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# 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

## 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

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