The total number of the integers inside of the T-shape is the T-total. On this simple 9 by 9 grid the T-total of the T-shape is 1+2+3+11+20=37.
The enlarged and circled number in the T-shape on the grid is the T-number. This investigation investigates the connections between the T-number and the T-total. To help me investigate and create more of an understanding I will change variables like the grid sizes and T-shape rotation.
90 180 270 360
Here is a simple 9 by 9 grid. On this grid I have shaded in three t-shapes. The first one I shades is the one that has the five numbers 40, 41, 42, 50 and 59.
The T-total is 40+41+42+50+59=232. If I alter the position of this shape one space to the left I will have a t-total of 39+40+41+49+58=227. Look at the last t-total of 232 and look at this one, there is a decrease of five, and if I move the T-shape two spaces to the right from the original position I get a T-total of 42+42+44+52+61=242. The T-total increases from the first by 10.
METHOD
For the pencil highlighted T-shape the T-number is 20 and the T-total is 37. For the red highlighted T-shape the T-number is 21 and the T-total is 42. Note that the difference is 5. I predict from this information that every time the T-number increases by one, the T-total increase by five just as it did on the grid in the plan. I will investigate in two ways. The first will be to discover the formula and the second shall be to prove the formula. These methods are called ‘relative formula’ and ‘difference in sequence’. In this investigation there are some independent variables such as,
The grid size in which the T- shape is situated in for example;
-9 by 9 grids, 6 by 6 etc.
The rotation of the T- shapes either,
-90 from vertical, 180 from vertical, 270 from vertical and 0 or 360 from vertical.
0
As well as the various positions of the T-shape.
The dependent variables in this investigation consist of,
The size of the T- shape
(3 across the top and 3 going down)
My method of calculation
-Numerical for example in the grid below I will take the five highlighted numbers and add them together as I did in the plan to find the T- total which in this case is 37.
-Then ill try the algebraic calculation, which is demonstrated below (using the information presented above).
N
N-19
N-18
N-17
N-9
____________________
5N-63
_ (This is the formula above and below is the numerical form of the formula.) _
355-63=292
I have found a formula, which is
5tn-number – 63= t-total
So how do we work out this formula and what can we do with it?
The formula begins with 5 times the t-number; the reason for this is because there is an increase of 5 for every t-number. We then -63, we work this calculation out by finding the difference between the t-number and another number in the t-shape. There is also an alternate way of finding -/+ 63. This method is to times 7 by the grid size. We can do this because the number you have to -/+ is always divisible by 7 in this case 9*7=63. This has to be done to all five numbers in the t-shape. In demonstration above is an algebraic investigation. There is also an alternate way of finding -/+ 63. This method is to times 7 by the grid size. We can do this because the number you have to -/+ is always divisible by 7 in this case 9*7=63. In demonstration above is an algebraic investigation.
Here is another example using a similar formula in a small sized grid of 6*6.
5tn-42=68
5*22-6*7=68
110-42=68
Now lets check if this is correct
9+
10+
11+
16+
. 22 _ .
68
Our formula proves to be effective in different shaped grids as well.
Grid size and translation.
Grid size.
Our first grid will be 4 by 4.
T-number= 15
T-total= 6+7+8+11+15= 47
7*4 (grid size)= 28
5*15-28=47
Translations and combination of translations.
On a 9 by 9 grid I have a vertical t-shape with a t-number of 41. I will now translate this shape to 180 degrees from its original position. Which makes the t-number 41. It would look like this. However as it has been reversed, in calculations we will have to reverse the method from – to +.
(Yellow highlighted shape/ 180 degrees translated shape.)
5tn+63=268
Is this formula correct?
T-number=41
T-total=58+59+60+50+41=268
Our reversing the minus sign has proven to work.
The formula is correct.
Now lets try it with the t-shape on its side. 270 degrees from its original position
The t-number is again 41
In this calculation we will use the method to find the difference to work out our t-total.
41-40
41-39
41-48
. 41-30.
5
5tn-7=198
So lets see if this worked.
T-number=41
T-total=41+40+39+48+30=198
It has proven to work.
So this show that the formula may alter from situation to situation but the basic introductory formula is always its template.
Conclusion
In this project I have found various ways in which to solve the problem that was presented with. This investigation named T-shape was and investigation where I had to find the formula for each circumstance for instance a different grid size, I was expected to work out the correct formula for the T-shape in this grid as well as find another correct formula for any different position that the t-shape could be in like 90 180 270 or its original position.
I found three formulas the first for the original shape position and original grid size was:
5tn-63=T-total
This was my original/introductory formula.
My second was the T-shape 180 from its original position the formula for this was:
5tn+63=T-total
And my third was the T-shape on its side/90 /270 the from its original position formula for this position was:
5tn-7=T-total.
Knowing all of this and presenting this to you as an investigation brings me to the end of the investigation.
