For other 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. To start the investigation a 10x10-numbered grid square is used as illustrated below in table 1:

Authors Avatar
Mill Hill County High School

Year 11

Mathematics GCSE Coursework

EDEXCEL 2003, SYLLABUS 1387/1388

F, I & H

Tejesh Patel

Class 11H

Assignment Part 1

Below is a 10x10 number grid:

The total on the numbers coloured in blue = 90 (i.e. 1+11+12+21+22+23)

Therefore the stair total in this 3-step stair = 90

Part 1 Objective:

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

To start the investigation a 10x10-numbered grid square is used as illustrated below in table 1:

From the complete 10 x 10 numbered grid square, we use part of it to carry out our initial investigation, for example the grid box on the right shows a slice of the 10 x 10 gird square, i.e. 6 boxes representing the numbers 1,11,12,21,22,23 (The 3-step stair)

From this basic numbered square that looks like stairs or steps we can start to establish if there is a pattern. If a pattern is found then we can use an algebra equation to represent this pattern and use the equation for a 10 x 10 numbered Grid Square.

By using the 3-step stair example we know there are [6] squares and lets assume in a 3-step stair the bottom grid box is equal to [x], therefore in our 3-step stair x = 21

Using the values in algebra the formula(s) would look like this:

The 1st square = 1 then the formula is x - 20 = 1

The 2nd square = 11 then the formula is x - 10 = 11

The 3rd square = 12 then the formula is x - 9 = 12

The 4th square = 21 it is simply just x = 21

The 5th square = 22 then the formula is x + 1 = 22

The 6th square = 23 then the formula is x + 2 = 23

The above algebra equations are shown below in our 3-step stair:

x - 20

x - 10

x - 11

X

x + 1

x + 2

By adding the values from the equation (-20 - 10 - 9 + 1 + 2) = [-36]

Thus we can use the algebra equation 6x - 36 = [the total value of squares in a 10x10 square]

Using the above logic and method we can use it in other grids, such as an 11x11 and a 12x12 numbered grid square. Using the same 3-step stair approach we can use the theory for the 11x11 and 12x12-numbered square to find a pattern.

Below are the results from the such an exercise on a 10x10, 11x11 and 12x12 numbered grid box

Table 4: 10 x 10 Grid

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

Table 5: Using the algebra equation and adding the values we get -36

x - 20

x - 10

x - 11

X

x + 1

x + 2

Table 6: 11 x 11 Grid

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

01

02

03

04

05

06

07

08

09

10

Table 7:

x - 32

x - 11

x - 10

X

x + 1

X + 2

Using the algebra equation and adding the values we get -40

Table 8: 12 x 12 Grid

2

3

4

5

6

7

8

9

0

1

2

3

4

5

6

7

8

9

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

00

01

02

03

04

05

06

07

08

09

10

11

12

13

14

15

16

17

18

19

20

Table 9:

x - 24

x - 12

x - 11

x

x + 1

x + 2

Using the algebra equation and adding the values we get -44

If we closely examine the results from the 3-step stairs from the 3 numbered grid squares, i.e. 10x10, 11x11 and 12x12 we can see there is a constant number that is consistent, which is [4]

We can draw up a table by using this constant number as illustrated below in table 10:

Table 10:

From the table on the left and using our results from the investigation we can see a pattern emerging. Every time the size of the grid box increases by 1, the value in the algebra formula increases by 4. For example for a 10x10 numbered grid using a 3-step stair the formula is 6x-36 where x is the number in the bottom left hand square, then we increase the grid size by 1, 11x11 and using the same 3-stepped stair approach the formula is 6x-40, etc. We can clearly see the constant number [4] is consistent every time the grid size increases by [1].

For any 3-step numbered gird box as shown in table 2 or in the six examples below, and [x] as the value from the bottom left hand square and the algebra theory used to calculate the equation the total value of the squares can be found. This theory has been put to the test using a 10x10, 11x11 and 12x12 gird squares. The results are conclusive and consistent, proving the theory to be accurate and reliable.

Using any of the algebra equation from the above table, e.g. 6x-36 or 6x-44 we can prove the results for any 3-step grid. The follow exercises in table 11 shows the expected results:

Table 11

33

FORMULA = 6x - 36

43

44

: ( 6 x 53) - 36 = 282

53

54

55

2: 33+43+44+53+54+55=282

77

FORMULA = 6x - 36

87

88

: (6 x 97) - 36 = 546

97

98

99

2: 77+87+88+97+98+00=546

3

FORMULA = 6x - 40

24

25

: (6 x 35) - 36 = 170

35

36

37

2: 13+24+25+35+36+37=170

51

FORMULA = 6x - 40

62

63

: (6 x 73) - 40 = 398

73

74

75

2: 51+62+63+73+74+75=398

21

FORMULA = 6x - 44

33

34

: (6 x 45) - 44 = 226

45

46

47

2: 21+33+34+45+46+47=226

53

FORMULA = 6x - 44

65

66

: (6 x 77) - 44 = 418

77

78

79

2: 53+65+66+77+78+79=418

From the above 6 examples, it clearly indicates that the algebra equation works and the theory is accurate.

In this exercise the method of adding the numbers was used to ensure that the algebra equation gives the correct results and in all cases it did, proving the equation to be correct.

From the above exercises and the results we can state the hypothesis as:

. Each time the step stair increases so does the amount of squares within the grid, which is called the formula of triangular numbers.
Join now!


2. Also as the size of the grid increase by x number of squares the value of the squares total will also change again known as the triangular numbers.

We continue with the exercise and so far we have our algebra equation is 6 x - 36 for a 3-step grid in a 10x10 boxed grid

6 x - 40 for an 11x11 boxed grid and 6 x - 44 for a 12x12 boxed grid.

We continue to proceed with the exercise to complete our formula (The General Formula). In the algebra equation we need ...

This is a preview of the whole essay