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Introduction

Tom Hartrick                07/10/2005

GCSE Mathematics Coursework

Introduction

In my mathematics GCSE coursework I shall be investigating the total and difference in sets of stairs in different grid sizes.  I will be investigating the relationship between stair totals on different grids.  This will also include different stair sizes.

I will choose to display my findings using a mixture of graphs, tables and grids.  I shall also try to work out the formula for each of the number of step sizes.

After I have finished investigating the different sized number stairs I will attempt to find out the different sized number grids.

At the end of my investigation I will conclude by demonstrating and explaining the relationship between all sizes of stairs.

Method

I shall be carrying out my stair challenge by finding out the correct information.  I shall do this by experimenting with the size of the grid corresponding to the size of the stairs used in the grid.

I shall try to work out the formulas by recognising any similarities in the results obtained.

Part 1- Three Step grid

I have investigated the three step stair (some of my findings can be viewed on the graph paper in the back).  During my investigation I believe that I have worked out the formula for the total inside the three step stairs on a 10 x 10 grid.

Middle      x+x+1+x+2+x+10+x+11+x+20

= 6x+44

1. Noticing that this is the total I realised that it is not so hard to work out the formula for the total.  Therefore due to the above diagram it is evident that the formula must be, 6x + 44.  It is easy to work out because there is six ‘x’s then you have to times x by 6 then you just add the other numbers together to get 44.
2. Lastly I have noticed that this formula does in fact work for any stairs anywhere in this number grid.

I noticed something about the common stair total.  It is evident that the larger the corner number is the larger the stair total is going to be.  The reason for this is that the higher ‘x’ is then the more it is multiplied.  Although the corner number affects the multiplication of x, it does not affect the number 4 which always remains the same no matter what the corner number is.  So, the three step stairs could be located anywhere in the grid and the only part of the formula that will change is the amount of times that you multiply x.  Here is a table simplifying this theory:

 X Total 25 194 63 422 58 394

Conclusion

In conclusion, in this part I have learnt that the formula for any three step total in a 10 x 10 grid is: 6x + 44.

Conclusion

X+x+1+x+2+x+3+x+4+x+g+x+g+1+x+g+2+x+g+3+x+g+g+x+g+g+1+x+g+g+2+x+g+g+g+x+g+g+g+1+x+g+g+g+g = 15x+20g+20

On all of the grids on which I tested this I found that this was constantly the formula. So the formula for the five step shape is; 20x+15g+20.  Here are all of the above results in the form of a table:

 Stair Shape Total 1 T = x 2 T = 3x + g + 1 3 T = 6x + 4g + 4 4 T = 10x + 10g + 10 5 T = 15x + 20g + 20

Formula Linking All of the Formulas

After now finding out all of the formulas for the entire stair shapes I shall now try to figure out a way to link the all into on formula.  This final formula is:

t = Tnx + ∑ Tg + ∑T

My workings out for this formula can be found in the back of his booklet on a piece of paper.

Conclusion

In conclusion, after working out these formulas including the final formula I have noticed a number of things.  Firstly all of the different sizes of stairs follow the same simple rule and that is; as you increase vertically you add the grid size number and then when you go along horizontally you always increase by one digit.  The larger the number stairs are the more they have increased both horizontally and vertically therefore giving a larger total inside the stairs.  The rule which I have just explained is applied in every grid and for any size number stairs.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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This mathematical investigation makes good use of algebraic expressions to analyse the relationship between numbers within varying grid sizes. It has a reasonable structure and the algebraic simplification is carried out effectively. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/07/2013

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