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• Level: GCSE
• Subject: Maths
• Word count: 1877

# This piece of coursework is called 'Opposite Corners' and is about taking squares of numbers from different sized number grids

Extracts from this document...

Introduction

Algebra Coursework

Introduction

This piece of coursework is called ‘Opposite Corners’ and is about taking squares of numbers from different sized number grids. I will be multiplying the opposite corners together and subtracting to find the difference.

I will make a prediction for each grid and use a few examples to find a formula to prove my prediction right.

Method

To calculate the difference of the squares drawn on the number grids, I will be multiplying each of the diagonal numbers. Then I will subtract the smaller number away from the larger number to find the difference of the square.

To do this I will use algebraic equations, I will use ‘N’ to represent the smallest number and ‘g’ to represent the grid size. I will then compare my results to ‘N’, then to complete the equation by expanding, simplifying and cancelling the brackets to get my final result.

Prediction

I predict that a 2x2 square from a 5 wide grid, will have a final difference of 5.

Proof:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Comparison:

N          N+1

x

N+5      N+6

N (N+6) = N² + 6N

(N+1) (N+5) = N² + 5N + N + 5

= N² + 6N + 5

Difference:

(N² + 6N + 5) – (N² + 6N)

= 5

What I have noticed:

Middle

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Comparison:

N          N+2

x

N+12      N+14

N (N+14) = N² + 14N

(N+2) (N+12) = N² + 12N + 2N + 24

= N² + 14N + 24

Difference:

(N² + 14N + 24) – (N² + 14N)

= 24

What I have noticed:

I have noticed that when a square that is 3x3, is taken from a 6x6 grid, the difference is always 24.

Prediction

I predict that a 4x4 square from a 6 wide grid, it will have a final difference of 54.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Comparison:

N          N+3

x

N+18      N+21

N (N+21) = N² + 21N

(N+3) (N+18) = N² + 18N + 3N + 54

= N² + 21N + 54

Difference:

(N² + 21N + 54) – (N² + 21N)

= 54

What I have noticed:

I have noticed that when a square that is 4x4, is taken from a 6x6 grid, the difference is always 54.

Prediction

I predict that a 5x5 square from a 6 wide grid, it will have a final difference of 96.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

Comparison:

N          N+4

x

N+24      N+28

N (N+28) = N² + 28N

(N+4) (N+24) = N² + 24N + 4N + 96

= N² + 28N + 96

Difference:

(N² + 28N + 96) – (N² + 28N)

= 96

What I have noticed:

I have noticed that when a square that is 5x5, is taken from a 6x6 grid, the difference is always 96.

I have found out the difference for a six wid grid is g(x-1)².

I will now see if the same eqyuation works for a seven wide grid.

I will now carry out the same investigation but on a 7 wide grid

Prediction

I predict that a 2x2 square from a 7 wide grid, it will have a final difference of 7.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

Conclusion

I have found out the difference for a seven wid grid is g(x-1)².

I now have found a formula that will work for and size number grid over five wide and square of 2x2. this formula is g(x-1)

Table of results:

5 wide grids

 Square size ,  Difference Gap in Difference Gap in Gap of Difference 2x2 5 +5 n/a 3x3 20 +15 +10 4x4 45 +25 +10 5x5 80 +35 +10

6 wide grids

 Square size ,  Difference Gap in Difference Gap in Gap of Difference 2x2 6 +6 n/a 3x3 24 +18 +12 4x4 54 +13 +12 5x5 96 +42 +12

7 wide grids

 Square size ,  Difference Gap in Difference Gap in Gap of Difference 2x2 7 +7 n/a 3x3 28 +21 +14 4x4 63 +35 +14 5x5 112 +49 +14

Conclusion

During my investigation, I have cancelled out, expanded and simplified brackets, and compared my results to ‘N’. I did this to find a final equation to be able to calculate the difference of any square of any size drawn on a number grid. I found this to be g (x-1) ². This formula works for any grid size over 5 and square size over 2x2.

If I had more time I would want to find out if the same formula works with a rectangle on a number grid and experiment with other quadrilaterals and see if the same equation works.

I would also see if I could find any other equation that would work.

Overall I think my investigation went well and I have found the final equation.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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