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  • Level: GCSE
  • Subject: Maths
  • Word count: 1729

Math Grid work

Extracts from this document...

Introduction

Number Grids Coursework – Maths – Mr Danes

Maths Coursework

Introduction

We are looking at number grids and are using the numbers in the edges of any square or rectangle. I am multiplying the opposite edged numbers and subtracting the smaller number from the bigger one.

This is a 10 x 10 square

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I pick any square from this grid and do the multiplying and subtracting and see what I get. I multiply the top right hand number to the bottom left and number and then subtract the number from the other two numbers multiplied from it.

2 x 2 square

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image00.png

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image12.png

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I notice that I get the same number for any same size square so to prove this I will use X and prove why the answer is always 10. I can use a square in which the boxes are the edges of the square on the grid.

X

image27.pngimage23.png

X + 1

image32.png

X + 10

X + 11image33.png

image35.pngimage34.png

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+1)(X+10)…which is = X2+11X+10…and…

X(X+11)…which is = X2 +11X

[X2+11X+10]- [X2 +11X] = 10

X can be any number and will always equal 10!

3 x 3 square

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image36.png

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image02.png

X

X + 2

X + 20

X + 22

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+2)(X+20)…which is = X2+22X+40…and…

X(X+22)…which is = X2 +22X

[X2+22X+40]-[X2 +22X] = 40

X can be any number and will always equal 40!

4 x 4 square

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image03.png

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image04.png

X

X + 3

X + 30

X + 33

...read more.

Middle

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X

image06.png

X + 4

X + 40

X + 44

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+4)(X+40)…which is = X2+44X+160…and…

X(X+44)…which is = X2 +44X

[X2+44X+160]-[X2 +44X] = 160

X can be any number and will always equal 160!

Working out a formula for any square size on a 10 by 10 number grid

image07.png

The sequence I get is 10, 40, 90 and then 160.

So I can do the use a square to find out a formula that works out N for any square size on a 10 by 10 number grid.

image08.png

I do what I do with these figures if they were numbers:

1. [X+(S-1)][X+10(S-1)

    I get X2+11X(S-1)+10(S-1)2

And…

2. X[X+11(S-1)]

    I get X2+ 11X(S-1)

I then subtract 2 from 1…

[X2+11X(S-1)+10(S-1) 2] -  [X2+ 11X(S-1)]

And whatever that is left over must be the formula, so therefore the formula to work out N for any square size in a 10 by 10 grid square is…

N= 10(S-1)2

Test

When the square size is 2…

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Using the formula, (2-1) 2 = 1.. x 10 = 10  

When the square size is 5…

2

image10.png

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Using the formula, (5-1) 2 = 16… x 10 = 160  

The formula works!

What happens if I change the grid size?

I think that I have come to the stage where I can use algebra to work out what the number each time will be. So, I don’t have to show examples with numbers all the time.

A 6 by 6 grid square on a 2x2 square

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image11.png

If I multiply this out if I were doing what I was doing with the numbers in the grid I would,

(X+1)(X+6)…which is = X2+7X+6…and…

X(X+7)…which is = X2 +7X

[X2+7X+6]-[X2 +7X]= 6

X can be any number and will always equal 6 on a 2 by 2 square on a 6 by 6 grid.

An 8 by 8 grid square on a 2x2 square

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...read more.

Conclusion

N= g(S-1)2

Test

When the square size is 2 and the grid size is 10

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image09.png

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Using the formula, (2-1) 2 = 1.. x 10 = 10  

When the square size is 3 and the grid size is 6

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image29.png

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Using the formula, (3-1) 2 = 4… x 6 = 24  

The formula works!

Final formula: for any size square/rectangle in any grid square

Common sense tells me that, this is basically replacing the S by the length and width of any square/rectangle.

image30.png

I do what I do with these figures if they were numbers:

1. [X+(w-1)][X+g(L-1)]

    I get X2+gX(L-1)+X(w-1)+g(w-1)(L-1)

And…

2. X[X+g(L-1)+(w-1)]

    I get X2+ gX(L-1)+X(w-1)

I then subtract 2 from 1…

[X2+gX(L-1)+X(w-1)+g(w-1)(L-1)]  - [X2+ gX(L-1)+X(w-1)]

The only thing left over is g(w-1)(L-1)  

And whatever that is left over must be the formula, so therefore the formula to work out N for any square/rectangle in a g by g grid square is…

N= g(w-1)(L-1)

Test

When the grid size is 10 and the length and width are 2

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Using the formula, 10(2-1)(2-1) = 10x1x1= 10

When the grid size is 5, length is 2 and width is 4

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image31.png

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Using the formula, 5(4-1)(2-1) = 5x3x1 = 15

The formula works!

***

By: Haroon Motara

...read more.

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