# To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).

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Introduction

Aim: To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).

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 |

On posed with this investigation I began to look at the relativity of the stair shape on the grid and its relation to the numbers it coincides with. I then began to imagine the grid as an axis whereby the x and y lay on the grid.

## Part 1

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

Firstly I had to find the relationship of the stair shape where it lay on the grid to the stair total. I began to look at different ways, I found that by working systematically on the grid i.e. starting by looking at horizontal, vertical and diagonal movements I would be able to find the common factors involved.

In the example we are given the numbers:

25, 26, 27, 35, 36, 45

They total to 194

To work out the placement of the stair shape I used the smallest number of the stair to translate the shape from, in this case 25.

Middle

182

33

34

23

24

25

44 | 188 | ||

34 | 35 | ||

24 | 25 | 26 |

45 | 194 | ||

35 | 36 | ||

25 | 26 | 27 |

46 | 200 | ||

36 | 37 | ||

26 | 27 | 28 |

Although there is no common number here at the end of the total for each stair shape there is a common number fond when the difference is calculated.

200-194=6

194-188=6

188-182=6

This means that for every horizontal movement either left or right on the grid the total difference between each stair shape for a 3-step stair would be 6.

Positive diagonal movement on the grid:

23 | 62 | ||

13 | 14 | ||

3 | 4 | 5 |

34 | 128 | ||

24 | 25 | ||

14 | 15 | 16 |

45 | 194 | ||

35 | 36 | ||

25 | 26 | 27 |

56 | 260 | ||

46 | 47 | ||

36 | 37 | 38 |

There is no common number found here at the end of each total. The common difference is:

260-194=66

194-128=66

128-62=66

This means that when the stair shape is moved diagonally in the “y=x” direction the difference increase is 66. The total stair shape total for the stair shape will increase by 66 for every increase or decrease by 1.

Negative diagonal movement on the grid:

36 | 140 | ||

26 | 27 | ||

16 | 17 | 18 |

45 | 194 | ||

35 | 36 | ||

25 | 26 | 27 |

54 | 248 | ||

44 | 45 | ||

34 | 35 | 36 |

63 | 302 | ||

53 | 54 | ||

43 | 44 | 45 |

302-248=54

248-194=54

194-140=54

The negative diagonal difference when translating the shape in a “y=-x” manner is 54. So for every movement up 1 on the grid in that direction you can add 54 to the total.

After working out the vertical, diagonal and horizontal difference in the entire 3-step stair shapes I found that there was a common difference with all the stair shapes and that the difference found related to 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.

To further my investigation I decided to observe the relationship between larger stair shapes and their position on the grid, also their relationship with each other.

2-step stair

Using the same process as the 3-step stair I began to look at the horizontal, vertical and diagonal differences. I first started with looking at the horizontal difference

35 | 86 | |

25 | 26 | |

36 | 89 | |

26 | 27 | |

34 | 83 | |

24 | 25 |

All the last numbers of the individual stair totals all have a multiple of three at the end.

89-86=3

86-83=3

The total difference overall is three and the multiple of the last number is also three. The horizontal difference is equalled to three.

Vertical difference:

45 | 116 | |

35 | 36 | |

35 | 86 | |

25 | 26 | |

25 | 56 | |

15 | 16 |

Conclusion

14

15

35 | 86 | |

25 | 26 |

46 | 119 | |

36 | 37 |

The numbers 3, 6 and 9 can be seen; they all are multiples of three. The total overall difference for the diagonal 2-step stair:

119-86=33

86-53=33

As can be seen the number 33 is also a multiple of 3.

Negative diagonal 2-step stair:

26 | 59 | |

16 | 17 |

35 | 86 | |

25 | 26 |

44 | 113 | |

34 | 35 |

As can be seen the numbers shown at the end are 3, 6 and 9, they are all multiples of three.

Working out the common difference:

113-86=27

86-59=27

Twenty-seven is seen to be the common difference tis however is also a multiple of three.

4 step-stair

Horizontal difference:

54 | 350 | |||

44 | 45 | |||

34 | 35 | 36 | ||

24 | 25 | 26 | 27 | |

55 | 360 | |||

45 | 46 | |||

35 | 36 | 37 | ||

25 | 26 | 27 | 28 |

56 | 370 | |||

46 | 47 | |||

36 | 37 | 38 | ||

26 | 27 | 28 | 29 |

They all are multiples of 10, so already you can see a common difference.

Working out the overall difference:

370-360=10

360-350=10

Vertical difference:

55 | 360 | |||

45 | 46 | |||

35 | 36 | 37 | ||

25 | 26 | 27 | 28 |

45 | 260 | |||||||||

35 | 36 | |||||||||

25 | 26 | 27 | ||||||||

15 | 16 | 17 | 18 | |||||||

35 | 160 | |||||||||

25 | 26 | |||||||||

15 | 16 | 17 | ||||||||

5 | 6 | 7 | 8 |

60 is a common number in this, the total difference worked out:

360-260=100

260-160=100

There is a difference of 100 for each progressional movement vertically up or down on the 4-step star grid.

Positive diagonal movement:

55 | 360 | |||

45 | 46 | |||

35 | 36 | 37 | ||

25 | 26 | 27 | 28 |

44 | 250 | |||

34 | 35 | |||

24 | 25 | 26 | ||

14 | 15 | 16 | 17 |

33 | 140 | |||

23 | 24 | |||

13 | 14 | 15 | ||

3 | 4 | 5 | 6 |

360-250=110

250-140=110

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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