• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 3711

Number Stairs

Extracts from this document...

Introduction

image32.pngimage31.pngimage30.pngimage29.png

image00.png


I have been given a number grid that counts in ascending order from one to a hundred, beginning at the bottom left hand corner to end at the top right corner with the number one hundred. With this grid I have been given the task of investigating the relationship/s between the grid size, the stair size, the stair total and the ‘n’ number.

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

The step shape given above is a 3-step stair, simply because it consists of 3 steps.

The stair total is labelled as being the sum of all of the numbers in the stair shape:

        24 + 25 + 26 + 34 + 35 + 44 = 212

                The Stair total for this 3-step shape is 212

The ‘n’ number is defined as being the smallest number of the stair shape, in the grid above it is specified as ‘ 24’.

I will systematically work my way through this problem to find appropriate algebraic solutions to simplify the workings of the stair totals. In doing so, I will use different size grids and use different size stairs. Again, investigating relationships and discovering formulae for each problem I encounter.

I am going to start with a 10 by 10 grid with a 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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

n

T

1

56

2

62

3

68

4

74

5

80

6

86

...read more.

Middle

Using the same method as the previous, I will develop a formula.

Again, you can see that the ‘T’ column is increasing by 6 each time the ‘n’ number is increased by 1.

The ‘T’ column increases by 6 each time, so we multiply the ‘n’ number by the increase:

  • 6n, and then add the remaining amount to end up with the ‘T’ number.
  • 6n + 36 = T

To check this formula I will calculate another sum to which I know the solution to:

n

T

1

42

2

48

3

54

4

60

5

66

37

258


image33.png

Using another method, I will ensure that my formula is correct and if possible simplify the expression given.

n+3G

n+3G+1

n+3G+2image12.png

n+2G

n+2G+1

n+2G+2

n+G

n+G+1

n+G+2

n

n+1

n+2

n+n+1+n+2+ n+G+n+G+1+n+2G=T

Simplifying the above equation using means of collecting up will give me:

6n+4G+4=T

                For this grid, it would be:        6n+4*8+4=T

I have developed formulas for each grid (8 by 8, 9 by 9 and 10 by 10), I have come to the simple conclusion that the general formula 6n+4G+4=T applies to all grids as long as the stair number remains the same at 3 (3-step stair).

Developing further on the generic subject matter, I will find solutions on moving the ‘n’ number either along the x

...read more.

Conclusion

n+5G+2

n+5G+3

n+5G+4

n+5G+5

n+5G+6

n+5G+7

n+5G+8

n+5G+9

n+4G

n+4G+1

n+4G+2

n+4G+3

n+4G+4

n+4G+5

n+4G+6

n+4G+7

n+4G+8

n+4G+9

n+3G

n+3G+1

N+3G+2

n+3G+3

n+3G+4

n+3G+5

n+3G+6

n+3G+7

n+3G+8

n+3G+9

n+2G

n+2G+1

N+2G+2

N+2G+3

n+2G+4

n+2G+5

n+2G+6

n+2G+7

n+2G+8

n+2G+9

n+G

n+G+1

N+G+2

N+G+3

n+G+4

n+G+5

n+G+6

n+G+7

n+G+8

n+G+9

n

n+1

n+2

N+3

n+4

n+5

n+6

n+7

n+8

n+9

Above I will develop a 6-step stair shape formula that will work with any grid:

N+n+1+n+2+ n+3+n+4++n+5+ n+G+ n+G+1+ n+G+2+ n+G+3+n+G+4+n+2G+ n+2G+1+n+2G+2+n+2G+3+ n+3G+n+3G+1+n+3G+2+n+4G+n+4G+1+n+5G=T

Using means of collecting up I can simplify this formula to:  21n+35G+35 = T

I will check that this formula is correct using the 10 by 10 grid:

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

35

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

n

T

1

406

2

427

3

448

4

467

5

486

6

505

                        Here I am using the formula to see whether it works…

  1. 21*1+35*10+35=406
  1. 21*2+35*10+35=427
  1. 21*3+35*10+35=448

From the sums I have calculated above you can see that the formula works and is accurate as it follows my table completely.

I assume and believe that the formula will work with all problems that use a 6-step stair.

...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.

    z ( n - 1 ) So, if for example we were given a 7 X 2 square taken from a 8 wide grid the calculation would be: 8(7-1) For any random square inside any random width grid. z ( n - 1 )( d - 1 )

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

    and so on Using the algebra equation and adding the values we get -110 If we closely examine the results from the 4-step stairs from the 3 numbered Grid Square, i.e. 10x10, 11x11 and 12x12 we can see that the constant value [10] We can then complete the rest of

  1. Number Stairs

    24 25 11 12 13 14 15 16 1 2 3 4 5 6 7 1+2+3+4+5+6+7+11+12+13+14+15+16+21+22+23+24+25+31+32+33+34+41+42+43+51+52+61=644 The Stair Total of this 7-Step Stair is =644 Using the formula= x(s(s+1)) + (11s3 - 11s) = t. 2 6 (1)(7(7+1)) + (11(73))-(s11)=t 2 6 (1)

  2. Algebra Investigation - Grid Square and Cube Relationships

    16 17 18 19 20 21 22 23 24 25 Experimental number box: Algebraic box: n n+w-1 n+5(h-1) n+5h-5+w-1 Which (when brackets are multiplied out) simplifies to: n n+w-1 n+5h-5 n+5h+w-6 Stage A: Top left number x Bottom right number = n(n+5h+w-6)

  1. Number stairs

    40 & 44 and show them as shown below: I know that the above values increase by the constant number [4] and also that in any 3-step grid square if the grid size increases by 1 then the constant number is added to the value (The Highest Common Factor)

  2. Maths Grid Investigation

    calculating the diagonal difference for a 6 x 6 grid inside a 8 x 8 grid. Justifying My Results 1 2 3 4 5 2 x 2 grid 3 x 3 gird 4 x 4 grid 5 x 5 grid 6 x 6 grid 8 x 1 8 x 4

  1. Maths - number grid

    490 640 810 2x2 3x3 4x4 5x5 6x6 7x7 8x8 9x9 10x10 Looking at the diagram, the top number indicates the difference between each defined difference found and the numbers along the bottom indicate the defined differences of each square investigated.

  2. Number Stairs

    And those are: Here we can see that stair number= 46 Whereas stair total= 46+47+48+54+55+62=312 But by using the nth term, stair total= (6x46) + 36= 312 Stair number=41 Whereas stair total= 41+42+43+49+50+57= 282 But by using the nth term, stair total= (6x41)

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