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  • Level: GCSE
  • Subject: Maths
  • Word count: 2132

Number Stairs

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Introduction

Year 11 Algebra coursework – Number Stairs        Laura Bird

I am going to investigate into the stair total of different sized stair shapes on different sized number grids, the stair total being the total of the numbers inside the stair shape. I will then see if I can produce an algebraic formula to calculate this total for any sized stair shape on any sized grid. I shall start by finding a formula for the total of numbers in a 3-step stair on a 10×10 grid (for example see fig. 1), based on one of the numbers inside the stair shape.

image42.jpg

Fig. 1

This is a 3-step stair. The stair total for this stair shape would be:

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

I started simple to form some simple equations to calculate the stair total.

Eg 1. For a 3-step stair on a 10×10 grid I came up with the following formula to calculate stair total, based on one of the numbers in the stair shape:

image00.png

The formula for finding the stair total for a 3-step stair on a 10×10 grid would be:

6x + 44        x being the number in the bottom left of the stair.

I then calculated similar formulas for different sized steps (see Egs) on different sized grids and displayed them in the Table 1 to see if I could notice any patterns:

image40.pngimage22.pngimage08.pngimage06.pngimage07.pngimage01.png

Egs:image41.png

image23.pngimage11.png

          3-step stair on a 9×9 grid

image03.pngimage04.pngimage02.png

image05.png

6x + 40

image06.pngimage07.pngimage01.pngimage10.pngimage09.pngimage08.png

image11.pngimage04.pngimage03.pngimage13.pngimage12.png

...read more.

Middle

image34.pngimage33.png

3-step stair

3-step stair = 6x + 4g +4

I then went on to calculate more of these formulas for other size step stairs on any size grids. I showed these formulas in Table 2:

Table 2

Step size

Formula - any size grid

1

x

2

3x + 1g + 1

3

6x + 4g + 4

4

10x + 10g +10

5

15x + 20g + 20

From Table 1 and Table 2 I can see that the co-efficient of x is always the same for each step size no matter what size grid it is on, and that it is only the end part of the formula (the + something bit) that changes on each size grid. The co-efficient of x is always a triangular number as the stair shapes make a triangle shape (Fig. 2), therefore the formula for finding the coefficient of x will be the same as the formula for finding triangular numbers:  n(n+1)   n being the step size.

2

Fig. 2

image36.png

For example for a 2-step stair the formula is always 3x plus something, therefore to find the co-efficient of x, in this case 3, I would substitute the step size, in this case 2, into the triangular formula:

2(2+1)           =        4+2        =        6        =        3

              2                  2                2

Therefore this equation works for a 2-step stair as the end result was 3. I then went on to check if this formula worked with other sized stairs.

...read more.

Conclusion

rd difference for the y values on a 9×9 grid was 10 as the y values for a 9×9 grid are all multiples of 10:

Step size (n)

1

2

3

4

5

y

0

10

40

100

200

1st difference        +10        +30        +60        +100        

2nd difference        +20        +30        +40

3rd difference        +10+10

This was the case, therefore the 3rd difference, which is also the number divided by 6 in my equation is always grid size plus 1, therefore in my equation would be:

[ n(n+1) ] x + [ ( g + 1 ) n³ - ( g + 1 )n ]      

[    2      ]        [ (            6    )        (    6     )  ]

Where n = step size, x = number in the bottom corner of the stair and g = grid size

I then simplified this equation:

[ n(n+1) ] x + [ g + 1( n³ - n ) ]

[     2      ]       [          6                   ]

This is the formula for any sized step stair on any size grid.

I then went on to try this formula on a different size step stair on a different size grid to prove and check that it works. The example I used was a 3-step stair on a 10×10 grid with the number in the bottom left of the step shape being 25. The stair total for this step stair should equal 194. I substituted my example into my formula to calculate the stair total to see if this answer was given:

[ n(n+1) ] x + [ g + 1( n³ - n ) ] = Stair Total

                        [      2     ]       [    6                          ]

[ 3(3+1) ] 25 + [ 10 + 1( 3³ - 3 ) ] = Stair Total

                      [       2    ]         [      6                          ]

[ 9+3 ] 25 + [ 11 (27 - 3 ) ] = Stair Total

                            [   2   ]         [  6                        ]

[ 12 ] 25 + [ ( 11 )24 ] = Stair Total

                               [  2  ]          [ (  6  )     ]

150 + 44 = Stair Total

194 = Stair Total

This proves that my formula works as it produced the correct stair total for a 3-step stair on a 10×10 grid.image39.png

...read more.

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