# For 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. Investigate further the relationship between the stair totals and other step stairs on other number grids.

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Introduction

Maths Coursework

By Tom Nixon

10

Middle

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

This is a 3-step stair.

The total of the numbers inside the stair shape is

25+26+27+35+36+45 = 194

The stair total for this 3-step stair is 194.

## PART 1

For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.

## PART 2

Investigate further the relationship between the stair totals and other step stairs on other number grids.

PART 1

The total of the squares inside the stair is all the squares added together. The stair number is the number in the bottom left hand corner of the stair. We can call this number n.

In order to see a pattern between the totals of the shapes, we can arrange information in a table.

n | 6n | +44 |

1 | 6 | 50 |

2 | 12 | 54 |

3 | 18 | 64 |

4 | 24 | 68 |

25 | 150 | 194 |

As you can see from the above table we can come to a formula of:

### T = 6n + 44

T is the stair total, n the stair number and 44 is the remaining number.

n+20 | ||

n+10 | n+11 | |

n | n+1 | n+2 |

T = n + (n+1) + (n+2) + (n+10) + (n+11) + (n+20)

T = 6n + 44

I have tested the formula above on other stairs on a 10 x 10 grid but not on other grid sizes.

e.g 1+2+3+11+12+21 = 50

T = 6n + 44

T = 6x1 + 44

## T = 50

On a 9x9 grid, instead of each square increasing by ten vertically, the numbers increase by 9. We can see that this reflects the size of the grid.

Using the same algebraic method as previously, we can work out the increase or decrease in the stair total when the grid is changed, with g being the grid number. (I.e. 9x9)

n+2g | ||

n+g | n+g+1 | |

n | n+1 | n+2 |

Conclusion

T = 6+7+8+16+17+26

### T= 80

Vector (b/0) vertical by b units

51 | ||

41 | 42 | |

31 | 32 | 33 |

21 | ||

11 | 12 | |

1 | 2 | 3 |

Each square has increased by 30, which is the number of places that the stair number has moved up by, times the grid size, which is 10.

We can arrange a formula which, when we insert into the original stair, will work out the total of the new stair.

When doing the formula, we can simply add on to the formula for working out the total of a 3-step stair on any grid.

n+2g +3b | ||

n+g +3b | n+g+1 +3b | |

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

T = (n+3b) + (n+1+3b) + (n+2+3b) + (n+g+1+3b) + (n+2gt+3b) + (n+g+3b)

T = 6n + 18b + 4g + 4

T equals 6 times the stair number, plus 18 times the number of units moved vertically, plus 4 times the grid number, in this instance it is 10, plus 4.

T = 6x1 + 18x10 + 4x10 + 4

T = 6 + 180 + 40 + 4

### T = 230

T = 31+41+51+32+33+42

### T = 230

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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