• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

I will attempt to discover a general formula that will find the difference between the product of the top left number and the bottom right and the product of top right and bottom left (diagonals) of any size square and any size grid.

Extracts from this document...

Introduction

Maths Coursework

In this investigation I will attempt to discover a general formula that will find the difference between the product of the top left number and the bottom right and the product of top right and bottom left (diagonals) of any size square and any size grid.

I will start on a 10 by 10 grid and a square of 4 numbers, by picking a square of 4 numbers and multiplying the top left number with the bottom right and then finding the difference between this and the product of top right and bottom left.

Eg.

3          3image00.pngimage01.png

       4

13         13

             14

In this case, it would be the difference between 3 x 14 and 4 x 13, which is 10.

2x2 square

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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

100

1

2

11

12

1 x 12 = 12

2 x 11 = 22

Difference between the product of the diagonals is 10

44

45

54

55

44 x 55 = 2420

45 x 54 = 2430

Difference = 10

77

78

87

88

77 x 88 = 6776

78 x 87 = 6786

Difference = 10

18

19

28

29

18 x 29 = 522

19 x 28 = 532

Difference = 10

Pattern: As you can

...read more.

Middle

121

11 is added as you go vertically down the grid

Here we see that the number added vertically is the width of the grid. As 10 is the width of the grid we can substitute this into the equation by representing the width as w

nimage08.pngimage12.png

n+1

n+w

n+w+1

n x (n+w+1)        = n2 + nw +n

image02.pngimage02.pngimage02.png

(n+1) x (n+w)     = n2 + nw + n + w

As n2 + nwcancels out,the difference will always be w, which is the width of the grid

3x3 square

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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

100

1

2

3

11

12

13

21

22

23

1 x 23 = 23

3 x 21 = 63

Difference = 40

71

72

73

81

82

83

91

92

93

71 x 93 = 6603

73 x 91 = 6643

Difference = 40

78

79

80

88

89

90

98

99

100

78 x 100 = 7800

80 x 98 = 7840

Difference = 40

8

9

10

18

19

20

28

29

30

8 x 30 = 240

10 x 28 = 280

Difference = 40

We can turn the values found in the 3x3 square into albegra:

nimage03.png

image04.png

n+2

Dd

n +2w

n +w+2

n x (n+2w+2)        =  n2 + 2nw +2n

image05.pngimage05.pngimage05.png

(n+2) x (n+2w)     =  n2 + 2nw + 2n + 4w

Here the difference will always be 4w

4x4 square

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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

100

1

2

3

4

11

12

13

14

21

22

23

24

31

32

33

34

1 x 34 = 34

4 x 31 = 124

Difference = 90

61

62

63

64

71

72

73

74

81

82

83

84

91

92

93

94

61 x 94 = 5734

64 x 91 = 5824

Difference = 90

67

68

69

70

77

78

79

80

87

88

89

90

97

98

99

100

67 x 100 = 6700

70 x 97 = 6790

Difference = 90

7

8

9

10

17

18

19

20

27

28

29

30

37

38

39

40

7 x 40 = 280

10 x 37 = 370

Difference = 90

nimage07.pngimage06.png

9

n+3

17

19

20

27

29

30

n +3w

100

100

n +3w+3

...read more.

Conclusion

If we represent the length of the square as l then the number added would be (l –1)

nimage12.pngimage08.png

n+(l –1)

n+w

n+w+(l –1)

nimage13.pngimage14.png

n+(l –1)

Dd

n +(l –1)w

n +w+(l –1)

nimage07.pngimage06.png

9

n+(l –1)

17

19

20

27

29

30

n +(l –1) w

100

100

n +3w+(l –1)

n (n +3w+(l –1))         = n2 + 3nw + (l –1)n

n+(l –1) x n +(l –1)w = n2 + 3nw + (l –1)n + (l –1) 2w

The difference of any size square of numbers is:

W(l –1)2i.e width of the grid x ((the length of the square - 1) squared)

Checking the formula

With the formula W(l –1)2we should be able to find the difference on any size grid and any size square.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

Grid 14x14, square 8x8

By using the formula, the difference should be:

W(l –1)2  = 14 (8-1) 2

               = 686

Check:

61x166 = 10126

68x159 = 10812

Difference does equal 686

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

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

Grid 7x7, square 7x7

By using the formula, the difference should be:

W(l –1)2  = 7 (7-1) 2

               = 252

Check:

1x49 = 49

7x43 = 301

Difference does equal 252

...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.

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 Number Stairs, Grids and Sequences essays

  1. Marked by a teacher

    Number grids. In this investigation I have been attempting to work out a ...

    4 star(s)

    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

  2. Marked by a teacher

    Number Grid Aim: The aim of this investigation is to formulate an algebraic equation ...

    3 star(s)

    24 - 13 = 11 2. 19 x 31 = 589 20 x 30 = 600 ?600 - 589 = 11 3. 48 x 60 = 2880 49x 59 = 2891 ?2891 - 2800 = 11 4. 85 x 97 = 8245 86 x 96 = 8256 ?

  1. Algebra Investigation - Grid Square and Cube Relationships

    = n�+11nw-11n Stage B: Bottom left number x Top right number = (n+10w-10)(n+w-1) = n�+nw-n+10nw+10w�-10w-10n-10w+10 = n�+11nw-11n-20w+10w�+10 Stage B - Stage A: = [ n�+11nw-11n-20w+10w�+10] - (n�+11nw-11n) = 10w�-20w+10 The result of these calculations means that for any size square box, the difference should be easily calculated using the formula.

  2. Number Grid Investigation.

    (4 X 72) - (12 X 64) = 480. This is correct. What next? I am now going to find a general formula for a grid with multiples. In the one above, multiples of 2, the formula was... 4Z(n-1)(d-1) The multiple number, 2, is squared to give 4 (as in 4Z) Mini prediction.

  1. What the 'L' - L shape investigation.

    by 7 -18 8 by 8 -21 9 by 9 -24 By looking at the results in my table I can see that the difference is always 3 and as Part 2 of this investigation is to find a relationship between the L-Sum, the L-Shape and the grid size I

  2. Investigation of diagonal difference.

    I will now calculate the diagonal difference of a 2x2 cutout using my n and G solutions. n n + 1 n + G n + G + 1 The diagonal difference of a 2 x 2 cutout on any size grid equals G.

  1. Maths - number grid

    are not in any particular order, I feel to establish a major trend I would have needed to start with a 3x2 and move up one each time, yet I feel this would be very time consuming and that through more investigating I will still be able to reach my

  2. Maths Coursework - Grid Size

    Then the height of the box will be altered. 'h' will represent the height of the square/rectangle. To prove my results I will test it on other grid sizes, square/rectangle positions and height of the square/rectangle. Finally I will alter the width of the square/rectangle.

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