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Number stairs.

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Introduction

GCSE MATHEMATICS COURSEWORK:

NUMBER STAIRS

NAME: PRATEEK BHANDARI

          FORM: 11C

          DATE:  7TH MARCH 2004

PART 1:

For the other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.

As described in the question I will be investigating the relationship between stair total and the position of the stair shape on a 10 by 10 grid. To find that out, I have decided to take three stairs on different positions on the grid and find their stair total. After that, I will find a formula through which I can calculate the stair total of any stair number.

Below is a three level stair shape on a 10 by 10 grid.

                                                If we write the same stair in terms n (stair number) then it

                                                It will be:

                                                                                            Stair number up by 20        

   stair number up by 10        stair number up by 11

   stair number         stair no. up by 2

                                                                                                             stair number up by 1.  

This shows that:        

  • Every time we move to the right we increase by 1.
  • Every time we move up we increase by 10. Mainly because of the grid size.
  • Similarly if we move 1 to the left it will decrease by 1.
  • By moving down one square it will decrease by 10.

The diagram shows:

  • n to n+1  is increased by one
  • n to n+2 is increased by two
  • n to n+10 is increased by ten
  • n to n+11 is increased by eleven
  • n to n+20 is increased by twenty.
...read more.

Middle

6 x 38 + 44 = 272

                                                                If reversed it would be:

6n + 44 = 272

                                                                6n = 272 – 44

                                                                6n = 228

                                                                  n = 38

STAIR 1:

:                                                               The total for this stair is:

                                                                1 + 2 + 3 + 11 + 12 + 21 = 50

                                                                Therefore this can be calculated in a similar way:

6 x 1 + 44 = 50

                                                                If reversed it would be:

6n + 44 = 50

                                                                6n = 50 – 44

                                                                6n = 6

                                                                  n = 1

In conclusion, I can now calculate the stair total of a three level stair on any position on the grid 10 by 10 through the general formula 6n + 44.

PART 2:

Investigate further the relationship between the stair total and other step stairs on other number grids.

  My objective in this section will be to further investigate the relationship between the stair total and other step stairs on other number grids. To do that I will need to find a general formula through which I can calculate any stair total or stair number on any number grid at any position.

 To find the formula I will first and foremost try the same formula in part 1 on a stair shape on an 8 by 8 number grid.                                                        

                                                                The stair total for this stair will be:                                                                                                                                

1 + 2 + 9 = 12                      

                                                                If I now use the formula 6n + 44 it will be:

                                                                6 x 1 + 44 = 50

                                                                This therefore clearly indicates that if we use the                                      

                                                                formula 6n + 44 on other number grids then it                

                                                                will not work.

The main reasons for this are:

  • The difference between the upper square and the stair number is of 8.
  • If I had used the same method i.e. by adding up all the numbers and the nth terms then I could have reached a formula for that particular kind of stair but not as a general formula. For this it would have been: n + n + 1 + n + 8 = 3n + 9.  Similarly if I had done this on a different stair on the same grid then it would have had a different formula. Therefore I am now going to look for a general formula through which I can calculate any stair total on any grid.
...read more.

Conclusion

        = 15x + 20 + 20g

15 x 5 + 20 + 20 x 9

        = 275

Below is stair 79 on a 12 by 12 grid. The value of g on this grid will become 12.

:                                                               The total for this stair is:

79 + 80 + 81 + 91 + 92 +103 = 526

                                                                If I calculate this stair total with the same formula

                                                                then it will be:

x + x +1 + x + 2 + x + g + x + g +1 + x + 2g

= 6x + 4 + 4g

                                                               = 6 x 79 + 4 + 4 x 12

                                                               = 526

Below is 24 on a 6 by 6 grid and the value of g will become 6.

                                                              The total for this stair is:                            

24 + 25 + 30 = 79

                                                              The stair total through the formula will be:  

                                                              x + x +1 + x g

                                                              = 3x + 1 + g

                                                              = 3 x  24 + 1 + 6    

                                                              = 79      

CONCLUSION:

  Altogether, I have found 2 formulas. One was for calculating the stair total of a three level stair on a 10 by 10 grid and the other was to calculate the stair total of any stair on any number grid and on any position.

The two formulas were:

6n + 44 = stair total

x(stair number) + numbers added to x + g(the value of the grid) = stair total

  If I had more time then I could have done the following changes:

  • Test it on more grids such as 20 by 20 or 30 by 30. By doing that I could have gone into more depth regarding the formula.
  • Find a more precise formula which would have fixed numbers such as the one in Part 1.
  • I could have also tried putting the stairs on upside down positions or any other positions.
  • I would have also tested stairs with the width of 6 squares, which would have made my task more difficult.

...read more.

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