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Mathematics Coursework Explaining The Relationship Of The 3 Step Stair

1184 words - 5 pages

GSCE COURSEWORKPART 1Random SampleI drew 5 3-step stairs all of them were the same size, but they differed in position.Comment:1) Notice how all the totals are even2) The totals increase the higher up the grid you go3) Across goes up about 42 (92-50)4) Up goes up about 420 (470-50)Systematic SampleI drew 5 stairs all on the bottom line going across, I did this to see if there was a pattern.Comment:There is clearly a pattern here; the totals are increasing by 6, we call this a linear sequence.From this information I predict that the next total will be 80 and the 8th will be 92.Both my predictions were correct.Algebraic ExpressionI drew 1 stair with Algebra on it.Total:(x)+(x+1) +(x+2) +(x+10) +(x+11) +(x+20) = 6X+44Comment:The reason it's 6x is because there are 6 stairs (x+x+x+x+x+x)It's 44 because the totals across for these stairs are 4 across and 40 upwards.4: 1+1+2=41: This 1 came from the 11, because you go 1 across from the 10 to get 1140: 20+10+10=4010: This 10 came from the 11, because you go 10 up from the 1 to get 11PART 2I have changed the grids to see whether or not there is a pattern. I kept the shape name as one all the time. I have done 5, 6,7and 8 grids.There is clearly a pattern here; the totals are increasing by 4; so each time I increase the grid size by 1 the total increases by 4, there is a common difference.From this information I predict that the 9th grid will have a total of 46.Results:My prediction was correct.Algebraic ExpressionI have drawn a 3-step stair with algebra on it, the g represents grid.Totals:(x)+(x+1) +(x+1+1) +(x+g) +(x+g+1) +(x+g+g) = 6x+4g+4Comment:The reason it's 6x is because there are 6 stairs, each one with one x on it (x+x+x+x+x+x=6x), x represents the name of the shape.The reason it's 4g is because say for example you had a 5 grid, notice how every time you go up the total increases in accordance with the grid number; e.g.:So basically if it were any grid the number on top of one would always increase by the grid number. That is why the g is situated on top of the x in my algebra stairs; and since there are 4g's we write it down in the formula as 4g.The reason it's +4 is simply because it's (1+1+1+1). These '1's' are on my algebra stairs because in any stairs no matter what grid it is, you always go one across.So, my formula is: 6x+4g+4I will now experiment it on 3 different grids to see if it works.I will try it on a 3 grid, a 7 grid and a 10 grid.My formula worked for all 3 grids, from this I can conclude that the formula, 6x+4g+4 will work on any grid provided that it's a 3 step-stair.The formula 6x+4g+4 is a linear formula. I have found a formula that will work on any 3-step stair, now I need to find a formula which will work on any sized step.I will now change the stairs size to try and find a formula which will work on any stair size.I will do a 3, 4, 5 and 6 step stairs on a 10 grid.From this information I have found a link between triangle numbers and the numbers above:Triangle numbers are;...

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