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# Opposite Corners

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Introduction

Opposite Corners

This piece of coursework uses a grid of numbers and a x by x box drawn around a set of numbers. I am trying to see if there is a pattern between the differences of the products.

Aim: My task is to investigate the differences of the products of the diagonally opposite corners of a rectangle, drawn on a 10x10 grid, with the squares numbered off 1 to 100.

I will aim to investigate the differences for rectangles and squares of different lengths, widths.

I plan to use algebra to prove any rules I discover which I will hopefully find by analysing my results that I will display in tables. I will test any rules, patterns and theories I find by using predictions and examples.

Middle

n        n+1

n+10        n+11

n(n+11)= n²+11n

(n+10)(n+1)=n²+10n+n

n²+11n+10

n²+11n

10

To find a formula for any square on a 10x10 grid I created this table from the parts above:

 Squares Difference Diff. /10 Square root of Diff/10 2x2 10 1 1 3x3 40 4 2 4x4 90 9 3 5x5 160 16 4

Looking at the last column I noticed that the numbers were 1 less than the dimension size in the 1st column.

Working backwards I took 1 dimension(d) then subtracted 1, squared it then multiplied by 10 to give this formula:

(d-1)²x10

I then proved my formula worked by testing it with a 3x3 box.

(3-1)²x10

2²x10

4x10=40

Rectangles:

Conclusion

n        n+1        n+2

n+10        n+11        n+12

n(n+12)=n²+12n

(n+10)(n+2)=n²+12n+20

n²+12n

n²+12n+20

20

I created this table for rectangles:

 Rectangle Difference Diff /10 2x3 20 2 2x4 30 3 3x4 60 6 3x5 80 8

I worked out from the table and other evidence the formula:

(w-1)(l-1)x10

w=width        l=length

Grid Size:

I started to try and find a formula for any grid size for a square and rectangle.

I saw that in both formulae there was a x10 at the end. I thought that this may be the grid size.

I tested my prediction with a 3x3 box and a 2x3 rectangle on a 20x20 grid.

3x3-        (3-1)²x20                                        2x3-        (2-1)(3-1)x20

2²X20                                                1x2x20

4x20=80                                                2x20=40

Actual Differences:

3x3-        1x43=43                                        2x3-        1x42=42

3x41=1232x41=82

8040

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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