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Mathematics - Investigating Stair Shapes

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Introduction

Mathematics - Investigating Stair Shapes Introduction This piece of coursework is going to investigate and look into the relationship between stair numbers, and the stair total. It is basically all about drawing patterns and conclusions from the relationships, and constructing formulas based on them. Initially, I will begin by looking into 3 stair shapes, and how the stair numbers relate to the stair total in a grid of that size. Then I will broaden the coursework by looking into different sized stair shapes, and investigating their relationships. Near to the end of this coursework, I will conclude my results by finding formulas for the individual sized stair shapes, and then I will find the general formula which works for any sized stair shape on any sized grid. Part 1 - 3 stair shapes If a stair shape is moved once to the right, its stair total is increased by 6. ...read more.

Middle

Here is what we can conclude from this: = x + 44 = 6 numbers = 6 x + 44 Test If x = 1... 1+2+3+11+12+21 = 50 (6x1 add 44 = 50) If x = 2... 2+3+4+12+13+22 = 56 (6x2 = 12 add 44 = 56) It is important to test these kind of formulas, as simple mistakes could be found, which could alter the answer drastically, although they could be rectified very easily, so it is best to check your answers at an early stage. Part 2 Here I will test different sized stair shapes, like 4, 5, and 6 stair shapes. When a 4 stair shape is moved to the left once, its stair total is decreased by 10. Similarly, when it is moved once to the right, its stair total is increased by 10. Stair number: 1 1+2+3+4+11+12+13+21+22+31 = 120 Stair number: 2 2+3+4+5+12+13+14+22+23+32 = 130 When a 4 stair number is moved down once, its stair total is decreased by 100. ...read more.

Conclusion

I have compiled them into a table, shown below: Stair Shape Formula 3 Stair Shape 6x + 44 4 Stair Shape 10x + 110 5 Stair Shape 15x + 220 6 Stair Shape 21x + 385 7 Stair Shape 28x + 616 8 Stair Shape 36x + 924 Using these formulas together, I managed to work out a general formula, which also takes into account the fact that there can be any grid size. We all started off with the formula: an3+bn2+cn+d A = a 6th of the third difference, which is 11 D = the 0th term, which is 0 I finally worked out the formula: This takes into account everything. Evaluation Because I found out the final formula, I believe that I did well to get that far. However, I do believe that if I had had enough time, I would have got to grips with the whole investigation a lot better, and would have then had the time and understanding to then expand on my work and hopefully get a better mark on it. ?? ?? ?? ?? ...read more.

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