Investigation into Numerical 4 by 4 Squares - page 3
Keywords: Investigation into Numerical 4 by 4 Squares maths Formula, maths sqaure formula
By Mose. on 01/09/2008
Level: GCSE Key Stage 4 (Years 10-11)
Page Number: 3 of 5 pages: 1 2 3 4 5
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.
x x+1
x+10 x+11
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.
x x+2
x+20 x+22
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.
x x+3
x+30 x+33
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.
x x+4
x+40 x+54
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 2 3
11 12 13
1 X 13 = 13 The difference between them equals twenty.
3 X 11 = 33
4 5 6
14 15 16
4 X 16 = 64 The difference between them equals twenty.
6 X 14 = 84
88 89 90
98 99 100
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 82 83
91 92 93
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 42 43 44
51 52 53 54
61 62 63 64
41 X 64 = 2624 The difference is equal to sixty.
44 X 61 = 2684
44 45 46 47
54 55 56 57
64 65 66 67
44 X 67 = 2948 The difference is equal to sixty.
47 X 64 = 3006
11 12 13 14
21 22 23 24
31 32 33 34
11
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