Number Grid Investigation

Introduction:

The coursework task is to investigate the patterns generated from using rules in a square grid. The grid provides a structured approach to learning number relationships.

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

In accordance with the task there are four steps to follow:

. A box is drawn round four numbers [see grid above].

2. Find the product of the top left number and the bottom right number.

3. Do the same with the top right number and the bottom left number.

4. Calculate the difference between these numbers.

From the example in the above grid: [45 x 56 = 2520 - 46 x 55 = 2530] I find that the product difference for the diagonal square [2x2] is 10.

I am going to start by investigating as to whether or not the location of the 2x2 square on the grid is significant.

A. 81 x 92 = 7452 - 82 x 91 = 7462. Product difference is 10.

B. 9 x 20 = 180 - 10 x 19 = 190. Product difference is 10.

C. 12 x 23 = 276 - 13 x 22 = 286. Product difference is 10.

From the worked examples A, B & C I find that the product difference is 10. Taking these results into account, I predict for any 2x2 square the result will always be 10.

Below is a table of results for 2x2 squares that were randomly chosen from the 10x10 grid.

Table 1. 2x2 results.

st No. multiplication

2nd No. multiplication

Difference

5 x 16 = 80

6 x 15 = 90

0

7 x 28 = 476

8 x 27 = 486

0

32 x 43 = 1376

33 x 42 = 1386

0

68 x 79 = 5372

69 x 78 = 5382

0

The results from T.1 show that the difference is a constant 10, so my prediction was correct.

In Mathematics, Algebra is designed to help solve certain types of problems; letters can be used to represent values, which are usually unknown. I will now attempt to prove my results algebraically. Let `N` represent the number in the top left of the square.

N

N +1

N+10

N+11

This can be expressed into the equation:

( N + 1) ( N + 10) - N ( N + 11)

Multiply out the brackets:

N² + 10N + N + 10 - N² - 11N

Simplified to:

N² + 11N + 10 - N² - 11N

= 10.

By using `N` as the variable and constants together, the results for a 2x2 square prove to be 10. When subtracted the `N` cancels out leaving the number 10 which is actually the size of the number grid I am working with, and also equal to the constant.

I can see from the results that for every multiplication and subtraction carried out the difference will always be 10. Can this be true for any square size ? I shall carry this formula on, and investigate further and see if this works for different sized selection squares e.g. 3x3, 4x4 ect; again I will select random squares on the grid.

3 x3 Squares:

A. ( 3 x 21) - ( 1 x 23 ) = 40

B. ( 44 x 62) - ( 42 x 64 ) = 40

C. ( 8 x 26 ) - ( 6 x 28 ) = 40

D. ( 56 x 76 ) - ( 56 x 78 ) = 40

For the 3x3 squares the constant difference is 40. From the worked examples A, B & C all demonstrate that if you put a 3x3 square anywhere on a 10x10 grid, the product difference will equal 40. Therefore, I can predict for any 3x3 square the result will always be 40.

Now I will show the table for a 3x3 and the algebraic formula. Let `N` represent the top left number in the 3x3 square.

N

N+2

N+20

N+22

This can be expressed into the equation:

( N +2) ( N + 20) - N ( N +22)

Multiply out the brackets:

N² +20N +2N +40 - N² - 22N = 40

This also can be simplified, consequently you are left with 40, which is the constant difference for a 3x3 square.

4x4 Squares:

I will now investigate for the last time with square boxes using a larger 4x4 square. I will again place them randomly on the 10x10 grid.

Introduction:

The coursework task is to investigate the patterns generated from using rules in a square grid. The grid provides a structured approach to learning number relationships.

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

In accordance with the task there are four steps to follow:

. A box is drawn round four numbers [see grid above].

2. Find the product of the top left number and the bottom right number.

3. Do the same with the top right number and the bottom left number.

4. Calculate the difference between these numbers.

From the example in the above grid: [45 x 56 = 2520 - 46 x 55 = 2530] I find that the product difference for the diagonal square [2x2] is 10.

I am going to start by investigating as to whether or not the location of the 2x2 square on the grid is significant.

A. 81 x 92 = 7452 - 82 x 91 = 7462. Product difference is 10.

B. 9 x 20 = 180 - 10 x 19 = 190. Product difference is 10.

C. 12 x 23 = 276 - 13 x 22 = 286. Product difference is 10.

From the worked examples A, B & C I find that the product difference is 10. Taking these results into account, I predict for any 2x2 square the result will always be 10.

Below is a table of results for 2x2 squares that were randomly chosen from the 10x10 grid.

Table 1. 2x2 results.

st No. multiplication

2nd No. multiplication

Difference

5 x 16 = 80

6 x 15 = 90

0

7 x 28 = 476

8 x 27 = 486

0

32 x 43 = 1376

33 x 42 = 1386

0

68 x 79 = 5372

69 x 78 = 5382

0

The results from T.1 show that the difference is a constant 10, so my prediction was correct.

In Mathematics, Algebra is designed to help solve certain types of problems; letters can be used to represent values, which are usually unknown. I will now attempt to prove my results algebraically. Let `N` represent the number in the top left of the square.

N

N +1

N+10

N+11

This can be expressed into the equation:

( N + 1) ( N + 10) - N ( N + 11)

Multiply out the brackets:

N² + 10N + N + 10 - N² - 11N

Simplified to:

N² + 11N + 10 - N² - 11N

= 10.

By using `N` as the variable and constants together, the results for a 2x2 square prove to be 10. When subtracted the `N` cancels out leaving the number 10 which is actually the size of the number grid I am working with, and also equal to the constant.

I can see from the results that for every multiplication and subtraction carried out the difference will always be 10. Can this be true for any square size ? I shall carry this formula on, and investigate further and see if this works for different sized selection squares e.g. 3x3, 4x4 ect; again I will select random squares on the grid.

3 x3 Squares:

A. ( 3 x 21) - ( 1 x 23 ) = 40

B. ( 44 x 62) - ( 42 x 64 ) = 40

C. ( 8 x 26 ) - ( 6 x 28 ) = 40

D. ( 56 x 76 ) - ( 56 x 78 ) = 40

For the 3x3 squares the constant difference is 40. From the worked examples A, B & C all demonstrate that if you put a 3x3 square anywhere on a 10x10 grid, the product difference will equal 40. Therefore, I can predict for any 3x3 square the result will always be 40.

Now I will show the table for a 3x3 and the algebraic formula. Let `N` represent the top left number in the 3x3 square.

N

N+2

N+20

N+22

This can be expressed into the equation:

( N +2) ( N + 20) - N ( N +22)

Multiply out the brackets:

N² +20N +2N +40 - N² - 22N = 40

This also can be simplified, consequently you are left with 40, which is the constant difference for a 3x3 square.

4x4 Squares:

I will now investigate for the last time with square boxes using a larger 4x4 square. I will again place them randomly on the 10x10 grid.