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• Level: GCSE
• Subject: Maths
• Word count: 8908

# Mathematical Coursework: 3-step 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 3-step 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 3-step 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

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1. 1+2+3+13+14+25= 58
 133 134 135 136 137 138 139 140 141 142 143 144 121 122 123 124 125 126 127 128 129 130 131 132 109 110 111 112 113 114 115 116 117 118 119 120 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
1. 2+3+4+14+15+26= 64
 133 134 135 136 137 138 139 140 141 142 143 144 121 122 123 124 125 126 127 128 129 130 131 132 109 110 111 112 113 114 115 116 117 118 119 120 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
1. 3+4+5+15+16+27= 70
 133 134 135 136 137 138 139 140 141 142 143 144 121 122 123 124 125 126 127 128 129 130 131 132 109 110 111 112 113 114 115 116 117 118 119 120 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
1. 4+5+6+16+17+28=76
 Term 1 2 3 4 5 total 58 64 70 76 82

Pattern= +6

As I now have found the pattern, this would allow me to predict the following 3-step 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 3-step stair shape.

 23 14 15 5 6 7

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 3-step stair by just adding 6.  However this technique would restrict me from picking out a random 3-step 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 3-step 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:

• 6N =6n x 1= 6
• 6+b=58

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:

• 58-6=52
• b= 52

To conclusion my new formula would be:

• 6n+52
 8cm by 8cm 9cm by 9cm 10cm by 10cm 11cm by 11cm 12cm by 12cm Difference Formula 6n+36 6n+40 6n+44 6n+48 6n+52 +4 Term 1 Total 42 46 50 54 58 +4 Term 2 Total 48 52 56 60 64 +4 Term 3 Total 54 58 62 66 70 +4 Term 4 Total 60 64 68 72 76 +4 Prediction for  Term 5 Total 66 70 74 78 82 +4 Pattern 6 6 6 6 6

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 3-step stairs in different grid you’ll have to. If I’m going down grids then:

However if I’m going up a grid

 11 1 2
 21 11 12 1 2 3
 31 21 22 11 12 13 1 2 3 4
 41 31 32 21 22 23 11 12 13 14 1 2 3 4 5
 11 1 2

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.

 N+10 n N+1

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.

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Conclusion

"1">

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8cm by 8cm grid- 4-step stair

 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8
 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8

9cm by 9cm grid

 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9

1+2+10=13

2+3+11=16

3+4+12=19

9cm by 9cm grid- 4-step stairs

 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9
 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59 60 61 62 63 46 47 48 49 50 51 52 53 54 37 38 39 40 41 42 43 44 45 28 29 30 31 32 33 34 35 36 19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9

The Calculations

1+2+3++410+11+12+19+20+28= 110

2+3+4+5+11+12+13+20+21+29= 120

3+4+5+6+12+13+14+21+22+30=130

9cm by 9cm grid- 5-step stairs

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