Investigate the relationship between the T-total and the T-number.

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T-total Coursework

  1. To investigate the relationship between the T-total and the T-number.
  2. Use grids of different sizes.  Translate the T-shape to different positions.  Investigate the relationship between the T-total and the T-number and the grid size.
  3. Use grids of different sizes.  Try other transformations and combinations of translations.  Investigate relationships between the T-total, the T-number and the translations.

Relationships between T-number (x) and T-total (t) on a 9 x 9 grid.

From the 9 by 9 grid we can see that the first T-shape highlighted in green has a T-number of 20 which is the number located at the bottom of the T-shape and the T-total (t) which is all the numbers in the T-shape added together equals 37 (20+11+1+2+3).  With the second T-shape with a T-number of 23, the T-total adds up to 52, you can see that the larger the T-number the larger the total.

If you plot all the other T-shapes and put the information into a table about the T-total and T-number you can really see a pattern and start to work out the 1st part of the formula.

The table proves that the bigger the T-number is bigger the T-total is larger; the T-numbers are arranged in order of size and the T-totals gradually get larger with the T-number.  From this we are able to work out some parts of the formula for a 9 by 9 grid. Taking the T-number of 20 as an example, we can say that the T-total is gained by:

t = (20 - 19) + (20 - 18) + (20 - 17) + (20 - 9) + (20 - 0) = 37

As there are 5 numbers in each T-shape, we have to use five lots of twenty. The number above the 20 is 11, which is 9 less than 20; the other numbers in the T-shape are 1,2 & 3, which are 19,18, & 17 less than 20.  

If we say that 20 is the T-number and the T-number is then represented by x. We can use x to replace 20 so that it would fit in with any T-shape:

t =  (x – 19) + (x – 18) + (x – 17)+ (x – 9) + (x + 0)

Which gives you five lots of x to give you 5x which is the 1st part of the formula and you have to take away 63 (which is the rest of the numbers in the sum).

Which gives us the formula of   t = 5x – 63

Finding relationships on grids with sizes other than 9 x 9

If we take this 8 x 8 grid with a T-number of 18 we get the T-total of 34. If you use the same method used in a 9 x 9 grid you get the formula of:

t = 5x – 56

You can also show this in a T-shape form: showing a quicker and faster way to create the formula.

Testing this out using 36 as x we get:

t = (5 x 36) – 56

t = 180 – 56

t = 124

19 + 20 + 21 + 28 + 36  = 124

On a 4 x 4 grid we can try the same method of generating a formula.

t = 5x – 28

If we test this:

t = (5 x 15) – 28

t = 75 – 28

t = 47

6 + 7 + 8 + 11 + 15  = 47

To find the T total for any grid sizes the formula is 5x  –  (a multiple of 7)

We should now try to find a general rule that works for a grid of any size.  We can use the letter g to represent the grid size. Which will then be added into the formula.

 

If we get the formula of a 3 by 3 grid you can use this as an easy start to work out a general formula.

t = 5x – 21 is the formula for a 3 by 3 grid.

To find the number that you take away by you look at all the multiples of 7, in this case it is 3 or represented as the g for grid size.

3 x 7 = 21

Or

g x 7 = 21

This grid shows the relation ship between g and the number you take away.

Join now!

If we use all the information of each section above together we can get a general formula for a grid of any size and any T-shape that we wish.

t = 5x – 7g

Translations

If we move the T-shape 1 Square to the right the T-total leads to an increase of +5. A translation of 1 square to the left leads to a decrease of –5. And if we translate the T-shape upwards 1 square this leads to a decrease of –45 if it is translated down it leads to an increase of ...

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