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  • Level: GCSE
  • Subject: Maths
  • Word count: 2784

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

Extracts from this document...

Introduction

image00.png

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:

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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.

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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

...read more.

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

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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:

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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

...read more.

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

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(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.

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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

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(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

...read more.

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