Maths Coursework

Number Grids

## Chapter One

For the first part of my maths G.C.S.E coursework I have been provided with a 10x10 number grid, which is numbered 1 to 100. I have been instructed to find the product of the top left number and the bottom right number and the same with the top right and bottom left number within selected squares used.

I am going to use this grid to examine at random various sizes of squares and rectangles. My objective here is to establish a trend to identify an overall formula.

I am firstly going to examine a series of 2x2 squares the primary one I will look at has been outlined in the number grid I was provided with, I will then select alternative 2x2 squares at random from the grid.

- 13x22 - 12x23

286 - 276

Difference = 10

58x67- 57x68

3886 - 3876

Difference = 10

- 35x44 – 34x45

1540 - 1530

Difference = 10

By looking at the three 2x2 squares chosen above it is possible to see each time that there is a difference of 10. So in conclusion to this I can say that any further investigations using 2x2 squares will always result in a difference of 10.

Looking at my results and the number grid at this stage, I feel I can suggest that the reason I may get the same result of 10 each time is one of two small theories, my first one being that it is possible to see that the selected 2x2 squares have a difference of 10 between the bottom and the top line of each set of numbers, before even using multiplication. And the second piece of theory for this difference could be that the grid I am using is a 10x10 number grid.

I am now going to use algebra to prove that the trend of 10 that I found will always be the found result when investigating any 2x2 squares in a 10x10 number grid.

r (r+1)

(r+10)(r+11)

(r+1)(r+10) –r (r+11)

= r (r+10) +1 (r+10) – r -11r

= r +10r +r+10 – r -11r

=r +11r+10 –r -11r

=10

This algebra proves my results that for any 2x2 square the answer for the defined difference will always be 10.

## Furthering my Investigation

I am now going to further my investigation by increasing the size of the squares. I am going to be repeating my process used previously but will be looking at randomly selected 3x3 squares, my aim being to see if I can find a trend and a pattern in my results.

I predict that there will be a difference of 30. I have come to this forecast by simply using my original findings from the 2x2 squares and multiplying by three for a prediction of my new trend for any 3x3 square.

E.g. (Previous findings) 10 x (Increased size) 3 = 30

I also found that from studying the 10x10grid that my prediction of 30 could be explained by taking the top line of numbers of a selected 3x3 square grid, ‘3’ digits and multiplying the difference of either side of the 3x3 square.

E.g. (Number of digits) 3 x 10 (Difference between each row) = 30

- 26x44 – 24x46

1144 - 1104

Difference = 40

- 80x98 – 78x100

7840 - 7800

Difference = 40

From the above it shows that my prediction was not correct, although I do not see any major patterns between my findings of the 2x2 squares and 3x3 squares, it is possible to see that the trends found for each size of square are multiples of 10.

I feel that the trends are multiples of 10 simply because I am using a 10x10 number grid.

Again I am going to use algebra to prove that the defined difference of my 3x3 squares is correct.

r (r+2)

(r+20)(r+22)

(r+2)(r+20)-(r+22)r

=r (r+20)+2(r+20) – r -22r

=r +20r+2r+40-r -22r

=40

I have now calculated a trend for my 2x2 squares and came to a difference of 10 and a trend for my 3x3 squares and came to a difference of 40. I am now going to continue my investigation by increasing again to a larger square.

Furthering my Investigation

I am now going to look at 4x4 squares randomly selected from my 10x10 number grid. I am doing this to help me reach my aim of calculating my predictions successfully and finding a pattern in my investigation.

6x33 – 3x36

198 - 108

Difference = 90

- 38x65 – 35x68

2470 -2380

Difference = 90

Below is the algebra for my 4x4 squares that will confirm that my defined difference of 90 is correct.

(r+3)(r+30) -r (r+33)

r(r+30) +3 (r+30) –r -33r

r +30r+3r+90-r -33r

=90

Now I have identified my findings for my 4x4 squares with a difference of 90. I still have not managed to find any major trends or patterns. As already stated the only trend from each defined difference that I can see is that all findings are multiples of 10.

Furthering my Investigation

At this stage in my investigation I have still not managed to find any major pattern in my findings. I am now going to increase my square size to 5x5 and I aim after this to hopefully see a pattern emerging from my defined differences.

- 60x96 – 56x100

5760 - 5600

Difference = 160

6x42 – 2x46

252 - 92

Difference = 160

By looking at these calculations it can be assumed that they are correct because they give the same defined difference of 160. To ensure my calculations are right I will again use algebra.

(r+4)(r+40)- r(r+44)

r(r+40) +4(r+40)- r -44r

r +40r + 4r+160- r -44r

=160

By investigating this 5x5 square and identifying a defined difference of 160, it is now possible for me to see a pattern in my findings. By reassessing my previous answers I have become aware of the following trend:

30 50 70

10 40 90 160

To confirm that this pattern I have found is truthful I will predict that in an 8x8 square the defined difference will be 490.

I am going to move on past 6x6 and7x7 squares as if my prediction is correct it will be unnecessary to carry out any extra calculations. If my prediction for the 8x8 square is correct I can simply use the pattern I have identified to find the defined differences of any 6x6 or 7x7 squares.

28x91 – 21x98

2548 - 2058

Difference = 490

I will again use algebra to prove my defined difference of 490 for any given 8x8 square is correct.

(r+7)(r+70)- r(r+77)

r(r+70) +7 (r+70)- r -77r

r +70r+7r+490-r -77r

=490

I now feel that I have identified and proven a major trend within my investigation of a 10x10 square number grid.

I feel confident with my trend and I will continue on to show the complete pattern identified starting from a 2x2 square to a 10x10square.

30 50 70 90 110 130 150 170

10 40 90 160 250 360 490 640 810

2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10

Looking at the diagram, the top number indicates the difference between each defined difference found and the numbers along the bottom indicate the defined differences of each square investigated.