Number Grid Maths Coursework.

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David Macmillan        

Number Grid Maths Coursework

Introduction

In this investigation I will look at different number grids and different sized rectangles within these grids and try to explain the patterns and give algebraic equations for the results that are found during this investigation. In a grid a rectangle is drawn and then the product of the opposite corners is found and the difference is calculated. From this, and further investigation I will find these patterns and work out equations which can then be proven by numerical examples.

Squares in a 10 x 10 grid

In this first section of the investigation I will investigate what patterns can be found with 2x2, 3x3, 4x4 etc. squares in a 10x10 grid.

Numerical Examples

Results

From the number grid and table above I can therefore say that with every 2x2 square in a 10x10 grid that the difference is 10. I can also say that for all 3x3 squares that the difference is 40 and for all 4x4 squares the difference is 90 in 10x10 grids. From this I can therefore say that all squares of the same size have the same difference when the top right and bottom left corner are multiplied and the answer is then subtracted from the product of the top left corner and bottom right corners.

An LxL square in a 10x10 grid

Since the difference results of any square is the same I will now make the size of the square variable and as it is variable and not a set number it will be known as L for the length of the side.

Numerical examples 

Using these results from the numerical patterns and the squares themselves an algebraic formula can be found to find the difference of each individual square. n will be used to represent the number in the top left corner of each square.

Here are the corners for a 2x2 square, which is why all they all have the same difference also.

The 3x3 square

The 4x4 square

But these formulae only work for each individual square but as all squares of the same size have the same difference then instead of using the numbers we can change it into algebra by replacing the length number with the letter L.

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These are the corners for a square of any length sides i.e. an LxL square

 

The corners are then multiplied as with the numerical examples which leaves us with two algebraic equations that need subtracting to find the difference. These are.

n x [n+11(L-1)]= n2 +11n(L-1)

And

[n+(L-1)]x[n+10(L-1)]= n2 +10n(L-1)+1n(L-1)+10(L-1) 2

n2 +10n(L-1)+1n(L-1)+10(L-1) 2 – [n2 +11n(L-1)]= 10(L-1) 2

When these two equations are then subtracted we are left with the difference for any square. So D=10(L-1) 2 .

Conclusion

From the equations and numerical examples above I can therefore conclude that the difference (D) for any square in a 10x10 grid is 10(L-1) 2 ...

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