41 X 5 = 205

6 X 50 = 300 The difference is one hundred and sixty.

46 X 10 = 460

51 X 95 = 4845 The difference is one hundred and sixty.

91 X 55 = 5005

I did an additional example to prove that my results were correct.

56 X 100 = 5600 My results were correct as the difference Is again

60 X 96 = 5760 one hundred and sixty.

I recorded my results in a table, in the event that I will be able to find a formula and trend pattern within my results.

I can see from my results that there is indeed a pattern occurring. The difference is the square of the previous number before it . For example 2 x 2 its square is 4. In the table if we ignore the zeros, I observed that 2 x 2’s square was the difference for a 3 x 3 square. This continued throughout the table as we see that the square of 3 x3 is indeed the difference for a 4 x 4 square. From my results I predict that the difference for a 6 x 6 square will be two hundred and fifty. I worked this out as 5 x 5 equals twenty five, therefore to follow the pattern of the table I multiplied this number by ten giving me two hundred and fifty.

I am now going to see if my prediction was correct for a 6 x6 sqaure.

5 x 60 = 300 The difference is two hundred and fifty.

10 x 55 = 550

50 x 95 = 4750 The difference is two hundred and fifty.

100 x 45 = 4500

23 x 78 = 1794 The difference is two hundred and fifty.

28 x 73 = 2044

This is clearly what I predicted as the difference is always equal to two hundred and fifty. I attempted another example to back up my results.

1 x 56 = 56 The difference again is always two hundred and fifty.

51 x 6 = 306

To make further investigations quicker. I worked out a formula from my table of results above. With the formula it will enable to see what the differences are for any square without having to work out the products of the top right and bottom left hand number, and the top left and bottom right hand numbers. I could see that the differences of each square is always the size of square penultimate. Therefore I worked out the formula to be this;

(n-1) ² x 10

To ensure that my formula worked I tested it on a 2 x 32 square of which the difference was 10.

(n-1) ² x 10 = (2-1) ² x 10 =

1² x 10 = 10

This shows me that my formula was correct, as it gave me the correct difference.

To further my investigation I am going to prove my results algebraically for a 2 x 2 square.

Therefore : x(x+11) = (x+1)(x+10) =

x² + 11x x² + 10x + x + 10 =

x² + 11x + 10

Here we can see that no matter what the value of x may be, the difference will always be ten.

I am now going to prove my results algebraically for a three by three square.

Therefore : x(x+20) (x+1)(x+10) =

x² + 20x x² + 20x + 2x + 40 =

x² + 22x + 40

Again this proves my results, as the difference will always be forty between the two numbers.

I am now going to prove my results algebraically for a four by four square.

Therefore : x(x+33) (x+3)(x+30) =

x² +33x x² + 30x + 3x + 90 =

x² + 33x + 90

Here the difference between the numbers is + 90, which proves and supports my results in the table.

To conclude I am going to prove my results algebraically for a five by five square.

Therefore : x(x+54) = (x+4)(x+40) =

x² + 54x x² + 40x + 4x + 160 =

x² + 44x + 160

Again the difference will always be +160. This example supports and proves my results.

To further my investigation, I am now going to find out what happens in a rectangle, starting with a three by two rectangle, and observe any patters or trends.

1 X 13 = 13 The difference between them equals twenty.

3 X 11 = 33

4 X 16 = 64 The difference between them equals twenty.

6 X 14 = 84

90 X 98 = 8820 The difference between them equals twenty.

88 X 100 = 8800

I have noticed that the difference is always equal to twenty in a three by two rectangle. To prove my results are correct, I am going to conclude by doing another example.

81 X 93 = 7533 The difference between them equals twenty

83 X 91 = 7553

I have now chosen to enlarge my rectangle to three by four.

41 X 64 = 2624 The difference is equal to sixty.

44 X 61 = 2684

44 X 67 = 2948 The difference is equal to sixty.

47 X 64 = 3006

11 X 34 = 374 The difference is equal to sixty.

14 X 31 = 434

To prove my results I did another example.

65 X 88 = 5720 The difference is again sixty.

68 X 85 = 5780

To further my investigations I am going to again increase the size of my rectangle to four by five.

1 X 35 = 35 Here the difference between the two numbers is one hundred

5 X 31 = 155 and twenty.

6 X 40 = 240 The difference between the two numbers is one hundred and

10 X 36 = 360 twenty.

44 X 78 =3432 The difference between the two numbers is one hundred and

48 X 74 = 3552 twenty

To prove my findings I did another example.

32 X 66 = 2112 The difference between the two numbers is again one hundred

36 X 62 = 2232 twenty.

I am now going to show results of a rectangle, five by six.

24 X 69 = 1656 The difference between the two numbers is two hundred.

29 X 64 = 1856

55 X 100 = 5500 The difference between the two numbers is two hundred.

60 X 95 = 5700

11 X 56 = 616 The difference between the two numbers is two hundred.

16 X 51 = 816

I proved my results with one further example.

5 X 50 = 250 Again we see that the difference between the two numbers

10 X 45 = 450 will always be two hundred.

I put my results in a table in order to enable me to discover a pattern or trend, and also to find a formula with the size of my rectangles.

My results have shown me exactly the same pattern as to what happened with using squares. The size of the rectangle equals the next rectangle’s product after it. For example the 2 X 3 rectangle multiply together to give the product of the next rectangle 3 X 4, which is 200 (multiplied by ten). From my table of results I am going to work out a formula, making it easier and quicker to work out differences of any size rectangle. This is what I came up with.

(L-1)(W-1) X 10

To ensure that my formula was correct and worked I tested it with an example. I used a 5 X 4 rectangle of which the difference is one hundred and twenty.

(L-1)(W-1) X 10 =

(5-1)(4-1) X 10 =

4 X 3 = 12 Therefore : 12 X 10 = 120

As you can see my formula clearly worked, and gave me the correct difference of one hundred and twenty.

I am now going to prove my results algebraically for a 2 X 3 rectangle.

Therefore : x(x+12) = (x+2)(x+10) =

x² + 12x x² + 20x + 2x +20 =

x² + 22x + 20

This clearly shows that no matter what the value of x maybe in a 2 X 3 rectangle the difference is always going to be twenty.