# Number Stairs - For part one, I am investigating the relationship between the stair total and the position of the stair shape on the grid.

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Introduction

## Part one

For part one, I am investigating the relationship between the stair total and the position of the stair shape on the grid.

I began my investigation by drawing a large 10x10 number grid like so:

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 |

## Key

- 3-step stair 1
- 3-step stair 2
- 3-step stair 3
- 3-step stair 4

### Calculations

3-step stair 1 - 25+26+27+35+36+45 = 194

3-step stair 2 – 65+66+67+75+76+85 = 434

3-step stair 3 – 61+62+63+71+71+81 = 410

3-step stair 4 – 21+22+23+31+31+41 = 170

As you can see, within this number grid I produced four 3-step stair diagrams, labelled 3-step stair 1, 3-step stair 2, 3-step stair 3 and 3-step stair 4. I then shaded them in individually with a key below to indicate the colour to the name.

I then added the numbers together in the individuals 3-step stairs to reveal four different sets of answers like shown when doing my calculations.

I have studied these answers and concluded that they are all even.

To investigate further, I looked at what would happen if I moved the 3-step stairs to the left and right and up and down. I predict that if I move the 3-step stair upwards, the stair total will be greater and if I was to move the 3-step stair downwards, the total will be less. If I moved the 3-step stair to the left, it would be less than moving it to the right.

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 |

## Key

- 3-step stair 5
- 3-step stair 6
- 3-step stair 7

- 3-step stair 8

### Calculations

3-step stair 5 – 28+18+19+8+9+10 = 92

3-step stair 6 – 98+88+89+78+79+80 = 512

3-step stair 7 – 91+81+82+71+72+73+ = 470

3-step stair 8 – 21+11+12+1+2+3 = 50

Middle

I will begin using the corner number and the number of squares in the 3-step stair to work out the stair total and I will use 3-step stair 8 as an example to calculate the formula.

6 = number of squares in the 3-step stair

n = corner number

t = total

3-step stair 8

Stair total = 50

31 | 32 | 33 | 34 |

21 | 22 | 23 | 24 |

11 | 12 | 13 | 14 |

1 | 2 | 3 | 4 |

6 x n – 10 = -4

6 x n + 10 = 16

6 x n + 20 = 26

6 x n + 30 = 36

6 x n + 40 = 46 – need four more to achieve the total

6 x n + 44 = 50

6n + 44 + t (the formula)

As you may see from above, I have moved onto trial and error to work out the formula. I understood that in a 2-step stair there are six squares. I predicted that the corner number had some part in the formula and so selected it to use in my formula seen as it was a low number and I was trying to reach a number that was larger than itself (50) therefore I began by multiplying. This formula would work for every 3-step square.

## Part two: 4-step stair

The aim of this section is to investigate further the relationship between the stair totals and other step stairs on other number grids.

Here is a 4-step stair number grid:

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 |

## Key

- 4-step stair 13
- 4-step stair 14
- 4-step stair 15

- 4-step stair 16

I have forwarded from part one to extended from part one and have moved onto a 4-step stair rather than a 3-step stair like before.

## Calculations

4-step stair 13 – 37+27+17+7+28 +18 +8 +19+9+10 = 180

4-step stair 14 – 97+87+77+67+88+78+68+79+69+70 = 780

4-step stair 15 – 91+81+71+61+82+72+62+73+63+64 = 720

4-step stair 16 – 31+21+11+1+22+12+2+13+3+4 = 120

780 – 720 = 60

180 – 120 = 60

780 – 180 = 600

720 – 120 = 600

60 x 10 = 600

Conclusion

6 = number of squares in the 3-step stair

n = corner number

t = total

+44 was the original number to be added in the original formula so seen as the number grid has just doubled then I will start of by doubling this number to +88:

6n + 88 = t

As an example, I will use 3-step stair 24:

3-step stair 24

Stair total = 100

62 | 64 | 66 | 68 |

42 | 44 | 46 | 48 |

22 | 24 | 26 | 28 |

2 | 4 | 6 | 8 |

(6 x 2) +88 = 100

This proves that my formula works for the two times table but now I must test to see whether it works on the three times table. From working out a formula for the three times table I will be able to work out a formula overall for all times tables.

273 | 276 | 279 | 282 | 285 | 288 | 291 | 294 | 297 | 300 |

243 | 246 | 249 | 252 | 255 | 258 | 261 | 264 | 267 | 270 |

213 | 216 | 219 | 222 | 225 | 228 | 231 | 234 | 237 | 240 |

183 | 186 | 189 | 192 | 195 | 198 | 201 | 204 | 207 | 210 |

153 | 156 | 159 | 162 | 165 | 168 | 171 | 174 | 177 | 180 |

123 | 126 | 129 | 132 | 135 | 138 | 141 | 144 | 147 | 150 |

93 | 96 | 99 | 102 | 105 | 108 | 111 | 114 | 117 | 120 |

63 | 66 | 69 | 72 | 75 | 78 | 81 | 84 | 87 | 90 |

33 | 36 | 39 | 42 | 45 | 48 | 51 | 54 | 57 | 60 |

3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 |

## Key

- 3-step stair 25
- 3-step stair 26
- 3-step stair 27

## 3-step stair 28

## Calculations

3-step stair 25 – 63+33+36+3+6+9= 150

3-step stair 26 – 84+54+57+24+27+30= 276

3-step stair 27 – 294+264+267+234+237+240= 1536

3-step stair 28 – 273+243+246+213+216+219= 1410

1536-1410= 126

276-150= 126

1536-276= 1260

276-150= 126

126 x 10 = 1260

As you may see, I am able to multiply the smallest total by ten to reveal the largest number. I cannot find any reasonable explanation to this and therefore it is an anomalous result. This may not affect my formula though:

6 = number of squares in the 3-step stair

n = corner number

t = total

As expected, I will be using this formula 6n + 132 = t on 3-step stair 25:

3-step stair 25

Stair total = 150

93 | 96 | 99 | 102 |

63 | 66 | 69 | 72 |

33 | 36 | 39 | 42 |

3 | 6 | 9 | 12 |

(6 x 3) + 132 = 150

6 = number of squares in the 3-step stair

n = corner number

t = total

a = times table

This proves that my formula of 6n + 132 works. I am now able to say that if ‘a’ represents the time’s table you are trying to work out then this is my overall formula for the times table number grids:

6n + 44a = t

C:\My Documents\maths cwk\Kelly Duggan\NDO\10Y1.doc Tuesday, 24 June 2003

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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