• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 2421

number stairs

Extracts from this document...

Introduction

GCSE Maths        

Number Stairs

Part 1:For the other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid.

Here we have a 10 by 10 grid. In the grid, there is a shape called the 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

The number at the bottom left of the 3-step stair is 25. This is called the Step-number.

The sum of all the numbers in the 3-step stair is 194.

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

This is called the Step-total.

I am going to investigate the relationship between the stair total and the position of the stair shape on the grid using the first step-number on the 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

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

The first 3-step stair is made up of

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

If I move the 3-step stair 1 unit to the right, the 3-step stair would be made up of

2 + 3 + 4 + 12 + 13 + 22 = 56

If I move the 3-step stair another unit to the right, the 3-step stair would be made up of

3 + 4 + 5 + 13 + 14 + 23 = 62

I have created a table of a few results of the 3-step stairs

Numbers in 3-step stair

Step-number

Step-total

1, 2, 3, 11, 12, 21

1

50

2, 3, 4, 12, 13, 22

2

56

3, 4, 5, 13, 14, 23

3

62

4, 5, 6, 14, 15, 24

4

68

5, 6, 7, 15, 16, 25

5

74

6, 7, 8, 16, 17, 26

6

80

From this information, you can see that;

  1. Each 3-step stair can be divided evenly by 2
  2. Each step-total increases by 6 starting from 50
  3. The step-number goes up by 1 and the step-total goes up by 6.
...read more.

Middle

43

44

45

46

47

48

49

50

31

32

33

34

35

36

37

38

39

40

n + 20

22

23

24

25

26

27

28

29

30

n + 10

n + 11

13

14

15

16

17

18

19

20

n

n + 1

n + 2

4

5

6

7

8

9

10

Since all the numbers in the stair-shape adds up to the step-total, if I simplify all the values in the 3-step stair, I should get a formula for the step-total

n + (n + 1) + (n + 2) + (n + 10) + (n + 11) + (n + 20)        =        6n + 44

Therefore, the step-total should equal to 6n + 44

I called step-total t

For stair-step 1, n = 1

t = 6n + 44

  = 6(1) + 44

  = 6 + 44

  = 50

I have proven that the formula works for the first 3-step stair. To verify that this formula works for every other 3-step stair, I will test the first example given, stair-step 25. The step-total should equal 194.

t = 6n + 44

  = 6(25) + 44

  = 150 + 44

  = 194

The formula works for all the step-numbers on the 10 by 10 grid. The relationship between the stair-totals and the position of the stair-shape on the grid is that if you multiply the step-number by 6 and the add 44, you should find the step-total.

Part 2:Investigate further the relationship between the stair totals and other step stairs on other number grids.

I have worked out the formula for the 10 by 10 grid to be t = 6n +44 but this formula only works for a 10 by 10 grid. I will investigate the relationship between the step-total, step-number and the grid size.

In the 10 by 10 grid I noticed that the numbers 1, 11 & 21 etc increases by 10. If you subtracted n from the number above n, you would get the grid size number Therefore, the step-total and the step-number relate with the grid size.

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

In this example, n is 25 and the number above n is 35. If I subtract n from 35, I would get 10. This is the grid size number.

The relationship between the step-number and the rest of the numbers are that;

The number 35 has a difference of 10 from the step-number. The number 45 has a different of 20 from the step-number, which is 2 multiplied by 10. The number 26 has a difference of 1 from the step number, which is 1 multiplied by 10, minus 9.The number 27 has a difference of 2 from the step number, which is 1 multiplied by 10, minus 8.

The number 36 has a difference of 11 from the step number, which is 1 multiplied by 10, plus 1

With this information, I will create a formula for each of the numbers in the 3 step-stair shape. The Grid size will be called g.

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

n+2g

46

47

48

49

50

31

32

33

34

n+g

n+(g+1)

37

38

39

40

21

22

23

24

n

n+(g-9)

n+(g-8)

28

29

30

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

n

= 25

n + ( g – 9)

= 25 + (10 – 9)

= 25 + 1

=26

n + ( g – 8 )

= 25 + (10 – 8)

= 25 + 2

=27

n + g

= 25 + 10

= 35

n + ( g + 1 )

= 25 + (10 + 1)

= 25 + 11

=36

n + 2g

= 25 + (10 x2)

= 25 + 20

=45

...read more.

Conclusion

n+6g – 14

From this information, I have come up with another equation that will work with a grid of any size;

6n + 6g – ( 16 ± 2d )

Where d is the difference of the grid from a 10 by 10 grid

I will test this new formula with the 10 by 10 grid and the 9 by 9 grid.

Step-number = 1

Grid size = 10 by 10

t = 6n + 6g – ( 16 ± 2d )

  = 6(1) + (6 x 10) – [16  ± (2 x 0)]

  = 66 – 16

  = 50

The new equation has worked for the 10 by 10 grid

Step-number = 1

Grid size = 9 by 9

t = 6n + 6g – ( 16 ± 2d )

  = 6(1) + (6 x 9) – [16 – (2 x 1)]

  = 60 – 14

  = 46

The new equation has also worked for the 9 by 9 grid

This formula should even work for a smaller scale, such as a 5 by 5 scale

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

The answer we should get is;

1 + 2 + 3 + 6 + 7 + 11 = 30

t = 6n + 6g – ( 16 ± 2d )

  = 6(1) + (6 x 5) – [16 – (2 x 5)]

  = 36 – 6

  = 30

I have proven that my formula works with any grid

Conclusion:

Here I have put all the formula I have come up with. This formula will apply to a grid of any size;

t = 6n + 6g – ( 16 ± 2d )

Where;

t   : step- total

n  : step-number

g  : grid size

d  : difference of grid size from a 10 by 10 grid

In this project I have found out many ways in which to solve the problem I have with the stair-shape being in various positions with different sizes of grids. The way I have made the calculations less difficult is by creating a main formula that works for all the different circumstances.

...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

    from 10 to 11 and then to 12, using the highest common factor for 3-step grids (4) the calculations are 9, 10 & 11 for example: 11 x 4 = 44 12 x 4 = 48 13 x 4 = 52 I can now use 4 as the constant number and [n] as the grid size in my General Formula.

  2. 100 Number Grid

    46 x 64 = 2944 48 x 62 = 2976 Product difference = 32 I predict that for any 3 x 3 square in an 8 x 8 number grid, the product difference will always be 32. I will prove this theory by using an algebraic equation.

  1. Number Stairs

    And those are: Stair number = 61 Stair total = (6x61) + 40 = 406, or alternatively, 61+62+63+70+71+79= 406 Stair number=7 Stair total= (6x7) + 40= 82 With out the nth term the stair total= 7+8+9+16+17+25=82 Stair number=55 With out the nth term stair total= 55+56+57+64+65+73= 370 With nth term, stair total= (6x55)

  2. number grid

    a a+2 a+20 a+22 I have used the letter 'a' for the top left number and added how much bigger the other key numbers are away from it. In my investigation I have to find the difference between, the product of the top left number and the bottom right number,

  1. number grid investigation]

    n n+1 n+2 n+3 n+10 n+11 n+12 n+13 n+20 n+21 n+22 n+23 n+30 n+31 n+32 n+33 Stage A: Top left number x Bottom right number = n(n+33) = n2+33n Stage B: Bottom left number x Top right number = (n+30)(n+3)= n2+3n+30n+90 = n2+33n+90 Stage B - Stage A: (n2+33n+90)-(n2+33n)

  2. Maths Number Grids/Sequences

    - ( 42 x 64 ) = 40 C. ( 8 x 26 ) - ( 6 x 28 ) = 40 D. ( 56 x 76 ) - ( 56 x 78 ) = 40 For the 3x3 squares the constant difference is 40. From the worked examples A, B & C all demonstrate that if you put

  1. Number Grid

    I will then look at the relationship between the two products, by finding the difference. I want to find out whether there is any pattern in a 3 x 3 box, in the same way there was in a 2 x 2 box.

  2. Number Stairs

    is the same as when not using the formula which proves that the formula 6(x)+44=(t) gives you the Stair Total of any 3step stair which travels from bottom right to top left in any position on the 10 by 10 number grid.

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