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  • Level: GCSE
  • Subject: Maths
  • Word count: 4076

Step-stair Investigation.

Extracts from this document...

Introduction

William Murray

Step-stair Investigation

Cousework Submission 8th December 2003

        For my GCSE Maths coursework I was asked to investigate the relationship between the stair total and the position of the stair shape on the grid. Secondly I was asked to investigate the relationship further between the stair totals and the other step stairs on other number grids. The number grid below has two examples of 3-step stairs. I will use Algebra as a way to find the relationship between the stair total and the position of the stair on the grid. 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

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The first thing I will do is find the formula for all 3-step stairs on a size 10 grid.  

I started off by making the bottom left hand number X. X is also the position of the stair on the grid. So in the diagram coloured red above X=15. I then added up the rest of the numbers in the three-step stair in terms of X. So 16= X+1, 17=X+2, 25=X+10 etc. The 3 step-stair in terms of X looks like this:

X+20

X+10

X+11

X

X+1

X+2

If you simplify all the Xs and all the numbers you end up with this: 6X + 44, X+X+1+X+2+X+10+X+11+X+20=6X + 44. By investigating the formula above you will find that it is the formula for all 3-step stairs on a size 10 grid. I worked this out by adding together all the numbers in the 3-step stair and then using the formula to see if the formula comes up with the total of all the numbers in the 3-step stair.

...read more.

Middle

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By using the formula 10X+10g+10=S, I worked out the total of the numbers inside the blue area of the 4-step stair.

(10*18)+(10*7)+10= 180 + 70+ 10= 260

Then  added up the numbers in the 4-step stair as I did before.

18+19+20+21+25+26+27+32+33+39= 260

The two examples above prove that the formula 10x+10g+10 calculates the total of the numbers inside the area covered by a 4-step stair on any grid size.

5 step stairs:

X+4g

X+3g

X+3g+1

X+2g

X+2g+1

X+2g+2

X+g

X+g+1

X+g+2

X+g+3

X

X+1

X+2

X+3

X+4

By adding all the Xs, all the gs and all the numbers together I got:

X+X+1+X+2+X+3+X+4+X+g+X+g+1+X+g+2+X+g+3+X+2g+X+2g+1+X+2g+2+X+ 3g+X+3g+1+X+4g = 15X+20g+20. This is the formula for all 5-step stairs on any size grid.

To prove this formula works for all size grids and therefore works in general I drew two different sized grids and did the following calculations:

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By using the formula 15X+20g+20=S, I worked out the total of the numbers inside the green area of the 5-step stair.

(15*25)+(20*10)+20=595

Then I added all the numbers in the 5-step stair to see if it came up with the same answer:

25+26+27+28+29+35+36+37+38+45+46+47+55+56+65=595.

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By using the formula 15X+20g+20=S, I worked out the total of the numbers inside the turquoise area of the 5-step stair:

(15*1)+(20*5)+20=15+100+20=135

I then added up all the numbers in the 5-step stair to see if it gave the same number:

1+2+3+4+5+6+7+8+9+11+12+13+16+17+21=135

These two diagrams prove that 15X+20g+20=S on all sized grids.

6-step stairs:

X+5g

X+4g

X+4g+1

X+3g

X+3g+1

X+3g+2

X+2g

X+2g+1

X+2g+2

X+2g+3

X+g

X+g+1

X+g+2

X+g+3

X+g+4

X

X+1

X+2

X+3

X+4

X+5

By adding all the Xs all the gs and all the numbers up together I got this:

X+X+1+X+2+X+3+X+4+X+5+X+g+X+g+1+X+g+2+X+g+3+X+g+4+X+2g+X+2g+1+X+2g+2+X+2g+3+X+2g+3+X+3g+X+3g+1+X+3g+2+X+4g+X+4g+1+X+5g = 21X+35g+35.

To prove that this formula works I drew up two grids and used the formula to calculate the total of the numbers inside the 6-step stair and saw if it was the right answer on both grids.

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I did this calculation to see if the formula works.

21X+35g+35 = (21*11) + (35*8) + 35 = 546

11+12+13+14+15+16+19+20+21+22+23+27+28+29+30+35+36+37+43+44+51= 546

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...read more.

Conclusion

(3-1) = 2

Σ Tr =  T1 + T2 = 1+3 = 4

r = 1

So, for a 3 step stair the value of (blank)g + (blank) = 4. If you look up the formula for a 3-step stair is 6X+4g+4. I then proposed that the formula:

image05.png

                   (n-1)           (n-1)image05.png

n(n+1)   +Σ   Tr =     g +     Σ  Tr =

      2           r = 1                         r = 1

Is the formula for any step stair on any sized grid.

I tried it on the 4 step stairs I had investigated before. The 3-step stair, the 4-step stair, the 5-step stair and the 6-step stair.

So, by replacing n with 3, for a 3-step stair I get this:

image06.png

                (3-1)                                          (3-1)

3(3+1)  +  Σ   Τr = T1 + T2   g   +     Σ    Tr = T1+T2

    2           r =1                                    r = 1

So by using this formula that I have explained above:

The formula gives: 6X+4g+4.  T1 + T2 = 4, (1+3).

This is the formula for a 3-step stair on any size grid. The formula works.

4-step stair:

                (4-1)                                            (4-1)image07.png

4(4+1)  +  Σ   Τr = T1 + T2  + T3   g   +     Σ    Tr = T1+T2 + T3

    2          r = 1                                             r = 1        

So by using this formula that I have exlained above:

The formula gives: 10X+10g+10.  T1 + T2 + T3 = 10 (1+3+6)

This is the formula for a 4-step stair on any size grid. The formula works.

 5-step stair:

                 (5-1)                                                 (5-1)image08.png

5(5+1)  +   Σ   Τr = T1 + T2  + T3 + T4  g   +     Σ    Tr = T1+T2 + T3 + T4

    2            r = 1                                                   r = 1        

By using the formula I have explained above;

The formula gives: 15X+20g+20. T1 + T2 + T3 + T4 = 20, (1+3+6+10)

This is the formula for a 5-step stair on any sized grid. The formula works.

6-step stair:

image09.pngimage09.png

                 (6-1)                                                           (6-1)

6(6+1)  +   Σ   Τr = T1 + T2  + T3 + T4 + T5    g   +     Σ    Tr = T1+T2 + T3 + T4 + T5

    2            r = 1                                                            r = 1        

By using the formula I have explained above:

The formula gives: 21X+35g+35. This is the formula for a 6-step stair on any sized grid. The formula works.

I have now proved that the formula I found works for any step stair size on any size grid.

image10.png

                   (n-1)          (n-1)image05.png

n(n+1)   +Σ    Tr =     g +      Σ     Tr =

      2           r = 1                        r = 1

This concludes my first coursework submission. Submitted on:

8th December 2003.

William Murray

...read more.

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