T-Shapes Coursework

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T-Shapes Coursework

This coursework is based upon a grid of numbers in which a ‘T’ shape is set out with 3 numbers at the top row and 3 down the middle like this:

           

It is a shape with 5 boxes.

We can call the highlighted number the, t-number. We can also call the sum of all the numbers inside the t-shape the, t-total.  This coursework investigates the relationships between the t-number and t-total, including different variables like various grid sizes and t-shape rotations.  

Now, to begin with, I will have a 9x9 grid.  

I have taken the t-shape, which consists of 40, 41, 42, 50 and 59.  Adding these up I get 232.  

If I move the t-shape one space left I get 39, 40, 41, 49 and 58.  The t-total is 227.  Notice the t-total has decreased by five.

If I move the t-shape two spaces right I get 42, 43, 44, 52 and 61.  The t-total is 242, increasing the original by ten.  

I moved the t one step right from the original, and predicted that the t-total from 232 would increase by five.  60+51+41+42+43=237.  My prediction was correct

If I have a t-total, from that I can work out another t-total if the new t-shape is directly to the left or right of the first. I know that moving a t-shape left or right I increase or decrease the first t-total by 5 or more.  This number could help me later.

If I took the t-shape consisting of 15,16,17,25 and 34, the total would be 107.  

On a grid if I moved this t one place up I’m left with a t consisting of 6,7,8,16 and 25.  The t-total is now 62.  107 – 62 = 45   By moving the t-shape up a step it decreases by 45.  

If I move the t-shape one place down from the original I now get 24,25,26,34 and 43.  This totals to 152.  152 – 107 = 45.

45 is a multiple of 9, 9 x 5 = 45.  Again, 5 has come in to the equation.  Now, if I have at least one t-total on a grid, I can work out what the t-total left, right, above or below that first t.  This is very useful knowledge.

If I am going to find a formula from the t-number to the t-total it will be easier to look at similarities if I put some results in a table.  These results are the t-number with the t-total.  Each time the T was moved one space right.

Every time the t-number increases by 1, the total increases by five.  I want to now try and find a formula.

I can predict what the 6th t-total will be if the next t-number is 25.

TT = 57 + 5.          57 + 5 = 62.

6 + 7 + 8 + 16 + 25 = 62

My prediction is correct.  The total is 62.  

Upon looking at numerous t-shapes I realised that the highlighted number is always the t-number take away 9.

25 – 9 = 16

So I eventually thought about taking the rest of the numbers in the t away from the t-number.

The t-number is 25.  If I take 16 from 25 I’m left with 9.  I can do the same to the rest and end up with

23 – 16 = 9

23 – 6 = 19

23 – 7 = 18

23 – 8 = 17

If I have ‘T’ with these results in it looks like:  

Note: In the t-shapes, the answers are written as

minuses because I was taking away the numbers

from the t-number so it therefore makes the answers

the opposite of their form.

Looking at this theory, I could add up the

minuses to give me a total of –63.  This

could become useful later.

I have obtained two possibly useful numbers, -63 and 5.  Perhaps I can use these numbers to generate a formula.  I went through a phrase of ‘trial and error’ with the two numbers to find the correct formula.  The formula is:

(Calling the t-number, ‘n’ and the t-total, ‘t’.)

n  x  5  -  63  =  t

Here are some examples of my formula in use.  

         

            31 x 5 – 63 = 92

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         12 + 13 + 14 + 22 + 31 = 92

            80 x 5 – 63 = 337

           61 + 62 + 63 + 71 + 80 =337

           

          38 x 5 – 63 = 127

              19 + 20 + 21 + 29 + 38 = 127

All the examples worked of ...

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