• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  • Level: GCSE
  • Subject: Maths
  • Word count: 2132

Number Stairs

Extracts from this document...

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.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Number Stairs, Grids and Sequences essays

  1. Number stairs

    together without a formula is= 1+2+3+16+17+31= 70 [The stair total for this 3-step stair is 70] My general formula to find the total values in a 3-step stair on a 15 by 15 grid is correct. This is because all the stair values in a 3-step stairs added together on

  2. Investigation of diagonal difference.

    a 2 x 3 cutout from a 10 x 10 grid and see if the end products are correct. n n + (X - 1) n + (Y - 1)G n + (Y - 1)G + (X - 1) 1 1 + (3 - 1)

  1. Number Grid Coursework

    value of z where: a is the top-left number in the box; (a + 1) is the top-right number in the box, because it is always "1 more" than a; (a + z) is the bottom-left number in the box, because it is always "the grid width (z)"

  2. Investigate the differences between products in a controlled sized grid.

    Investigation on a 2 by 2 box in a 9 by 9 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

  1. Number Stairs

    depending on the relevant stair number for the 8x8 grid. Which are from 1 to 5. 42 48 54 60 66 +6 +6 +6 +6 It is clear that the difference between the numbers is 6. This means that the formula, to solve the total, must include a multiple of 6.

  2. Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted.

    Rectangles Now I will investigate rectangular number grids. I have chose to keep the width (W) the same but vary the depth (D) of the grid. 2 x 3 Grid 1 2 3 4 5 6 2 x 4 Grid 1 2 3 4 5 6 7 8 2 x 5 Grid 1 2 3 4 5 6

  1. Maths-Number Grid

    73 74 75 83 84 85 93 94 95 3 � 3 Grids:- 1 2 3 11 12 13 21 22 23 74 75 76 84 85 86 94 95 96 52 53 54 62 63 64 72 73 74 From the above grid I have found out that my

  2. 100 Number Grid

    I will prove this prediction using an algebraic formula: X X + 1 X + 2 X + 10 X + 11 X + 12 X + 20 X + 21 X + 22 Step 1. x (x + 22)

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work