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

Mathematical Coursework: 3-step stairs

Extracts from this document...

Introduction

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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.

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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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...read more.

Middle

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  1. 1+2+3+13+14+25= 58

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  1. 2+3+4+14+15+26= 64

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  1. 3+4+5+15+16+27= 70

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  1. 4+5+6+16+17+28=76

Term

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total

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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.                                                                    

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  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.

image14.png

image15.png

image26.png

image18.pngimage17.png

image19.png

image02.png

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

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Term 2 Total

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Term 3 Total

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Term 4 Total

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Prediction for  Term 5 Total

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Pattern

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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:

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However if I’m going up a grid

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image29.png

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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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...read more.

Conclusion

"1">

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image46.png

8cm by 8cm grid- 4-step stair

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9cm by 9cm grid

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image49.png

1+2+10=13

2+3+11=16

3+4+12=19

image50.png

9cm by 9cm grid- 4-step stairs

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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

image51.png

9cm by 9cm grid- 5-step stairs

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...read more.

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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