GCSE MATHS COURSE WORK

The square grid is represented below:

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Find the product of the top left number and the bottom right number in the square grid, do the same with top right and bottom left numbers then calculate the differences between these products.

2 X 2

We take the squire number 1, S1

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Gcse Maths Course Work

(1 X12) – (2X11) = -10

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(2X13) – (12 X 3) = -10

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(21X32) – (22X31) = -10

3X3

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(1X23)  – (21X3) = -40

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(2X24)  – (22X4) = -40

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(11X33)- (13X31) = -40

4X4

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(1X34) – (31X4) = -90

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(2X35) – (32X5) = -90

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(11X44)- (41X14) = -90

5X5

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(1X45)- (41X5) = -160

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(2X46) – (42X6) = -160

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(11X55) – (51X15) =-160

From the above observations we can conclude that the value for

2X2=-10

3X3=-40

4X4=-90

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Gcse Maths Course Work

5X5=-160

Therefore we can conclude that for any 2X2 square the value will be -10 and also for the others, we therefore try to get a formula that will aid us to get these differences, the formula can therefore be derived from the following process

The differences between

2X2 and 3X3 = 30

3X3 and 4X4 = 50

4X4 and 5X5 = 70

Let P1 be the 2X2 grid, and P2 be the 3X3 grid and also P3 and P4 be the 4X4 and 5X5 grid.

Then P1 = -10

P2 = P1 – (20X 2) +10

From the above we can calculate P2 which will give us P2 = -10 –(20X2) +10 = -40

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For the P3 grids the formula will be the same as the one for P2 but in this case it will be as follows

P3 = P2 – (20 X 3) + 10, when we calculate the results will be

P3 = -40 – (20X3) +10 = -90

Therefore for the squire grids the general formula will take the following form

Pn = Pn-1-(20Xn) +10

The rectangular

Choose rectangular grid 2×3, 3×4, 4×5 and 5×6 do the same find the formula

2×3

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(1X13)- (11X3) =-20

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(2X14)- (12X4) =-20

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(3X15)- (13X5) =-20

3X4

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(1X24)- (21X4) =-60

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(2X25)- (22X5) =-60

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(11X34)- (31X14) =-60

4X5

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(1X35)- (31X15) =-120

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(2X36) – (32X6) =-120

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(11X45)- (41X15) =-120

From the above results we can note that

P1 which represent the 2X3 grid gives us a value of -20, P2 which is the 3X4 grid gives us a value of -60 and finally P3 which represents the 4X5 grid gives us a value of -120.

Therefore the results will be as follows

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Gcse Maths Course Work

P1 = -20

P2 = -60

P3 = -120

We can therefore analyze the results as follows

P1=-20

P2= P1 – (20X2)

P3= P2 – (20X3)

Therefore the formula can be simplified as follows

Pn = P n-1 – (20Xn), this formula therefore represent the rectangular grids where the columns exceed the rows by only one unit.

This formula however applies to rectangular grids whose columns exceed the rows by only one unit in the grid, however we will investigate the relationship that exist in the rectangles whose rows exceed the columns by one unit, therefore we will consider a 3X2, 4X3 and 5X4.

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3X2

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(1X22) – (21X2) =- 22

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(2X23)- (22X3) = -22

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(3X24)-(23X4)=-20

4X3

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(33X1) – (31X3) = -60

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(34X2) – (32X4) = -60

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(3X35) – (33X5) = -60

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5X4

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(1X44)- (41X4) =-120

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(2X45) – (42X5) = -120

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(3X46) – (43X6) = -120

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From the above calculations we can conclude that the results we get for a 3X4 grid is the same as the results for a 4X3 grid, also in the case of a 5X4 grid it is the same as that of a 4X5 grid. Therefore we can conclude that the resultant general formula for the rectangular grid we got from the previous formula will also apply to our above calculations and therefore Pn = P n-1 – (20Xn).

In this case n represents the number of columns and that the number of rows exceeds the columns by one or the number of columns exceeds the rows by one.

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