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  • Level: GCSE
  • Subject: Maths
  • Word count: 3089

To find any relationships and patterns in the total of all the numbers in 'Number Stairs', that might occur if this stair was in a different place on the grid.

Extracts from this document...

Introduction

Number Stairs Course Work

image00.png

Aim:

To find any relationships and patterns in the total of all the numbers in ‘Number Stairs’, that might occur if this stair was in a different place on the grid.

The 10 by 10 grid below is where I am to get all the information I need from to enable me to carry out this investigation.

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

Part One

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

91

92

93

94

95

96

97

98

99

100

81

82

83

84

85

86

87

88

89

90

71image01.png

72

73

74

75

76

77

78

79

80

61image09.png

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

By looking at this grid, the stair total is:

25 + (25+1) + (25+2) + (25+3) + (25+10) + (25+11) + (25+20)

If we call 25, the number in the corner of the stair, ‘x’ then we get:

X + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11) + (x + 20) =

6x + 1 + 2 + 3 + 10 + 11 + 20 =

6x + 44

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

The stair total for this three-step stair is 194

This formula should work with every number stair that can fit onto a 10 by 10 grid. To support my judgement I will repeat the formula another two times on the same three-stair grid, but in a different position to the previous three stairs.

91

92

93

94

95

96

97

98

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100

81

82

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85

86

87

88

89

90

71

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73

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75

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77

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79

80

61image01.png

62

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66

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68

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70

51

52

53

54

55

56

57

58

59

60

41image13.png

42

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44

45

46

47

48

49

50

31

32

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36

37

38

39

40

21

22

23

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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

By looking at this grid, the stair total is:

57 + (57+1) + (57+2) + (57+3) + (57+10) + (57+11) + (57+20)

If we call 57, the number in the corner of the stair, ‘x’ then we get:

X + (x + 1) + (x + 2) + (x + 3) + (x + 10) + (x + 11)

...read more.

Middle

5

6

7

8

6x

6        56        62        68        74        80        86

12

18

24

30

36

42

48

6x doesn’t give me the correct answer so I add 44 to 6x and get the Stair Total:

Stairs number (x)

1

2

3

4

5

6

7

8

Stairs total (tx)

50

56

62

68

74

80

86

92

6x

6

12

18

24

30

36

42

48

+44

44

44

44

44

44

44

44

44

6x+44

50

56

62

68

74

80

86

92

Testing the Formula

To ensure the formula works in every case, I tested it in five situations where

‘x’ was different each time.

Using the formula 6x + 44

If n = 1 then

·        6 x 1 + 44 = 6 + 44 = 50

If x = 12 then

·        6 x 12 + 44 = 72 + 44 = 116

If x = 23 then

·        6 x 23 + 44 = 138 + 44 = 182

If x = 34 then

·        6 x 45 + 44 = 270 + 44 = 314

All the results using the formula are correct, so I can come to the conclusion that the formula for the total of a stair:

·        On a 10 by 10 grid

·        Which travels downwards from left to right

·        With a height of 3 squares

Is: 6n + 44

In every case of a three step, the difference in position has a different total of 6.

Part Two (Extension)

Investigate further the relationship between the stair total and other step

stairs on the number grids.

For past of this investigation I am going to find out the totals for each step, starting from 2 step, to a 10step. As I found a formula for part one, I will continue to use that, in this part of this investigation, to see if there is any initial relationship.  To see if there are any relationships between the number of stairs and there total, I will then put the results into a table, to make it clearer.

Two and Three Stepimage03.png

91image15.png

92

93

94

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96

97

98

99

100

81

82

83

84

85

86

87

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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

41image03.png

42

43

44

45

46

47

48

49

50

31image02.png

32

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34

35

36

37

38

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40

21

22

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25

26

27

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29

30

11

12

13

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19

20

1

2

3

4

5

6

7

8

9

10

Two Step

‘x’ =43

X + (x + 1) + (x + 10) =

3x+11

=43+44+53=140

The stairs total for this two-step stair is 140

Three Step

‘X’=17

X + (X + 1) +  (X + 2) (X + 3) (X + 10) (X + 11) + (X + 20) =

6x+44

=17+18+19+27+28+37 = 146

The stairs total for this two-step stair is 146

Four and Five Step

91image03.png

92

93

94

95

96

97

98

99

100

81image04.png

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

41image03.png

42

43

44

45

46

47

48

49

50

31

32

33

34

35

36

37

38

39

40

21image05.png

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

Four Step

‘X’=33

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

10x+110

=33+34+35+36+43+44+45+53+54+63= 440

The stairs total for this two-step stair is 440

Step Five

‘X’=56

X + (X + 1) +  (X + 2) (X + 3) (X + 4) (X + 10) + (X + 11) + (X + 12) + (X + 20) + (X + 21) + (X + 22) + (X + 30) + (X + 31) + (X + 40) =

15x+220

=56+57+58+59+60+66+67+68+69+76+77+78+86+87+96= 1060

The stairs total for this two-step stair is 1060

Six Step

91

92

93

94

95

96

97

98

99

100

81image06.png

82

83

84

85

86

87

88

89

90

71

72

73

74

75

76

77

78

79

80

61image07.png

62

63

64

65

66

67

68

69

70

51

52

53

54

55

56

57

58

59

60

41

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44

45

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47

48

49

50

31

32

33

34

35

36

37

38

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40

21

22

23

24

25

26

27

28

29

30

11

12

13

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15

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17

18

19

20

1

2

3

4

5

6

7

8

9

10

...read more.

Conclusion

=1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+21+22+23+24+25+26+27+28+31+32+33+34+35+36+36+37+41+42+43+44+45+46+51+52+53+54+55+61+62+63+64+71+72+73+81+82+91=1906

The stairs total for this two-step stair is 1906

Having found out the stairs totals and the formula’s for the steps of 2-10, I am then going to put my findings into a table, as I am not able to see any instant relationship between the group of results as they are scattered all about.

Stairs Number

Formulae

Stairs Total

2

3x+11

140

3

6x+44

146

4

10x+110

440

5

15x+220

1060

6

21x+385

1330

7

28x+616

952

8

36x+924

1752

9

45x+1320

1814

10

55x+1815

1906

Having interpreted the results of the formulas and stairs totals into a table, I am now able to see a relationship between the formulas for working out each step. This being that the formulae’s for each step are triangular numbers.  

The first 10 triangle numbers are: 3,6,10,15,21,28,36,45,55

As you can see, they all are in the formulas.

Starting from the basic two stairs, the formula (relationship) is the first triangular number, and continues through out the table, and beyond ten stairs.

As you carry on with the number stairs, for example working out the 11th stair formula, you will notice that it will be the 11th triangular number, this will be for all of the formulas.

The table below states this:

Stairs Number

Formulae

Stairs Total

2

3x+11

140

3

6x+44

146

4

10x+110

440

5

15x+220

1060

6

21x+385

1330

7

28x+616

952

8

36x+924

1752

9

45x+1320

1814

10

55x+1815

1906

11

66x+{}

12

78x+{}

13

91x+{}

14

105x+{}

15

120x+{}

16

136x+{}

17

153x+{}

18

171x+{}

19

190x+{}

20

210x+{}

Nadine Okyere        11Mi        February 2003

...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.

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