grid square maths coursework GCSE grade A - page 1
Keywords: grid square maths coursework grade A mathematics GCSE
By Matt on 05/11/2006 13:22:38
Level: GCSE Key Stage 4 (Years 10-11)
Page Number: 1 of 6 pages: 1 2 3 4 5 6
Maths Coursework-
Investigating the difference between the products of the diagonals in a square (and rectangle) placed anywhere in a number grid.
Introduction
In this piece of coursework I am going to investigate the difference between the products of the diagonals in a square (and rectangle) placed anywhere on a number grid. During this investigation I am to try and answer the following questions:
Is the difference the same anywhere on a number grid?
What happens to the difference when the size of the square or rectangle changes?
Can I spot any patterns?
Can I make any predictions?
Can I find a formula to help me make my predictions?
Can I prove that my formula works?
Can I derive my formula?
Can I see how the differences in a rectangle alter the formula?
Investigating the products of the diagonals in a square (and rectangle) placed on a 10X10 number grid.
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
2 3
12 13
2X13=26
3X12=36
__
10 (difference of 10)
14 15
24 25
14X25=350
15X24=360
___
10 (difference of 10)
88 89
98 99
88X99=8712
89X98=8722
____
10 (difference of 10)
I have now done 3 small investigations to find out whether the difference of the products of the diagonals in a 2X2 square worked anywhere on a grid. I found out that it does. The difference is always 10 no matter where the square is put.
I also found that the formula:
n n+1
n+10 n+11
Is relevant for a 2X2 square anywhere on the 10X10 grid.
Example: (n) (n+11) = n2 + 11n
(n+1) (n+10) = n2 + 11n + 10
___________
10 (difference of 10)
3X3 squares
58 59 60
68 69 70
78 79 80
58X80= 4640
60X78= 4680
____
40 (difference of 40)
To help me prove this works anywhere I made a formula for this one too.
n n+2
n+20 n+22
(n) (n+22) =n2+22n
(n+2) (n+20) =n2 + 22n + 40
__________
40 (difference of 40)
4X4 Squares
1 2 3 4
11 12 13 14
21 22 23 24
31 32 33 34
1X34= 34
4X31= 124
___
90 (difference of 90)
N n+3
n+30 n+33
This formula works for anywhere on the grid,
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