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Introduction

Algebraic Investigation 1: Square Boxes on a 10x10 Grid

In this first investigation, the difference in products of the alternate corners of a square, equal-sided box on a 10x10 gridsquare will be investigated. It is believed that the products and their differences should demonstrate a constant pattern no matter what dimensions are used; as long as they remain equal. In order to prove this, both a numeric and algebraic method will be used in order to calculate this difference. The numeric method will help establish a baseline set of numbers for testing, and to help in the establishment of a set of algebraic formulae for use on an n x n gridsquare.

In the example gridsquare below, the following method is used in order to calculate the difference between the products of opposite corners.

 (a) (b) (c) (d)

Stage A:        Top left number x Bottom right number =        (a) multiplied by (d)

Stage B:        Bottom left number x Top right number =         (c) multiplied by (b)

Stage B – Stage A:        (c)(b) - (a)(d)          =  The difference

The overall, 10 x 10 grid that is used for the first investigation will be a standard, cardinal gridsquare, which progresses in increments of 1. The formulae calculated will mainly be applicable to this grid, as other formats of gridsquares will require others formulae to provide valid results.

 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 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

This first investigation will focus only on gridsquares with equal widths and heights, which will at this stage be represented by the universal, constant term ‘w’.

The top left number in the grid (letter (a) in the above example) will be represented by the term ‘n’, which will be referred to in this manner in all proceeding investigations also.

This is only the first section of the investigation.

Middle

In this section of the investigation, additional variables (constant terms) will be used due to the change in the number of factors involved.

‘n’ will continue to represent the top left number in the grid. All other formulae in the grids will refer to n.

Due to the height and width of the boxes no longer being equal values, ‘w’ will represent the width of the number box, whereas ‘h’ will refer to the height,

Part B:                Changing the Height, ‘h’

To provide a comparison for finding the height formula, summary boxes will be included, showing the algebraic results of the last section.

As in this part of the investigation, additional factors are concerned; the overall formula will contain numerous terms. The rules of brackets and of ‘bidmas’ will always apply to the formulae, helping to create a workable equation.

It should be possible to create an algebra grid detailing the numbers to be expected on any height x width grid on a 10 x 10 gridsquare. After multiplying the alternate corners and subtracting, an overall formula should be gained that can be used to calculate the difference in any h x w box; always on a 10 x 10 grid.

Part A:2 x Width Rectangles

2x2 Rectangles

Firstly, a rectangle with applicable numbers from the grid will be selected as a baseline model for testing.

 15 16 25 26
 n n+1 n+10 n+11

Stage A:        Top left number x Bottom right number =        n(n+11) =        n2+11n

Stage B:        Bottom left number x Top right number =         (n+10)(n+1)=        n2+1n+10n+10

=        n2+11n+10

Stage B – Stage A:        (n2+11n+10)-(n2+11n) = 10

When finding the general formula for any number (n), both answers begin with the equation n2+11n, which signifies that they can be manipulated easily.

Conclusion

Formula 1:                Bottom Right (BR)  =  Top Right (TR)  +  Bottom Left (BL)

As also shown by the summary boxes and examples above, the formula for the top right number remains constant, and is linked with the width, w, of the box in the following way:

Formula 2:                Top Right (TR)  =  n+ Increment size (s) x (Width -1)

It is also evident from the examples calculated that the bottom left number is also linked with the height, w, of the box using a formula that remains constant:

Formula 3:                Bottom Left (BL)  =  n+ Increment size x Gridsize x (height -1)

Using these rules, it is possible to establish an algebraic box that could be used to calculate the difference for any hxw box on any gxg grid.

 n ~ n+s(w-1) ~ ~ ~ n+gs(h-1) ~ n+s(w-1)+gs(h-1)

Which through simple algebraic process can simplify into:

 n ~ n+sw-s ~ ~ ~ n+ghs-sg ~ n+sw+ghs-gs-s

Stage A:        Top left number x Bottom right number

=        n(n+sw+ghs-gs-s)

=        n2+nsw+ghns-gns-ns

Stage B:        Bottom left number x Top right number

=         (n+ghs-gs)(n+sw-s)

=        n2+nsw-ns+ghns+ghs2w-ghs2-gns+gs2w+gs2

=        n2+nsw+ghns-gns-ns+ghs2w-ghs2+gs2w+gs2

Stage B – Stage A:        (n2+nsw+ghns-gns-ns+ghs2w-ghs2+gs2w+gs2)-(n2+nsw+ghns-gns-ns)

=        ghs2w-ghs2+gs2w+gs2

=         s2ghw-s2gh+s2gw+s2g

When finding the general formula for any number (n) any height (h), any width (w), and with any gird size, both answers begin with the equation n2+nsw+ghns-gns-ns, which signifies that they can be manipulated easily. Because the second answer has +ghs2w-ghs2+gs2w+gs2 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of s2ghw-s2gh+s2gw+s2g will always be present.

Charankamal Singh Theora

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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