• 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
  13. 13
    13
  14. 14
    14
  15. 15
    15
  • Level: GCSE
  • Subject: Maths
  • Word count: 5017

Maths Grids Totals

Extracts from this document...

Introduction

Module 4 Coursework

I am going to investigate squares of different sizes and on different grids.

I am going to draw a square around numbers on a grid, and find the product of the top-left and bottom-right numbers, and the top-right and bottom-left numbers. I will then calculate the difference.

10 x 10 grids

2 x 2 squares

12

13

22

23

12 x 23 = 276

13 x 22 = 286

286 – 276 = 10.

55

56

65

66

55 x 66 = 3630

56 x 65 = 3640

3640 – 3630 = 10.

25

26

35

36

25 x 36 = 900

26 x 35 = 910

910 – 900 = 10

I have found that that there is a difference of 10 on any 2 x 2 square, on a 10 x 10 grid.

3 x 3 squares

36

37

38

46

47

48

56

57

58

36 x 58 = 2088

38 x 56 = 2128

2128 – 2088 = 40.

77

78

79

87

88

89

97

98

99

77 x 99 = 7623

79 x 97 = 7663

7663 – 7623 = 40.

4 x 4 squares

25

26

27

28

35

36

37

38

45

46

47

48

55

56

57

58

25 x 58 = 1450

28 x 55 = 1540

1540 – 1450 = 90.

5 x 5 squares

32

33

34

35

36

42

43

44

45

46

52

53

54

55

56

62

63

64

65

66

72

73

74

75

76

32 x 76 = 2432

36 x 72 = 3592

3592 – 2432 = 160.

I have now found out the differences for 2x2, 3x3, 4x4 and 5x5 squares on a 10x10, which will be shown in the following table:

Size

Difference

2 x 2

10

3 x 3

40

4 x 4

90

5 x 5

160

All the differences are square numbers of the previous number multiplied by 10 (e.g. 3 x 3 = 10 x 22 = 10 x 4 = 40). This gives me the formula 10(n-1)2.

Using this formula, I predict that the difference for an 8 x 8 square will be 10(8-1)2 = 10 x 72 = 10 x 49 = 490.

I will take a random 8 x 8 square from the 10 x 10 grid:

12

13

14

15

16

17

18

19

22

23

24

25

26

27

28

29

32

33

34

35

36

37

38

39

42

43

44

45

46

47

48

49

52

53

54

55

56

57

58

59

62

63

64

65

66

67

68

69

72

73

74

75

76

77

78

79

82

83

84

85

86

87

88

89

12 x 89 = 1068

19 x 82 = 1558

1558 – 1068 = 490.

My prediction was correct; therefore this proves that the formula 10(n-1)2 is correct.

n

n

x

x+ (n-1)

x+ 10(n-1)

x+ 11(n-1)

I am going to prove 10(n-1)2 works, algebraically. First, I am going to show an n x n square on a 10 x 10 grid (algebraically).

“x” is the number in the top-left corner. The number to the right of it is “x+ (n-1)” because it is “x” added to the length of the square (take away one).

...read more.

Middle

37

38

39

40

41

42

43

46

47

48

49

50

51

52

53

54

57

58

59

60

61

62

63

64

65

68

69

70

71

72

73

74

75

76

79

80

81

82

83

84

85

86

87

90

91

92

93

94

95

96

97

98

101

102

103

104

105

106

107

108

109

112

113

114

115

116

117

118

119

120

32 x 112 = 3584

24 x 120 = 2880

3584 – 2880 = 704. The formula was correct.

I am now going to prove the formula algebraically.

“x” is the number in the top-left corner. The number to the right of it is “x+ (n-1)” because it is “x” added to the length of the square (take away one). Because the square is on an 11 x 11 grid, the formula for the number underneath the square is “x” added to (11 multiplied by “the squares length”-1). The bottom-right number is “x+ 12(n-1)” because it is “x+ 11(n-1) added to another (n-1), so therefore this becomes “x+ 10(n-1).

n

n

x

x+ (n-1)

x+ 11(n-1)

x + 12(n-1)

[x + (n-1)][x + 11(n-1)]= x2 + 11x(n-1) + x(n-1) + 11(n-1)2.

x[x+12(n-1)] = x2 + 12x(n-1).

This becomes: x2 + 11x(n-1) + x(n-1) + 11(n-1)2x2 – 12x(n-1).

The x2 and –x2 cancel each other out, while the 11x(n-1) and x(n-1) add together to cancel out the -12x(n-1). This leaves 11(n-1)2, which is the overall formula for an 11 x 11 grid.

A 10 x 10 grid’s formula is 10(n-1)2, a 9 x 9 grid is 9(n-1)2 and an 11 x 11 grid is

11(n-1)2. The size of the grid corresponds directly to the first digit of the formula.

This means the formulae can be re-written as g(n-1)2 (this is the overall rule for any square on any grid).

I am now going to predict differences for squares on a 14 x 14 grid, using the formula g(n-1)2. First, I am going to predict a 12 x 12 square on a 14 x 14 grid. The formula, with the numbers in place, is 14(12-1)2 = 14 X 112 = 14 X 121 = 1694.

I chose the following square:

29

30

31

32

33

34

35

36

37

38

39

40

43

44

45

46

47

48

49

50

51

52

53

54

57

58

59

60

61

62

63

64

65

66

67

68

71

72

73

74

75

76

77

78

79

80

81

82

85

86

87

88

89

90

91

92

93

94

95

96

99

100

101

102

103

104

105

106

107

108

109

110

113

114

115

116

117

118

119

120

121

122

123

124

127

128

129

130

131

132

133

134

135

136

137

138

141

142

143

144

145

146

147

148

149

150

151

152

155

156

157

158

159

160

161

162

163

164

165

166

169

170

171

172

173

174

175

176

177

178

179

180

183

184

185

186

187

188

189

190

191

192

193

194

40 x 183 = 7320.

29 x 194 = 5626

7320 – 5626 = 1694.

The formula worked. Just to make sure it works universally, I will choose another square, however this time it will be 7 x 7.

A 7 x 7 square on the grid will have the formula 14(7-1)2 = 14 X 62 = 14 X 36 = 504. I chose the following square:

64

65

66

67

68

69

70

78

79

80

81

82

83

84

92

93

94

95

96

97

98

106

107

108

109

110

111

112

120

121

122

123

124

125

126

134

135

136

137

138

139

140

148

149

150

151

152

153

154

70 X 148 = 10360

64 X 154 = 9856

10360 – 9856 = 504.

I have found that the formula g(n-1)2 works, and I will now prove it algebraically.

n

n

x

x+(n-1)

x+

g(n-1)

x+ g(n-1) + (n-1)

[x + (n-1)][x + g(n-1)] = x2 + xg(n-1) + x(n-1) + g(n-1)2

x[x + g(n-1) + (n-1)] = x2 + xg(n-1) + x(n-1).

This becomes x2 + xg(n-1) + x(n-1) + g(n-1)2 – x2 – xg(n-1) – x(n-1).

After cancelling out, only g(n-1)2 is left, which proves that it is the overall formula, not only for squares of different sizes, but also for different sized grids.

image01.png

Now that I have found the rule for squares, I am going to try and find a rule for rectangles. To start off, I am going to use rectangles on a 10 x 10 gird.

I am going to use the smallest possible rectangles to start off with (2 x 3 rectangles):

11

12

13

21

22

23

13 x 21 = 273

11 x 23 = 253

                         273 – 253 = 20.

66

67

68

76

77

78

68 x 76 = 5168

66 x 78 = 5148

                  5168 – 5148 = 20.

33

34

43

44

53

54

34 x 53 = 1802

33 x 54 = 1782

1802 – 1782 = 20.

7

8

17

18

27

28

8 x 27 = 216

7 x 28 = 196

216 – 196 = 20.

All 2 x 3 rectangles have a difference of 20, regardless of whether they are bigger horizontally or vertically.

I will now get the differences for other sized rectangles:

3 x 4

22

23

24

25

32

33

34

35

42

43

44

45

25 x 42 = 1050

22 x 45 = 990

1050 – 990 = 60.

3 x 4

58

59

60

68

69

70

78

79

80

88

89

90

60 x 88 = 5280

58 x 90 = 5220

5280 – 5220 = 60.

4 x 5

2

3

4

5

12

13

14

15

22

23

24

25

32

33

34

35

42

43

44

45

5 x 42 = 210

2 x 45 = 90

210 – 90 = 120.

4 x 5

66

67

68

69

70

76

77

78

79

80

86

87

88

89

90

96

97

98

99

100

70 x 96 = 6720

66 x 100 = 6600

6720 – 6600 = 120.
5 x 6

51

52

53

54

55

56

61

62

63

64

65

66

71

72

73

74

75

76

81

82

83

84

85

86

91

92

93

94

95

96

56 x 91 = 5096

51 x 96 = 4896

5096 – 4896 = 200.

5 x 6

16

17

18

19

20

26

27

28

29

30

36

37

38

39

40

46

47

48

49

50

56

57

58

59

60

66

67

68

69

70

20x 66 = 1320

16 x 70 = 1120

1320 – 1120 = 200.

6 x 7

34

35

36

37

38

39

44

45

46

47

48

49

54

55

56

57

58

59

64

65

66

67

68

69

74

75

76

77

78

79

84

85

86

87

88

89

94

95

96

97

98

99

39 x 94 = 3666

34 x 99 = 3366

3666 – 3366 = 300.


After trying out rectangles of 2 x 3, 3 x 4, 4 x 5, 5 x 6 and 6 x 7 size, I assembled the differences and put it into a table:

Size

Difference

2 x 3

20

3 x 4

60

4 x 5

120

5 x 6

200

6 x 7

300

The formula for squares was g(n-1)2 which is equal to g(n-1)(n-1). The g represented the grid size, and the n represented the width and height of the square. It could be re-written as g(h-1)(w-1), with h representing the height and w representing the width. This means the overall rule could be g(h-1)(w-1). I am going to test it in several ways. First, I am going to try it with rectangles I have already investigated:

For a 2 x 3 rectangle on a 10 x 10 grid, the rule would be 10(2-1)(3-1) which is 10 x 1 x 2 = 10 x 2 = 20. This is correct: I will now try the rule with a bigger rectangle.

A 5 x 6 rectangle is 10(5-1)(6-1), which become 10 x 4 x 5, which is 40 x 5 = 200. This is also correct. However, there is a possibility that this rule is not universal and that it only applies to rectangles with 1 square difference between the height and width.

I am now going to investigate the differences for other rectangles (i.e. rectangles with more than 1 square difference between the height and width (e.g. 2 x 4, 3 x 6 etc.)).

23

24

25

26

33

34

35

36

26 x 33 = 858

23 x 36 = 828

858 – 828 = 30.

1

2

11

12

21

22

31

32

2 x 31 = 62

1 x 32 = 32

62 – 32 = 30

36

37

38

39

40

46

47

48

49

50

56

57

58

59

60

...read more.

Conclusion

This leave g(h-1)(w-1), which is the universal rule for any square or rectangle on any size grid.


Matrices

A Matrix is a rectangular order of numbers (it can also be a square), i.e. in this case, it is the squares that are found on the grids (that are used to find the differences).
There are different things that can be done with matrices of different sizes. These include addition, subtraction, and multiplication.

Addition of a matrix is adding up the corresponding values of the matrices.

e.g. 2 matrixes on a 10 x 10 grid are:

image02.png

12

13

22

23

54

55

64

65

image02.png

        and


54 + 12

55 + 13

22 + 64

23 + 65

The addition of these matrices would be: image03.pngimage03.png

…which in turn would become:


image03.pngimage03.png

66

68

86

88

This is basic addition for matrices.

Matrix addition can be applied to any square (or rectangle for that matter) of any size, as I am going to show with a 5 x 8 rectangle on an 11 x 11 grid.

13

14

15

16

17

18

19

20

24image03.png

25

26

27

28

29

30

31image03.png

35

36

37

38

39

40

41

42

46

47

48

49

50

51

52

53

57

58

59

60

61

62

63

64

68

69

70

71

72

73

74

75

79

80

81

82

83

84

85

86image03.png

90

91

92

93

94

95

96

97

101

102

103

104

105

106

107

108

112

113

114

115

116

117

118

119

image03.png

          And



This becomes

81

83

85

87

89

91

93

95

103

105

107

109

111

113

115

117

125

127

129

131

133

135

137

139

147

149

151

153

155

157

159

161

169

171

173

175

177

179

181

183

I am going to show how algebraically, Matrix addition works:


Let a = top-left number of first matrix.

Written algebraically, the first matrix can be written as

a

a+1

a+2

a+3

a+4

a+5

a+6

a+7

a+g

a+g+1

a+g+2

a+g+3

a+g+4

a+g+5

a+g+6

a+g+7

a+2g

a+2g+1

a+2g+2

a+2g+3

a+2g+4

a+2g+5

a+2g+6

a+2g+7

a+3g

a+3g+1

a+3g+2

a+3g+3

a+3g+4

a+3g+5

a+3g+6

a+3g+7

a+4g

a+4g+1

a+4g+2

a+4g+3

a+4g+4

a+4g+5

a+4g+6

a+4g+7

There is a “g” in the cells: “g” stands for the size of the grid on which the squares are.

The second matrix would be exactly the same as this one, but with “b” replacing every “a”.


--

...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. GCSE Maths Sequences Coursework

    see here that there is not must of a pattern in the 1st difference or the 2nd difference, but when I calculate the 3rd difference, I can see that it goes up in 8's, therefore this is a cubic sequence and has an Nth term.

  2. Number Grid Coursework

    For simplicity, d = [p - 1], and e = [q - 1]: (a + d)(a + 10e) - a(a + 10e + d) = 10de a a+[p-1] a + 10[q - 1] a + 10[q - 1] + [p - 1] This means that I predict that with a

  1. How many squares in a chessboard n x n

    GENERATING DATA AND STRUCTURE m x n 1 x 2 Interesting that why are we left with this part. All will m x n be revealed in conditions for a 1 x 3 rectangle m x n + (n -1)(m -1) 2 x 3 + (3 - 1)(2 - 1)

  2. Investigation of diagonal difference.

    relationship between the length of the cutout, the top right corner, and the bottom right corner. The number added needed to make up the value of the top right and bottom right corner is always 1 less than the length of the cutout.

  1. Number Grids Investigation Coursework

    If I put these figures into my formula, I would get: D = w (n - 1)2 (n = 3) (w = 7) D = 7(3 - 1)2 = 7 x 22 = 7 x 4 = 28 If I draw out the start of a 7 x 7 grid,

  2. Algebra Investigation - Grid Square and Cube Relationships

    n+4(w-1)+4g(h-1) Simplifies to: n n+4w-4 n+4gh-4g n+4w+4gh-4g-4 Stage A: Top left number x Bottom right number = n(n+4w+4gh-4g-4) = n2+4nw+4ghn-4gn-4n Stage B: Bottom left number x Top right number = (n+4gh-4g)(n+4w-4) = n2+4nw-4n+4ghn+16ghw-16gh-4gn-16gw+16g = n2+4nw+4ghn-4gn-4n+16ghw-16gh-16gw+16g Stage B - Stage A: (n2+4nw+4ghn-4gn-4n+16ghw-16gh-16gw+16g)-(n2+4nw+4ghn-4gn-4n)

  1. Number Grids

    11 12 13 14 21 22 23 24 31 32 33 34 4 x 31 = 124 1 x 34 = 34 124 - 34 = 90 44 45 46 47 54 55 56 57 64 65 66 67 74 75 76 77 47 x 74 = 3478 44 x

  2. Number Grid Maths Coursework.

    on to see what differences are made to the formula and patterns if rectangles are used instead of squares in a 10x10 grid. Numerical examples 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

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