Mathematical Coursework: 3step stairs
Extracts from this document...
Introduction
91  92  93  94  95  96  97  98  99  100 
81  82  83  84  85  86  87  88  89  90 
71  72  73  74  75  76  77  78  79  80 
61  62  63  64  65  66  67  68  69  70 
51  52  53  54  55  56  57  58  59  60 
41  42  43  44  45  46  47  48  49  50 
31  32  33  34  35  36  37  38  39  40 
21  22  23  24  25  26  27  28  29  30 
11  12  13  14  15  16  17  18  19  20 
1  2  3  4  5  6  7  8  9  10 
In order to understand the relationship between the stairs total and the position of the stairs shape on the grid. Foremost, I needed to distinguish if the stair total had something in common. As I believe this would enable me to find a pattern, which would eventually develop into an algebraic equation. To be able to do this I had to experiment with the grid, by shifting the 3step stairs into different direction et cetera. In addition I would’ve seen if the pattern and algebraic equation would’ve work in different size grid such as 9cm by 9cm or in different size stair shape such as 4 step stairs.
In order to find the pattern I will have to draw various grids, this will enable me to eventually discover the pattern. Firstly, for my first experimental grid I decide the start in the bottom corner and randomly select a few other 3step stairs.
91  92  93  94  95  96  97  98  99  90 
81  82  83  84  85  86  87  88  89  80 
71  72  73  74  75  76  77  78  79  70 
61  62  63  64  65  66  67  68  69  60 
51  52  53  54  55  56  57  58  59  50 
41  42  43  44  45  46  47  48  49  40 
31  32  33  34  35  36  37  38  39  30 
21  22  23  24  25  26  27  28  29  20 
11  12  13  14  15  16  17  18  19  10 
1  2  3  4  5  6  7  8  9  10 
Because I haven’t discovered the algebraic equation, this would allow me to find the total stair number in less than minutes. I would have to find the total by using the basic method of adding.
The Calculations
1+2+3+11+12+21= 505+6+7+15+16+25=74
83+73+63+64+65+74=42247+48+49+57+58+67=326
91  92  93  94  95  96  97  98  99  100  
81  82  83  84  85  86  87  88  89  90  
71  72  73  74  75  76  77  78  79  80  
61  62  63  64  65  66  67  68  69  70  
51  52  53  54  55  56  57  58  59  60  
41  42  43  44  45  46  47  48  49  50  
31  32  33  34  35  36  37  38  39  40  
21  22  23  24  25  26  27  28  29 ...read more.Middle
125 126 127 128 129 130 131 132 109 110 111 112 113 114 115 116 117 118 119 120 97 98 99 100 101 102 103 104 105 106 107 108 85 86 87 88 89 90 91 92 93 94 95 96 73 74 75 76 77 78 79 80 81 82 83 84 61 62 63 64 65 66 67 68 69 70 71 72 49 50 51 52 53 54 55 56 57 58 59 60 37 38 39 40 41 42 43 44 45 46 47 48 25 26 27 28 29 30 31 32 33 34 35 36 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12
Pattern= +6 As I now have found the pattern, this would allow me to predict the following 3step stairs if I was still using the method. In order to check if my prediction is right I would have to again add up the numbers in the 3step stair shape.
5+6+7+14+15+23 = 82 This proves my Prediction to be correct. Therefore I’m able to find the total number of the 3step stair by just adding 6. However this technique would restrict me from picking out a random 3step stair shape out of the 9cm by 9cm grid. As I would have to follow the grid side ways from term 1 towards e.g. term 20. Thus making it’s time consuming. After analysing the grids, the table of results and my prediction I have summed up an algebraic equation which would allow me to find out the total of any 3step stair shape in a matter of minutes. The criterion of why I haven’t explained what B stands for is because I haven’t found it yet. Nevertheless using the annotated notes on the formula my formula would look like this now:
Now I would need to find the value of b in order to use my formula in future calculations. Therefore I will take the pattern number of the total:
To conclusion my new formula would be:
After analysing the summary of results and the diagrams above I’ve discovered a link between the difference and the change in formula. As you can see in the highlighted part in the table is the 10cm by 10cm grid. I have discovered the formula for the grid in part one. And looking at the other grids I noticed that when I went one grid down from 10cm by 10cm grid the formula went down by 4. However when I went up a grid the formula went up by four. Therefore I believe the formula of finding a 3step stairs in different grid you’ll have to. If I’m going down grids then: However if I’m going up a grid
Firstly the formula when finding the stairs total in the 10cm by 10cm grid is as following: 6n+44 N being the bottom corner number in this case number one. However this formula can’t be used when dealing with other grids. To Conclude I predict the diagram below might be the algebraic formula I’m looking for.
The criterion of why I suggested the algebraic formula might possible is the correct one when dealing with the 10cm by 10cm grid. Because if you add n by 1 it will add up to 2, this is the number on the above stair shape. Also if you add n (which is 1) by 10 it will add up to 11, this is also the number that occurs on the above stair shape. To check if my algebraic formula is correct I will randomly selected another stair shape in the 10cm by 10 cm grid. As I haven’t discovered my formula yet I will just experiment with the stair shape and see how my formula would look like.
