• 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
  9. 9
    9
  10. 10
    10
  11. 11
    11
  12. 12
    12
  13. 13
    13
  14. 14
    14
  15. 15
    15
  16. 16
    16
  17. 17
    17
  18. 18
    18
  19. 19
    19
  20. 20
    20
  21. 21
    21
  22. 22
    22
  23. 23
    23
  24. 24
    24
  25. 25
    25
  26. 26
    26
  27. 27
    27
  28. 28
    28
  • Level: GCSE
  • Subject: Maths
  • Word count: 4455

Mathematics - Number Stairs

Extracts from this document...

Introduction

Ben Foster        GCSE Mathematics Coursework 02/02/2008

Ben Foster

-------- School---------

Centre Number --------

GCSE Coursework: Number Stairs

Teacher: -------


Table of Contents

Aim

First Part

3 Step-Staircase / Grids of width 10

Second Part

3 Step-Staircase

3 Step-Staircase / Grid Width 8

3 Step-Staircase / Grid Width 9

3 Step-Staircase / Grid Width 11

3 Step-Staircase / Grid Width 12

2 Step-Staircase

2-Step Staircase/ Grid Width 8

2-Step Staircase/ Grid Width 9

2-Step Staircase/ Grid Width 10

2-Step Staircase/ Grid Width 11

2-Step Staircase/ Grid Width 12

1 Step-Staircase

4 Step-Staircase

4 Step-Staircase / Grid Width 8

4 Step-Staircase / Grid Width 9

4 Step-Staircase / Grid Width 10

4 Step-Staircase / Grid Width 11

4 Step-Staircase / Grid Width 12

5 Step-Staircase

5 Step-Staircase / Grid Width 8

5 Step-Staircase / Grid Width 9

5 Step-Staircase / Grid Width 10

5 Step-Staircase / Grid Width 11

5 Step-Staircase / Grid Width 12

The Final Formula

Finding the “n” term of the formula

Finding the “g” term of the formula

Finding the number term of the formula

Aim

In this investigation, I will be aiming to find out the formulas of the combinations of the size of number stairs and the size of grid widths. Eventually I will convene a formula connecting the stair totals and other steps on other number grids.

First Part

3 Step-Staircase / Grids of width 10

image00.png

21

11

12

1

2

3

n

1

2

3

4

5

T

50

56

62

68

74

There is a pattern here, so a formula may be constructed:

It is going up by 6 every time thus making it “6n”

If the 1st term is 50 then the 0th term is 44

So therefore the formula here is T = 6n+44

To make sure this formula works, the n term can be substituted with a number, for example, 20

T = 6 x 31 + 44 = 230

So:

51

41

42

31

32

33

31 + 32 + 33 + 42 + 41 + 51 = 230

So therefore my prediction works but I must prove it algebraically:

n+20

n+10

n+11

n

n+1

n+2


n + (n+1) + (n+2)

...read more.

Middle

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

5

4 Step-Staircase        

4 Step-Staircase / Grid Width 8

25

17

18

9

10

11

1

2

3

4

n

1

2

3

4

5

T

100

110

120

130

140

Suspected formula: T = 10n + 90

Prediction / Test: 10 x 25 + 90 = 340

49

41

42

33

34

35

25

26

27

28

25 + 26 + 27 + 28 + 33 + 34 + 35 + 41 + 42 + 49 = 340

Algebraic Proof:

n+24

n+16

n+17

n+8

n+9

n+10

n

n+1

n+2

n+3

n + (n+1) + (n+2) + (n+3) + (n+8) + (n+9) + (n+10) + (n+16) + (n+17) + (n+24) = 10n + 90


8

9

10

11

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

 T = 10n + 90

5

4 Step-Staircase / Grid Width 9

28

19

20

10

11

12

1

2

3

4

n

1

2

3

4

5

T

110

120

130

140

150

Suspected formula: T = 10n + 100

Prediction / Test: 10x 24 + 100 = 340

51

42

43

33

34

35

24

25

26

27

24 + 25 + 26 + 27 + 33 + 34 + 35 + 42 + 43 + 51 = 340

Algebraic Proof:

n+27

n+18

n+19

n+9

n+10

n+11

n

n+1

n+2

n+3

n + (n+1) + (n+2) + (n+3) + (n+9) + (n+10) + (n+11) + (n+18) + (n+19) + (n+27) = 10n + 100

8

9

10

11

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

 T = 10n + 90

 T = 10n + 100

5


4 Step-Staircase / Grid Width 10

31

21

22

11

12

13

1

2

3

4

n

1

2

3

4

5

T

120

130

140

150

160

Suspected formula: T = 10n + 110

Prediction / Test: 10 x 23 + 110 = 340

53

43

44

33

34

35

23

24

25

26

23 + 24 + 25 + 26 + 33 + 34 + 35 + 43 + 44 + 53 = 340

Algebraic Proof:

n+30

n+20

n+21

n+10

n+11

n+12

n

n+1

n+2

n+3

n + (n+1) + (n+2) + (n+3) + (n+10) + (n+11) + (n+12) + (n+20) + (n+21) + (n+30) = 10n + 110

8

9

10

11

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

 T = 10n + 90

 T = 10n + 100

T = 10n + 110 

5

From here, I see a pattern of a jump size of 10 so I predict for grid widths 11 and 12 will be T = 10n + 120 and T = 10n + 130.


4 Step-Staircase / Grid Width 11

34

23

24

12

13

14

1

2

3

4

n

1

2

3

4

5

T

130

140

150

160

170

Suspected formula: T = 10n + 120

Prediction / Test: 10 x 47 + 120 = 590

80

69

70

58

59

60

47

48

49

50

47 + 48 + 49 + 50 + 58 + 59 + 60 + 69 + 70 + 80 = 590

Algebraic Proof:

n+33

n+22

n+23

n+11

n+12

n+13

n

n+1

n+2

n+3

n + (n+1) + (n+2) + (n+3) + (n+11) + (n+12) + (n+13) + (n+22) + (n+23) + (n+33) = 10n + 120

8

9

10

11

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

 T = 10n + 90

 T = 10n + 100

T = 10n + 110 

T = 10n + 120 

5


4 Step-Staircase / Grid Width 12

37

25

26

13

14

15

1

2

3

4

n

1

2

3

4

5

T

140

150

160

170

180

Suspected formula: T = 10n + 130

Prediction / Test: 10 x 15 + 130 = 280

51

39

40

27

28

29

15

16

17

18

15 + 16 + 17 + 18 + 27 + 28 + 29 + 39 + 40 + 51 = 280

Algebraic Proof:

n+36

n+24

n+25

n+12

n+13

n+14

n

n+1

n+2

n+3

n + (n+1) + (n+2) + (n+3) + (n+12) + (n+13) + (n+14) + (n+24) + (n+25) + (n+36) = 10n + 130

8

9

10

11

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

 T = 10n + 90

 T = 10n + 100

T = 10n + 110 

T = 10n + 120 

 T = 10n + 130

5


5 Step-Staircase        

5 Step-Staircase / Grid Width 8

33

25

26

17

18

19

9

10

11

12

1

2

3

4

5

n

1

2

3

4

5

T

195

210

225

240

255

Suspected formula: T = 15n + 180

Prediction / Test: 15 x 34 + 180 = 690

66

58

59

50

51

52

42

43

44

45

34

35

36

37

38

34 + 35 + 36 + 37 + 38 + 42 + 43 + 44 + 45 + 50 + 51 + 52 + 58 + 59 + 66 = 690

Algebraic Proof:

n+32

n+24

n+25

n+16

n+17

n+18

n+8

n+9

n+10

n+11

n

n+1

n+2

n+3

n+4

n + (n+1) + (n+2) + (n+3) + (n+4) + (n+8) + (n+9) + (n+10) + (n+11) + (n+16) + (n+17) + (n+18) + (n+24) + (n+25) + (n+32) = 15n + 180

8

9

10

11

12

1

 T = n

 T = n

 T = n

T = n 

T = n 

2

 T = 3n + 9

T = 3n + 10 

 T = 3n + 11

 T = 3n + 12

T = 3n + 13

3

T = 6n + 36 

 T = 6n + 40

T = 6n + 44 

T = 6n + 48 

T = 6n + 52 

4

 T = 10n + 90

 T = 10n + 100

T = 10n + 110 

T = 10n + 120 

 T = 10n + 130

5

T = 15n + 180 


5 Step-Staircase / Grid Width 9

37

28

29

19

20

21

10

11

12

13

1

2

3

4

5

n

1

2

3

4

5

T

215

230

245

260

275

Suspected formula: T = 15n + 200

Prediction / Test: 15 x 39 + 200 = 785

75

66

67

57

58

59

48

49

50

51

39

40

41

42

43

39 + 40 + 41 + 42 + 43 + 48 + 49 + 50 + 51 + 57 + 58 + 59 + 66 + 67 + 75 = 785

Algebraic Proof:

n+36

n+27

n+28

n+18

n+19

n+20

n+9

n+10

n+11

n+12

n

n+1

n+2

n+3

n+4

...read more.

Conclusion

Staircase Size

1

2

3

4

Formula

0g+0

g+1

4g+4

10g+10

Row 1

               g+1          3g+3           6g+6

Row 2

                         2                   3

Row 3

1

After finding out the method of differences, it turns out to be exactly the same sequence for finding out the “g” term considering the differences between the staircase sizes of the number terms follow the same sequence as the “g” term!

So now, there is a constant difference of 1, therefore, 1 ÷ 6 = 1/6s³

Now, I would expect a quadratic but since the sequence is exactly the same as the “g” term formula, I am predicting an automatic linear sequence:

Staircase Size

1

2

3

4

1/6³ Formula

0-(1/6 x 1³)

1-(1/6 x 2³)

4-(1/6 x 3³)

10-(1/6 x 4³)

Formula Solution

[-1/6]

[-1/3]

[-1/2]

[-2/3]

Row 1

                    [-1/6]              [-1/6]               [-1/6]

As predicted, the second part to the formula is -1/6.

The overall product is now: TS = n(0.5s² + 0.5s) + g(1/6s³ - 1/6s) + (1/6s³ - 1/6s)image03.jpg

To neaten the formula up a bit:

This seems ludicrous at first so I will now substitute the letters with numbers to prove this formula works.

With “n” being 1, “s” being 3 and “g” being 10, I should come up with a total step number of 50.

image04.jpg

image05.jpgimage06.jpgimage07.jpgimage09.jpgimage08.jpg

Staircase Size = 3, Grid width = 10, bottom left number = 1

1 + 2 + 3 + 11 + 12 + 21 = 50

So I have seen that this works fully and now that I have the overall formula connecting the staircase sizes, the grid widths and the bottom left number of the staircases together; the total of the numbers in any staircase of any size of any grid width can be worked out by using this formula:image03.jpg

...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 Grid Investigation.

    My findings show also by using algebra in the equations, I can construct an algebraic formula. By using `N` in the top left of the algebraic table, the equation, when fully worked out will always give the correct constant answer in any square selection size in any size grid.

  2. Number stairs

    THE GENERAL FORMULA FOR ANY 3-STEP STAIR GRID SIZE IS: T=15x + 20 (n+1) SIX- STEP STAIRS: To start my investigation I am going to start by using a 10 by 10 Number grid below: Below is a portion of a 6 step- stair in algebraic terms: The total of

  1. Number Grids Investigation Coursework

    This is it: 14 16 34 36 So the difference between the products of opposite corners in this square would be: (top right x bottom left) - (top left x bottom right) = 16 x 34 - 14 x 36 = 544 - 504 = 40 I will use algebra

  2. Number Grid Investigation.

    = 20. Product difference = 20. Again, both times the Product difference of a 3 X 2 square is 20. I will now prove algebraically that the product difference will always be 20 with a 3 X 2 square. x x + 2 x 10 x + 12 X(x + 12)

  1. Number Grid Coursework

    Generalisation Using this apparent relationship, it can be assumed that, when a 2x2 box is placed anywhere on the grid, the difference of the two products will be z for all possible widths. Therefore, the following equation should be satisfied with any real value of a and any real

  2. Maths-Number Grid

    46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95

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

    values we get -200 12 x 12 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 38 39 40 41 42

  2. Maths - number grid

    Again I am going to use algebra to prove that the defined difference of my 3x3 squares is correct. r (r+2) (r+20)(r+22) (r+2)(r+20)-(r+22)r =r (r+20)+2(r+20) - r -22r =r +20r+2r+40-r -22r =40 I have now calculated a trend for my 2x2 squares and came to a difference of 10 and

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