Stair Shape on a 10x10 Number Grid
Introduction
For this investigation I will be investigating the relationship
between no. total and stair no. on a 10x10 number grid. I will also be
comparing results when the stairs are moved around the grid, finding
patterns that may occur.
Method
A diagram of the stair shape is shown below. There are three numbers
on the bottom layer, two on the second and one on the top, starting
from left. The stair no. will be identified by the number in the
bottom left hand corner of the stair. The grid starts from one in the
bottom left hand corner and ascends to 100 in the top right. This is
done so the stairs are in an up-right position rather than up side
down. As shown below.
I began by planning the first set of stairs…
I shaded in no.s 1,2, 3, 4 and 5 to make the outline of the stair case
clear. The no. in the bottom left hand corner is1, therefore this
stair will be known as stair shape no. 1…
Then, to work out the total no. I added up all the numbers in the
stair shape, using a calculator, to ensure the results are accurate…
The answer = 50
Therefore;
Stair no. = 1 Stair total = 50
I proceeded to apply this method to the next few stair no.s
To be able to compare the relationship between stair no. and total
horizontally across the grid, I am going to chose the next four stair
no.s to the right of 1.i.e stair no.s 1-5.This will make the formulae
easier to work out.I found the following results;
Stair no.
1
2
3
4
5
Stair total
50
56
62
68
72 74
+6 +6 +6 +4? +6
For stair no.5 the total did not follow the pattern.I will check it's
total for a second time…
Given Equation we have to find out the summation of natural numbers starting from ‘a’ to ‘n’.
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I will use arithmetic and algebra to investigate the relationships between the grid and the stair further. The variables used will be: Position of stair on grid = X Sum of all the numbers within the stair = S Step Size= n Grid size= g 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11
60 1,45 0,56 0,90 0,84 1,00 0,05 0,59 0,77 0,40 80 1,45 0,62 2,00 0,65 0,65
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