• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Number Stairs Investigation – Course Work

Extracts from this document...

Introduction

Number Stairs Investigation – Course Work

Aim

The aim of this coursework is to find relationships and patterns in the total of

all the numbers in ‘Number Stairs’ such as the one below. For example, the total

of the number stair shaded in black is:

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

I have to investigate any relationships that might occur if this stair was in a

different place on the grid.

Part One

For other 3-stepped stairs, investigate the relationship between the stair total

and the position of the stair shape on the grid.

By looking at this grid, the stair total is:

25 + (25+1) + (25+2) + (25+3) + (25+10) + (25+11) + (25+20)

If we call 25, the number in the corner of the stair, ‘n’ then we get:

n + (n + 1) + (n + 2) + (n + 3) + (n + 10) + (n + 11) + (n + 20) =

6n + 1 + 2 + 3 + 10 + 11 + 20 =

6n + 44

This formula should work with every number stair that can fit onto a 10 by 10

grid. I can say the total for square n (Tn) is, “n + n + 1 + n + 2 + n + 3 + n +

10 + n + 11 + n + 20”, because where ever n is on the grid, the number of the

square:

·        One place right of it will be n +1

·        Two places right of it will be n +2

...read more.

Middle

·        6 x 23 + 44 = 138 + 44 = 182

If n = 34 then

·        6 x 45 + 44 = 270 + 44 = 314

If n = 56 then

·        6 x 56 + 44 = 336 + 44 = 380

If n = 67 then

·        6 x 67 + 44 = 402 + 44 = 446

If n = 78 then

·        6 x 78 + 44 = 468 + 44 = 512

Stair Number (n)        1        12        23        34        45        56        67        78

Stair Total (Tn)        50        116        182        248        314        380        446        512

6n + 44        50        116        182        248        314        380        446        512

All the results using the formula are correct, so I can come to the conclusion

that the formula for the total of a stair:

·        On a 10 by 10 grid

·        Which travels downwards from left to right

·        With a height of 3 squares

Is:

6n + 44

Part Two (Extension)

Investigate further the relationship between the stair total and other step

stairs on the number grids.

I am going to investigate patterns and relationships in:

·        The position of the stairs

·        The height of the stairs

·        The width of the grid

I will then put everything together and produce a universal formula.

Throughout this section, the symbols for the variable inputs will be as follows:

Tn = Total of stair

n = Stair number (the number in the bottom left had corner of the stair)

h = Number of squares high the stair is

w = Width of grid (number of squares)

Relationships between different stair heights on a 10 by 10 grid

...read more.

Conclusion

3rd triangle would multiply n). The formula for triangle numbers is:

h2 + h

2

So the first part of the formula will be:

n (h2 + h)

   2

If I include this in a table, I may get some better results:

Height (h)        1        2        3        4        5        6

Stair Number (n)        1        1        1        1        1        1

Triangle Number of h        1        3        6        10        15        21

Triangle Number of h x n        1        3        6        10        15        21

11h3 – 11h        0        66        264        660        1320        2310

(11h3 11h)ק6        0        11        44        110        220        385

(Triangle Number of h x n) + (11h3 11h)ק6        1        14        50        120        235        406

Total (T1)        1        14        50        120        235        406

I included 11h3 11h from my last table. I divided 11h3 11h by 6. This is

because 6 was the number that got 11h3 11h closest to the required total. I

noticed that if you add the Triangle Number of h x n and (11h3 11h)ק6

together you get the Total. So my formula for any height Number stair on a 10 by

10 grid is:

n (h2 + h) + (11h3 11h)

2                 6

Testing the formula

Tn = Total of stair

n = Stair number (the number in the bottom left had corner of the stair)

h = Number of squares high the stair is

w = Width of grid (number of squares)

Test One – Square 25, Height 3

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

T25 = 25 x ((32 + 3) ק 2) + ((11 x 33 11 x 3) ק 6)

T25 = 25 x (11 ק 3) + (297 33) ק 6

T25 = 25 x 6 + 44

T25 = 194

Test Two Square 68, Height 3

Total = 68 + 69 + 70 + 78 + 79 + 88 = 194

... thats all ive don

...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

    if it is a 10 x 10 [10] or 11 x 11 [11] represented as algebra to give us the total of the numbers added together To start my investigation for the general formula I need to establish the highest common factors, by using our values above, which are 36,

  2. Number Grids Investigation Coursework

    + 160 - a2 - 44a = 160 As I have proved this, I will now try a 4 x 4 square. 40 42 44 46 60 62 64 66 80 82 84 86 100 102 104 106 The difference between the products of opposite corners, in this example, equals: (top right x bottom left)

  1. Investigation of diagonal difference.

    the amount of G's needed in the bottom two corners of a cutout. The height of the cutout takeaway 1 defines the amount of G's needed in the bottom two corners, and since this is a trend for all the different size cutouts I can now substitute this number with an algebraic expression.

  2. Algebra Investigation - Grid Square and Cube Relationships

    matter what number is chosen to begin with (n), a difference of 30 will always be present. Any rectangular box with width 'w', and a height of 2 It is possible to ascertain from the above examples that each box follows a certain trend (each progressive increase in width shows an overall increase in the difference by +10).

  1. Number Grid Investigation

    n n + (a-1) n + 8(a-1) n + (a-1) + 8(a-1) n x [n +(a-1)+8(a-1)] = n2+(a-1)+8n(a-1) n+(a-1) x [n +8(a-1)]= n2+8n+8(a-1)2 The difference between these is 8(a-1)2 As you can see the table is almost indistinguishable from the 10 x 10 algebra grid I completed earlier.

  2. For other 3-step stairs, investigate the relationship between the stair total and the position ...

    17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

  1. Number Stairs

    The following table shows the stair total (T) depending on the relevant stair number for the 9x9 grid. Which are from 1 to 5. N T 1 46 2 52 3 58 4 64 5 70 46 52 58 64 70 +6 +6 +6 +6 It is clear that the difference between the numbers is 6.

  2. Step-stair Investigation.

    28 15 16 17 18 19 20 21 8 9 10 11 12 13 14 1 2 3 4 5 6 7 By using the formula 10X+10g+10=S, I worked out the total of the numbers inside the blue area of the 4-step stair.

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