# Number Stairs Coursework

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Introduction

Number Stairs Coursework

Mathematics

Number stairs coursework

Aim:

I have been given a 10 by 10 number grid (as shown below) with a stair shape drawn on it. The stair shape is a 3–step stair and the total of the numbers inside it is 194.

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 |

I am required to carry out the followings:

- Part 1: To investigate the relationship between the stair total and the position of the stair shape on the grid for other 3-step stairs.

- Part 2: To investigate further the relationship between the stair totals and other step stairs on other number grids.

Plan for Part 1

I shall now investigate part 1 where I will show and explain the relationship between the stair total and the position of the stair shape on the grid for other 3- step stairs. The stair shape on the diagram is a 3 step stair because both the length and the width of the stair are 3 steps. Part 1 is basically all about drawing patterns and conclusions from the relationships, and constructing formulas based on them. To do this, I will change the position of the 3-step stair from the bottom left hand corner (see blue on diagram) from positions 1-5 (see green), vertically across the grid and calculate the total of the numbers inside it.

Middle

Total = an + b Therefore a= 6

6+b = 50

b = 50-6

b = 44

Therefore, 6n+44 can be used as a formula for every position possible on the number grid.

Testing my Predictions

To prove that my formula works, I can test my predictions by firstly manually calculating the total of the 3-step stair, and then algebraically calculating the total using my formula finally I will randomly select three positions to test. If my formula works I should have the same total for both of the stairs;

- Position 58 = 58 + 59 + 60 + 68 + 69 + 78 = 392

By using my formula (6n + 44), I should get the same total, thus if n = 58,

6 x 58 + 44 = 392

- Position 16 = 16 + 17 + 18 + 26 + 27+ 36 = 140

By using my formula (6n + 44), I should get the same total, thus if n = 16,

6 x 16 + 44 = 140

- Position 32 = 32+ 33 + 34 + + 27+ 36 = 140

By using my formula (6n + 44), I should get the same total, thus if n = 16,

6 x 16 + 44 = 140

Plan for Part 2

For part 2 of this task I shall investigate further the relationship between the stair totals and other step stairs on other number grids. To do this, I will change the size of the number stair (from a 1-step stair to a 5-step stair) and calculate the total of the algebraic letters inside it. This can indicate what the relationship between the totals of the number stairs are compared to their sizes.

Conclusion

Extending the Investigation

There are a number of possible ways to extend this investigation. For example, it is possible to determine formulas for any step stair on say an 8 by 8 grid and then comparing them with those from a 10 by 10 grid. This can be further extended to finding an overall formula for any step stair on say an 8 by 8 number grid as shown below:

I shall first investigate what the formulas are for other step stairs in order to find an overall formula for any step stair possible on an 8 by 8 grid. I then compare this to other step stairs.

1-Step stair

n |

Total: n

Formula: 1n

2-Step stair

n+9 | |

n | n +1 |

Total: n + n + 1 + n + 9

Formula: 3n + 10

3-Step stair

n + 17 | ||

n +9 | n + 10 | |

n | n +1 | n + 2 |

Total: n + n + 1+ n + 2 + n + 9 + n + 10

n + 17

Formula: 6n + 39

4-Step stair

n + 25 | |||

n +17 | n +18 | ||

n +9 | n + 10 | n + 11 | |

n | n +1 | n + 2 | n + 3 |

Total: n + n + 1+ n + 2+ n + 3 + n + 9 + n + 10 + n + 11+ n + 17 + n + 18 + n + 25

Formula: 10n + 96

5-Step stair

n +33 | ||||

n +25 | n+ 26 | |||

n +17 | n +18 | n + 19 | ||

n +9 | n + 10 | n + 11 | n + 12 | |

n | n +1 | n + 2 | n + 3 | n + 4 |

Total: n + n + 1+ n + 2+ n + 3 + n + 4 + n + 9 + n + 10+ n + 11+ n + 12 + n + 17 + n + 18+ n + 19 + n + 25 + n + 26 + n + 33

Formula: 15n + 190

By looking at the number stairs I can see another pattern similar to the numbers on a 10 by 10 grid.

10 by 10 grid 8 by 8 grid

Step Stair (s) | Formulas |

1 | n |

2 | 3n + 11 |

3 | 6n + 44 |

4 | 10n + 110 |

5 | 15n + 220 |

Step Stair (s) | Formulas |

1 | n |

2 | 3n + 10 |

3 | 6n + 39 |

4 | 10n + 96 |

5 | 15n + 190 |

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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