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  • Level: GCSE
  • Subject: Maths
  • Word count: 2163

T-Shapes Investigation

Extracts from this document...

Introduction

Introduction

Here is a number grid with a T-shape inside:

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This number grid is a 3 X 3 number grid because it has 3 columns across:

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and three rows down:

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The T-shape in the middle contains three squares at the top going across and two squares dropping vertically from the central square creating a T-shape.

The number at the base of the T-shape is called the T-number:

In this piece of course work I am going to signify the T-number in algebraic expression as 'x':

The T-total is given by the sum of the numbers within the T shape.

For example if I were to use a 3 X 3 number grid the T would look like this:

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Now if add all the numbers in the T shape it would total 19.  This is the T-total for this T-shape.

The T-shape

In this section I am going to try to find a formula which shows how to calculate the T-total and will try to create formulae for vectors.

As I said in the introduction I will assume the T number is x.  Here is a T- shape in a 5 X 5 grid:

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In this T-shape the T-number is 18 and the T-total is 55.  If I wanted to express this T-shape in algebraic fashion, using x as the T-number, I would put it like this:

        x-2n-1                x-2n                x-2n+1

...read more.

Middle

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In this T-shape x=18 and n=5.  This means to find the T-total we have to use the formula

T= 5 x 18 - 7 x 5

T= 90 - 7 x 5

T = 90 - 35

T = 55.

To prove my formula, the sum of the numbers in the T  (7 + 8 + 9 + 13 + 18) should equal 55.

The sum of the numbers 7, 8, 9, 13 and 18 does equal 55 and therefore my formula is correct.

I am now going to introduce a new letter so that I can use vectors.  A vector is used when a T-shape is moved to a different position in a number grid.  First I am going show how to find a formula for moving a T-shape one place horizontally across.  

If I use a 6 X 6 number grid and want to move a T-shape one place across and then find the T-total, it would be a lot easier to do this using a formula.

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The T-shape is to be moved from the position in the above grid across one place to the following position

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The letter I am going to introduce is 'a'.  This letter will stand for how many spaces the T-shape has been moved across.  As we move the T-shape across one space we must add 1 to each of the numbers in the T-shape.  There are 5 numbers in the T-shape so we add 1 each of the 5 numbers which gives 5 x 1, which equals 5.

In this example a=1.  If the T-shape were moved 2 places across then we would add 2 to each number, which gives 5x2 = 10.  The T-total would increase by 10.  Therefore if 'a' represents the number of places the T-shape is moved, we can add '5 x a' to the original T-total to find a new T-total when the T-shape is moved 'a' spaces across

Therefore we have the formula 5x - 7n + 5a to find the T-total anywhere on a horizontal line.

To get the formula for the T-total when the T-shape is moved vertically we have to introduce the new letter 'b'.

This letter represents how many spaces the T-number goes vertically downwards.

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The T-shape in the grid above is moved down 1 space in the grid below.                

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...read more.

Conclusion

T=x + x+n + x+2n-1 + x+2n + x+2n+1 = 5x + 7n

This can be tested on the 3x3 grid below.

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The T-total from the formula 5x +7n, where x = 2 and n = 3 is 5x2 +7x3 = 31.  

2+5+7+8+9 = 31, therefore the formula is correct.

We do no have to find another formula for vectors (moving the rotated T-shape horizontally and vertically) as none of the factors have changed from the from the formulae we identified earlier in the section dealing with vectors.

This means that the final formula for a T-shape rotated 180 degrees is 5x + 7n + 5a + 5bn.

Rotating the T-shape 270 degrees

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The T-shape in this grid has been rotated 270 degrees.  In this T-shape the T-number is 4 because it is at the base of the T.

This can be shown algebraically as follows:

                                        X+2-n

                        X        x+1        x+2

                                        X+2+n

If we add these expressions together to find the equation for the T-number it would be

T = x + x+1 + x+2 + x+2-n + x+2+n = 5x + 7

This can be tested on the 3x3 grid below.

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The T-total from the formula 5x +7, where x = 4 is 5x4 +7 = 27  

3+4+5+6+9 = 27, therefore the formula is correct.

We do no have to find another formula for vectors (moving the rotated T-shape horizontally and vertically) as none of the factors have changed from the from the formulae we identified earlier in the section dealing with vectors.

This means that the final formula for a T-shape rotated 270 degrees is 5x + 7 + 5a + 5bn.

...read more.

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