A2+0.5A))) – 7M (E3). S is the number of small squares in one big square, E is the scale factor, T is the small square at the bottom left of the T number square and A is the scale factor – 1.

Part 3. The investigation into combinations of transformations.

I now want to find the formula that links any combination of translations, reflections and enlargements. To do this I will combine all my formulae found in the previous parts.

Tt + 5X + Y (5M). (Any translation)

Tt + 15R + D (10R). (Any reflection on the y-axis)

Tt + 19M + D (10M). (Any reflection in the x-axis)

5(S (T + (3M + R) – M (0.5A2+0.5A) + R (0.5A2+0.5A))) – 7M (E3). (Any enlargement)

To do this I wanted to have a clause in my formula for each transformation so that they could be applied one by one. I also had to allow for the changing order of the transformations (if the shape was enlarged before translated the formula would be different to if it was translated and then enlarged as the T total will be different). I therefore realised that the transformation depended on the t number at the start. This meant that I had to change the Tt part of my formulae into the total after the previous transformation. I also realised that I couldn’t add the clauses together as then my total would be to large, I needed them to ‘follow on” from each other

Therefore my formula read in words:

NTt = ((T total + first transformation clause) + second transformation clause) + third transformation clause) etc.

I would define the transformation clause as the part of the formula that is specific to the transformation e.g. in a translation it would be the (+ 5X +Y (5M)). In other words the remainder of the formula after the Tt has been removed. In the case of an enlargement this doesn’t work because the formula doesn’t use the previous T total. Therefore the transformation clause for an enlargement remains the entire formula. I also saw that the enlargement must be the final transformation as the other formulae only apply to the original T shape.

This however cannot be converted into algebra and so must remain in words to be filled in for specific cases. For example if I wanted to reflect my shape in the x-axis at a distance of 0 and the translate it (3, -2) and then enlarge it by scale factor 2 my formula would be:

NTt = (((Tt +19M + D (10M) + 5X +Y (5M)) + 5(S (T + (3M + R) – M (0.5A2+0.5A) + R (0.5A2+0.5A))) – 7M (E3).

If I started with a T number of 22 then

NTt = (((40 +190) + 5*3 -100) + 5(4 (-14 + (31) – 11) – 70 (23).

NTt = 210 - 85 + 5(4(17) - 11) - 560

NTt = 125 + 285 - 560

NTt = -150

However this is incorrect. It is because the formula for enlargement does not work if the shape has been inverted. Therefore my formula cannot mix x-axis reflections with enlargements.

Limitations of Formulae

I have put below all the limitations of my formulae:

The formula 5Tn - 7M (T number to Total) only applies when the shape is upright
The formulae Tt + 5X + Y (5M). (Translation), Tt + 15R + D (10R). (Y-axis reflection) and Tt + 19M + D (10M). (X-axis reflection) only apply to an original sized shape
The formula 5(S (T + (3M + R) – M (0.5A2+0.5A) + R (0.5A2+0.5A))) – 7M (E3). Can only be used in an upright shape and cannot make the shape smaller than 1 number per square
NTt = ((T total + first transformation clause) + second transformation clause) _ + third transformation clause) etc. Cannot work for an enlargement after an x-axis reflection and in it enlargements can only be the final transformation.