• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 1703

3 Step Stairs

Extracts from this document...

Introduction

Part 1

3 Step Stairs

Structures

This is a 3 step stair on a 10 x 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

All 3 step stairs consist of 6 squares and its total is found by adding all the numbers inside the squares.

Total = (25+26+27+35+36+45)

Total = 194

This 3 step stair is in a certain position on the grid. This position is called P1 (see below)

This position is called P1

This position is called P2

This position is called P3

This position is called P4

Formulas

There are 4 different formulas to find the total of the 3 step stairs, each one depending on the position of the stairs.

e.g. total = number added + ((number of squares x squares right) + (10number of squares x squares up))

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

Number added = (1+2+3+11+12+21)

Number added = 50

Number of squares = 6

Squares right = 0

Squares up = 0

Abbreviations:

Total = n

Number of squares = s

...read more.

Middle

This position is called P1

This position is called P2

This position is called P3

This position is called P4

Formulas

There are 4 different formulas to find the total of the 4 step stairs, each one depending on the position of the stairs.

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

Number added = (1+2+3+411+12+13+21+22+31)

Number added = 120

Number of squares = 10

Squares right = 0

Squares up = 0

For this position, in 4 step stairs, the formula would be:

n = 120 + ( ( s x r ) + ( 10s x u ) )

For this position, in 4 step stairs, the formula would be:

n = 220 + ( ( s x r ) + ( 10s x u ) )

For this position, in 4 step stairs, the formula would be:

n = 230 + ( ( s x r ) + ( 10s x u ) )

For this position, in 4 step stairs, the formula would be:

n = 130 + ( ( s x r ) + ( 10s x u ) )

2 Step Stairs

...read more.

Conclusion

For this position, in 2 step stairs, the formula would be:

n = 24 + ( ( s x r ) + ( 10s x u ) )

For this position, in 2 step stairs, the formula would be:

n = 24 + ( ( s x r ) + ( 10s x u ) )

For this position, in 2 step stairs, the formula would be:

n = 24 + ( ( s x r ) + ( 10s x u ) )

Proof

This is a randomly selected 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

I can work out the total by using my formula:

n = 50 + ( ( s x r ) + ( 10s x u ) )

= 50 + ( ( 42 ) + ( 360 ) )

N = 452

Simple total = 68+69+70+78+79+88

= 452

This is a randomly selected 4 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

I can work out the total by using my formula:

n = 120 + ( ( s x r ) + ( 10s x u ) )

= 120 + ( ( 60 ) + ( 600 )

N = 780

Simple total = 67+68+69+70+77+78+79+87+88+97

= 780

This is a randomly selected 2 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

I can work out the total by using my formula:

n = 14 + ( ( s x r ) + ( 10s x u ) )

= 14 + ( ( 18 ) + ( 180 )

N = 212

Simple total = 67+68+77

= 212

Maths Investigation

Number Stairs

Alex O'Carroll 10a

...read more.

This student written piece of work is one of many that can be found in our GCSE T-Total 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 T-Total essays

  1. T-Total Maths

    Formula: T=5N-7 I tested that when: T-number= 42 T-total= 204 Below is a T-shape, and in each cell how number is connected with T-number on a 7 by 7 number grid. To prove the formula: T= N+N+1+N+9+N+2+N-5 T= 5N-7 How the formula works there are some example shown in below

  2. Investigate a formula to see how many squares would be needed to make any ...

    (i) First Differences (a + b): 0 4 8 12 16 20 . . . . . . (ii) Second Differences (2a): 4 4 4 4 4 . . . . . . . . (iii) The bottom row of differences indicates a constant number, which shows there is a pattern.

  1. Step Stairs

    Firstly, each time the shape moves along one square, for example from step shape one to step shape two the shape total increases by six. Also as the stair steps move up rows the shape total increases by 60. I have also noticed that there is a connection between N,

  2. I am going to investigate how changing the number of tiles at the centre ...

    I will now use it to make predictions for diagram number, N = 8. Prediction If C = 8 and N = 2. If C = 8 and N = 5 B = 2(2 x 5 + 8 - 1)

  1. Black and white squares

    White Squares Cumulative Number of White Squares Number of Black Squares Cumulative Number of Black Squares 1 1 1 0 0 1 1 2 5 6 4 4 1 2 3 13 19 8 12 5 7 4 25 44 12 24 13 20 5 41 85 16 40 25

  2. Number Stairs

    6 7 Total of position 6: 6+7+16 = 29 Position Seven : 17 7 8 Total of position 7: 7+8+17 = 32 Position number Calculation Position Total Difference 1 1+2+11 14 2 2+3+12 17 +3 3 3+4+13 20 +3 4 4+5+14 23 +3 5 5+6+15 26 +3 6 6+7+16 29

  1. Number Stairs Investigation

    Part 2 Investigate Further: Investigate further the relationship between the stair totals and other step stairs on the other number grids. I am now investigating the formula for 4-step stairs on a 10 x 10 grid. Results Table: X T (total)

  2. A dark cross-shape has been surrounded by white squares to create a bigger cross-shape. ...

    I think this will mainly be in the form of a multiple of 4. Another relationship with the results is that the original cross-shapes total squares is the same number as the total number of squares on the previous shape.

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