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  • Level: GCSE
  • Subject: Maths
  • Word count: 1364

To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).

Extracts from this document...

Introduction

Aim: To investigate the relationship between a 10 by 10-number grid and various stair shapes (as in the example below).

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

On posed with this investigation I began to look at the relativity of the stair shape on the grid and its relation to the numbers it coincides with. I then began to imagine the grid as an axis whereby the x and y lay on the grid.

Part 1

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

Firstly I had to find the relationship of the stair shape where it lay on the grid to the stair total. I began to look at different ways, I found that by working systematically on the grid i.e. starting by looking at horizontal, vertical and diagonal movements I would be able to find the common factors involved.

In the example we are given the numbers:

25, 26, 27, 35, 36, 45

They total to 194

To work out the placement of the stair shape I used the smallest number of the stair to translate the shape from, in this case 25.

...read more.

Middle

182

33

34

23

24

25

44

188

34

35

24

25

26

45

194

35

36

25

26

27

46

200

36

37

26

27

28

Although there is no common number here at the end of the total for each stair shape there is a common number fond when the difference is calculated.

200-194=6

194-188=6

188-182=6

This means that for every horizontal movement either left or right on the grid the total difference between each stair shape for a 3-step stair would be 6.

Positive diagonal movement on the grid:

23

62

13

14

3

4

5

34

128

24

25

14

15

16

45

194

35

36

25

26

27

56

260

46

47

36

37

38

There is no common number found here at the end of each total. The common difference is:

260-194=66

194-128=66

128-62=66

This means that when the stair shape is moved diagonally in the “y=x” direction the difference increase is 66. The total stair shape total for the stair shape will increase by 66 for every increase or decrease by 1.

Negative diagonal movement on the grid:

36

140

26

27

16

17

18

45

194

35

36

25

26

27

54

248

44

45

34

35

36

63

302

53

54

43

44

45

302-248=54

248-194=54

194-140=54

The negative diagonal difference when translating the shape in a “y=-x” manner is 54. So for every movement up 1 on the grid in that direction you can add 54 to the total.

After working out the vertical, diagonal and horizontal difference in the entire 3-step stair shapes I found that there was a common difference with all the stair shapes and that the difference found related to 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.

To further my investigation I decided to observe the relationship between larger stair shapes and their position on the grid, also their relationship with each other.

2-step stair

Using the same process as the 3-step stair I began to look at the horizontal, vertical and diagonal differences. I first started with looking at the horizontal difference

35

86

25

26

36

89

26

27

34

83

24

25

All the last numbers of the individual stair totals all have a multiple of three at the end.

89-86=3

86-83=3

The total difference overall is three and the multiple of the last number is also three. The horizontal difference is equalled to three.

Vertical difference:

45

116

35

36

35

86

25

26

25

56

15

16

...read more.

Conclusion

n>3

14

15

35

86

25

26

46

119

36

37

The numbers 3, 6 and 9 can be seen; they all are multiples of three. The total overall difference for the diagonal 2-step stair:

119-86=33

86-53=33

As can be seen the number 33 is also a multiple of 3.

Negative diagonal 2-step stair:

26

59

16

17

35

86

25

26

44

113

34

35

As can be seen the numbers shown at the end are 3, 6 and 9, they are all multiples of three.

Working out the common difference:

113-86=27

 86-59=27

Twenty-seven is seen to be the common difference tis however is also a multiple of three.

4 step-stair

 Horizontal difference:

54

350

44

45

34

35

36

24

25

26

27

55

360

45

46

35

36

37

25

26

27

28

56

370

46

47

36

37

38

26

27

28

29

They all are multiples of 10, so already you can see a common difference.

Working out the overall difference:

370-360=10

360-350=10

 Vertical difference:

55

360

45

46

35

36

37

25

26

27

28

45

260

35

36

25

26

27

15

16

17

18

35

160

25

26

15

16

17

5

6

7

8

60 is a common number in this, the total difference worked out:

360-260=100

260-160=100

There is a difference of 100 for each progressional movement vertically up or down on the 4-step star grid.

Positive diagonal movement:

55

360

45

46

35

36

37

25

26

27

28

44

250

34

35

24

25

26

14

15

16

17

33

140

23

24

13

14

15

3

4

5

6

360-250=110

250-140=110

...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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