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• Level: GCSE
• Subject: Maths
• Word count: 3157

Random Sample.

Extracts from this document...

Introduction

GSCE COURSEWORK

PART 1

Random Sample

 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

I drew 5 3-step stairs all of them were the same size, but they differed in position.

 Number Of Shape Calculations Total 1 1+2+3+11+12+21 50 8 8+9+10+18+19+28 92 34 34+35+36+44+45+54 248 71 71+72+73+81+82+91 470 78 78+79+80+88+89+98 512

Comment:

1) Notice how all the totals are even

2) The totals increase the higher up the grid you go

3) Across goes up about 42 (92-50)

4) Up goes up about 420 (470-50)

Systematic Sample

 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

I drew 5 stairs all on the bottom line going across, I did this to see if there was a pattern.

 Number Of Shape Calculations Total 1 1+2+3+11+12+21 50 2 2+3+4+12+13+22 56 3 3+4+5+13+14+23 62 4 4+5+6+14+15+24 68 5 5+6+7+15+16+25 74

Comment:

There is clearly a pattern here; the totals are increasing by 6, we call this a linear sequence.

From this information I predict that the next total will be 80 and the 8th will be 92.

 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

Results:

 Number Of Shape Calculations Total 6 6+7+8+16+17+26 80 8 8+9+10+18+19+28 92

Both my predictions were correct.

Algebraic Expression

 X+20 X+10 X+11 X X+1 X+2

I drew 1 stair with Algebra on it.

Total:

(x)+(x+1) +(x+2) +(x+10) +(x+11) +(x+20) = 6X+44

Comment:

The reason it’s 6x is because there are 6 stairs (x+x+x+x+x+x)

It’s 44 because the totals across for these stairs are 4 acrossand 40 upwards.

4: 1+1+2=4

1: This 1 came from the 11, because you go 1 across from the 10 to get 11

40: 20+10+10=40

10: This 10 came from the 11, because you go 10 up from the 1 to get 11

PART 2

 21 22 23 24 25 16 17 18 19 20 11 12 13 14 15 6 7 8 9 10 1 2 3 4 5
 31 32 33 34 35 36 25 26 27 28 29 30 19 20 21 22 23 24 13 14 15 16 17 18 7 8 9 10 11 12 1 2 3 4 5 6
 43 44 45 46 47 48 49 36 37 38 39 40 41 42 29 30 31 32 33 34 35 22 23 24 25 26 27 28 15 16 17 18 19 20 21 8 9 10 11 12 13 14 1 2 3 4 5 6 7

 57 58 59 60 61 62 63 64 49 50 51 52 53 54 55 56 41 42 43 44 45 46 47 48 33 34 35 36 37 38 39 40 25 26 27 28 29 30 31 32 17 18 19 20 21 22 23 24 9 10 11 12 13 14 15 16 1 2 3 4 5 6 7 8

Middle

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46

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37

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44

45

28

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Results:

 Name Of Grid Calculations Total 9 1+2+3+10+11+19 46

My prediction was correct.

Algebraic Expression

 x+g+g x+g x+g+1 x x+1 x+1+1

I have drawn a 3-step stair with algebra on it, the g represents grid.

Totals:

(x)+(x+1) +(x+1+1) +(x+g) +(x+g+1) +(x+g+g) = 6x+4g+4

Comment:

The reason it’s 6x is because there are 6 stairs, each one with one x on it (x+x+x+x+x+x=6x), x represents the name of the shape.

The reason it’s 4g is because say for example you had a 5 grid, notice how every time you go up the total increases in accordance with the grid number; e.g.:

 21 22 23 24 25 16 17 18 19 20 11 12 13 14 15 6 7 8 9 10 1 2 3 4 5

+5

So basically if it were any grid the number on top of one would always increase by the grid number. That is why the g is situated on top of the x in my algebra stairs; and since there are 4g’s we write it down in the formula as 4g.

The reason it’s +4 is simply because it’s (1+1+1+1). These ‘1’s’ are on my algebra stairs because in any stairs no matter what grid it is, you always go one across.

 21 22 23 24 25 16 17 18 19 20 11 12 13 14 15 6 7 8 9 10 1 2 3 4 5

+1

So, my formula is: 6x+4g+4

I will now experiment it on 3 different grids to see if it works.

I will try it on a 3 grid, a 7 grid and a 10 grid.

 7 8 9 4 5 6 1 2 3

 Name Of Grid Calculations Total 3 1+2+3+4+5+7 22
 Name Of Grid Key Calculations Total 3 x=1        g=3 (6x1)+(4x3)+4 22
 43 44 45 46 47 48 49 36 37 38 39 40 41 42 29 30 31 32 33 34 35 22 23 24 25 26 27 28 15 16 17 18 19 20 21 8 9 10 11 12 13 14 1 2 3 4 5 6 7
 Name Of Grid Calculations Total 7 5+6+7+12+13+19 62
 Name Of Grid Key Calculations Total 7 x=5       g=7 (6x5)+(4x7)+4 62
 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
 Name Of Grid Calculations Total 10 45+46+47+55+56+65 314
 Name Of Grid Key Calculations Total 10 x=45       g=10 (6x45)+(4x10)+4 314

My formula worked for all 3 grids, from this I can conclude that the formula, 6x+4g+4 will work on any grid provided that it’s a 3 step-stair.

The formula 6x+4g+4 is a linear formula. I have found a formula that will work on any 3-step stair, now I need to find a formula which will work on any sized step.

I will now change the stairs size to try and find a formula which will work on any stair size.

I will do a 3, 4, 5 and 6 step stairs on a 10 grid.

3-STEP STAIR

 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

4-STEP STAIR

 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

Conclusion

6

10 x 3= 3 x 10          Notice how these are triangle numbers. However, it’s the value of n+1.

20 x 3= 4 x 15          So we substitute it with the triangle numbers formula except we use (n+1):

35 x 3= 5 x 21          [n (n+1) (n+1) +1)]/2, but remember we have to divide by 3 because we

multiplied by 3 earlier so the formula is:  [n (n+1) (n+2)]/6

In our series the 4 is from a 3 step, the nth step would need to be (n-1) in

the above formula (Blue). We substitute n with (n-1):

[(n-1) ((n-1) +1) ((n-1) +2)]/6

We can simplify the formula into this:

[n (n+1) (n-1)]/6

So the universal formula is:   [n (n+1)]/2 x + [n (n+1) (n-1)]/6 g+ [n (n+1) (n-1)]/6

n= step size

x= number of shape

g= grid number

I will now test to see if it works, I want to get the formula for a 3-step stair, so I do the following:

[3(3+1)]/2 x + [3(3+1)(3-1)]/6 g + [3(3+1) (3-1)]/6  which equals:

6x+4g+4 – My formula works for the 3-step stairs.

I will do the same, but this time for a 7 step stairs:

[7(7+1)]/2 x + [7(7+1) (7-1)]/6 g + [7(7+1) (7-1)]/6 which equals:

28x+56g+56

Clearly my formula works, through the use of algebra and my insight into the relationships of series and triangle numbers I was able to achieve this. I needed to use a variety of methods and techniques to complete the universal formula, always linking it back with previous formulas and tables.

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

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