Investigation to find out a formula for rectangles on different grid sizes.
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Introduction
Mariela Mezquita 10ª
Math, Mr Bhoja
30 April 2008
COURSEWORK
Aim: To find out a formula for rectangles on different grid sizes.
That is done by multiplying the corners of the rectangle and then we subtract them.
Hypothesis:
Eg. (on a 10x10 grid size with a 2x3 rectangle)
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 | ||
72 | 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 | ||
3 x 11= 33
1 x 13=13
33-13 = 20
Then, move on the rectangle.
A 2 x 3 cube…
1 | 2 | 3 |
11 | 12 | 13 |
2 | 3 | 4 |
12 | 13 | 14 |
3 | 4 | 5 |
13 | 14 | 15 |
As we can see, when we have a 2 x 3 rectangle, the final result will always be 20.
Explanation:
If we substitute the
Middle
1 | 2 | 3 | 4 |
11 | 12 | 13 | 14 |
21 | 22 | 23 | 24 |
2 | 3 | 4 | 5 |
12 | 13 | 14 | 15 |
22 | 23 | 24 | 25 |
It will always be 60..
Explanation:
x | x+3 | ||||
x+20 | x+23 | ||||
x(x+23)
x+3(x+20)
expand
x(x+23) = x²+23x
x+3(x+20) = x²+20x+3x+60
x²+23x
x²+20x+3x+60
=60
x² and 23x cancel each other out, leaving us with a result of 60.
If we change the
Conclusion
So:
3x4 = 6n
FORMULA ABOVE ONLY APLLIES FOR A 4 X 5 RECTANGLE ON ANY GRID SIZE
So:
4x5= 12n
FORMULA ABOVE ONLY APLLIES FOR A 5 X 6 RECTANGLE ON ANY GRID SIZE
So:
5x6= 20n
FORMULA ABOVE ONLY APLLIES FOR A 6 X 7 RECTANGLE ON ANY GRID SIZE
So:
6x7= 30n
Overall Formula:
N(H-1)(L-1)
Extension:
In my extension I will draw crosses instead of rectangles. Multiply the opposite sizes and try to find out a formula.
Hypothesis: I think it will be similar to the rectangle formula but a little bit more complicated.
Example:
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 | ||
72 | 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 | ||
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