T-Total and T-Numbers Course work



PART 1




1) The relationship between the T-Total and the T-Number.

Here is a grid of nine by nine with the numbers starting from 1 to 81. There is a shape in the grid called the t-shape. This is highlighted in the colour red. This is shown below:





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The number 20 at the bottom of the t-shape is the t-number. All the numbers highlighted will be the t-total. In this section, there is an investigation between the t-total and the t-number.

T – Shapes



For this t-shape the
T-number is 20
And the
T-total is37




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For this t-shape the
T-number is 21
and the
T-total is 42 (all the t- numbers added up)
As you can see from this information is that every time the t-number goes up one the t-total goes up five.




This helps us because when I want to translate a t-shape to another position. Say I move it to here




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I all ready know the anwser to the one in red. To work out the one in blue all I have to do is work out the difference in the t-number and in this case it is 54. I then time the 54 by 5 because it rises five ever time the t- number goes up. Then I + the t-total from the original t-shape and I come out with the t-total for the green t-shape. This is another way to work out the t-total.
What I need now is a formula for the relationship between the t-total and the t-number. I have found a formula which is 5 x-number-63 = t-total.



The question is how did I work out this formula and what can I do with it?
The formula starts with 5 x the t-number. This is because there is a rise in the t-total by 5 for every t-number. I then – 63, which I do by working out the difference between the t-number and another number in the t-shape. This has to be done to the other 4 numbers in the t-shape. Here is an example:




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The t- shape has a t-number of 32. Now to work out the difference between the t-number and the rest of the numbers in this t-shape




Working out: -
32-13=19
32-14=18
32-15=17
32-23= 9
TOTAL= 63

This will happen to all the shapes this way up. To prove this I will do another.




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The t-number is 70. Now to work out the difference between the t-number and the rest of the numbers in this t-shape




Working Out: -




70-51=19
70-52=18
70-53=17
70-61=9
TOTAL=63



The number turns out to be 63. This is where the 63 came from in this equation. Another place this 63 comes from. This is 9x7=63. The nine in this comes from the size of the grid this one been nine. If the grid size Ire 10 by 10 then it would be 10x7.


At the end of this piece of coursework when I but all the formulas together I realise that the number I minus or plus by is divisible by seven. This is where I get the seven. The seven works with all the same sizes. The other method will also work with a different size grid.
If I add these two together, I have our formula.




5 x n – 63 = t-total





Here is an example of using the formula:


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5 x57-63= t-total
5 x57-63= 222

Check





T-total = 38+39+40+48+57=222




this formula has proven to work.




Here I will finely use an algebra equation to test my formula.


The number at the bottom of the T-shape is called the T-number.


The T-number for this T-grid is 20 or T


T (20)


If you take the other numbers in the T-Shape away from the T-Number you get a T-


Shape like this.


T-19


T-18


T-17


T-9


T


You will notice that the centre column of the T-Shape is going up in 9’s because of the table size. With the table set out like this a formula can be worked out to find any T-Total on this size grid.


This is shown in the working below:-


T-total = T-19+T-18+T-17+T-9+T


= 5T-63


Now to test this formula to see if it works


For T-total, I will use the letter X


For the T-Number, I will use the letter T


So X = 5T-63


T = 20


X = 5 x20-63


= 100-63


X= 37


My algebra formula has worked and my formula has worked ether-using algebra for number equations.


PART 2


2. This next section involves using grids of different sizes