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

Number stairs

Extracts from this document...

Introduction

NUMBER STAIRS

This is a 10 by 10 Number 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

Fig. 1

Marked on the grid is a stair shape; this is called a 3-Step Stair.

The total of the numbers inside the stair shape is:

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

The Stair Total for this 3-step stair is 194.

Part 1

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

Part 2

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

4w+1

4w+2

4w+3

4w+4

5w

3w+1

3w+2

3w+3

3w+4

4w

2w+1

2w+2

2w+3

2w+4

3w

1w+1

1w+2

1w+3

1w+4

2w

0w+1

0w+2

0w+3

0w+4

1w

w = width

Fig. 1

On every size of number grid the formula in each block above will be the right formula for the corresponding number on any grid. However, on larger or smaller number grids some blocks may not exist or more blocks may exist. The formulae in the furthest right hand side column are always in the furthest right hand side column on every size of grid as these define what the number is at the end of each row. However, the height and width of the number grid may be more than, or less than, five.

...read more.

Middle

11

12

13

14

15

16

17

18

19

20

1

2

3

4

5

6

7

8

9

10

Fig. 1

The formula for a 3-step stair is:

(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+2w]

This can be simplified to:

6wy-2w+6x+4

From these formulae, when x is substituted with 5 and y is substituted with 3 (5 and 3 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 194 is calculated.

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

Also from these formulae, when x is substituted with 7 and y is substituted with 7 (7 and 7 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 446 is calculated.

67+68+69+77+78+87 = 446

This confirms the formula 6wy-2w+6x+4 can be used to calculate the stair total of a 3-step stair on a 10 by 10 grid.

57

58

59

60

61

62

63

50

51

52

53

54

55

56

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

Fig. 1

From the formula (6wy-2w+6x+4), when x is substituted with 2 and y is substituted with 2 (2 and 2 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 86 is calculated.

9+10+11+16+17+23 = 86

From the formula (6wy-2w+6x+4), when x is substituted with 4 and y is substituted with 6 (5 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 266 is calculated.

39+40+41+46+47+53 = 266

This confirms that the formula 6wy-2w+6x+4 can be used to calculate the stair total of a 3-step stair on a number grid of any 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

...read more.

Conclusion

10wy-0w+10x+10 can be used to calculate the stair total of a 4-step stair on a number grid of any size.

73

74

75

76

77

78

79

80

65

66

67

68

69

70

71

72

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

Fig. 1

The formula for the 5-step stair is:

(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+2] + [(wy-w+x)+3] + [(wy-w+x)+4] +

[(wy-w+x)+w] + [(wy-w+x)+w+1] + [(wy-w+x)+w+2] + [(wy-w+x)+w+3] +

[(wy-w+x)+2w] + [(wy-w+x)+2w+1] + [(wy-w+x)+2w+2] + [(wy-w+x)+3w] +

[(wy-w+x)+3w+1] + [(wy-w+x)+4w]

This can be simplified to:

15wy+5w+15x+20

From these formulae, when x is substituted with 3 and y is substituted with 4 (3 and 4 being the coordinates of the keystone of the step stair on the grid above), an answer of 585 is calculated.

27+28+29+30+31+35+36+37+38+43+44+45+51+52+59 = 585

This confirms the formula 15wy+5w+15x+20 can be used to calculate the stair total of a 5-step stair on a number grid of any size.

73

74

75

76

77

78

79

80

65

66

67

68

69

70

71

72

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

Fig. 1

The formula for the 2-step stair is:

(wy-w+x) + [(wy-w+x)+1] + [(wy-w+x)+w]

This can be simplified to:

3wy-2w+3x+1

From these formulae, when x is substituted with 2 and y is substituted with 3 (2 and 3 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 63 is calculated.

18+19+26 = 63

This confirms the formula 3wy-2w+3x+1 can be used to calculate the stair total of a 2-step stair on a number grid of any size.

73

74

75

76

77

78

79

80

65

66

67

68

69

70

71

72

57

58

59

60

61

62

63

64

49

50

51

52

53

54

55

56

41

42

43

44

...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

    find the general formula FOUR-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 4 step- stair in algebraic terms: The total of the terms added in the second portion of the 4-step stair is: T= X + (x+3n)

  2. Maths - number grid

    the same answers to that of what I found in chapter on. I can now conclude that in my 10x12 number grid the formula is 10(r-1) , and will continue to be this, as long as the number grid remains 10 long, as I have found that the width can alternate without affecting the formula.

  1. Maths Grids Totals

    and the x(w-1). This leaves 10(h-1)(w-1), which is the universal rule for any square and rectangle on a 10 x 10 grid. I know that this formula is correct: I have just found that the formula 10(h-1)(w-1) for 10 x 10 grids is correct: now I need to find the rule for all sized grids.

  2. Maths coursework. For my extension piece I decided to investigate stairs that ascend along ...

    formula for calculating the stair total if you know the stair number of a 2-stair shape on a 5 x 5 grid, I will draw out the 2-stair shape on a 5 x 5 grid in terms of 'n'. I have realised that the square above the stair-number is (n + 5)

  1. Number Stairs

    +40=370 Here is an alternative way to find the stair total of the 9x9 grid by using further algebraic method. As we can see here n=stair number, and the 3x3 stair case from the 9x9 grid can be substituted in to the formula staircase for the 9x9 grid.

  2. number grid

    The Difference for any 2 X 3 grid is always 20 within a 10 X 10 grid. 2 X 4 Grid Here is an algebraic grid for a 2 X 4 grid within a 10 X 10 grid. a a+3 a+10 a+13 Therefore: a(a + 13)

  1. Mathematics - Number Stairs

    14 1 2 3 n 1 2 3 4 5 T 58 64 70 76 82 Suspected formula: T = 6n + 52 Prediction / Test: 6 x 20 + 52 = 172 44 32 33 20 21 22 20 + 21 + 22 + 32 + 33 + 44

  2. Mathematical Coursework: 3-step stairs

    going on the 2 et cetera. I choose this method, as I believed it would allow me to find the pattern. To prove my assumption I would now make a Table of Results which will state my findings. Term 1 2 3 4 5 total 50 56 62 68 74 The Pattern = 6 As I

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