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

My course work in maths is going to consist of opposite corners and/or hidden faces.

Extracts from this document...

Introduction

My course work in maths is going to consist of opposite corners and/or hidden faces. For my first course in mathematics I have been given the task of investigating the difference between the products of the numbers in the opposite corners of my rectangle that can be drawn on a 10 by 10 square.

I will start off by showing some examples of different rectangles and what the sum of the corners will be. This will help me to determine whether or not there is a pattern.

Example

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* 12=12

     2* 11=22

22-12=10

This example shows us that between the squares 1,2,11,12 the sum will be 10.

We will now go on to see what a square between 1,2,21,22 will be.

...read more.

Middle

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

3*14=42

4*13=52

52-42=10

For now it seems as though the pattern is working the same for the rest of the table. We need to justify our conclusion by drawing another two 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

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

3*24=76

4*23=96

96-76 = 20

Again the same results seem to be occurring in this row but to be sure we will do one more example.

Example

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

3*34= 102

4*33= 132

132 – 102 = 30

As is confirmed by my results any rectangle that is 2 along and 2 down will be 10, then 2 along 3 down will be 20 and so on and so forth.

Now that we have a small result

...read more.

Conclusion

Example

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*33=33

3*31=93

93-33=60


This confirms my prediction.

Now that I have gathered some information I can create I table of my results

and of further predictions.

2*2=10

3*2=20

4*2=30

2*3=20

3*3=40

4*3=60

2*4=30

3*4=60

4*4=90

2*5=40

3*5=80

4*5=120

2*6=50

3*6=100

4*6=150

2*7=60

3*7=120

4*7=180

2*8=70

3*8=140

4*8=210

2*9=80

3*9=160

4*9=240

2*10=90

3*10=180

4*10=270

I could carry on but I am sure that you can see from the table that a 5*2 would be 40 then a 5*3 would be 80 and so on.

Justification

The formula for the sum of the opposite corners minus the sum of the two opposite corners is :-

(10x – 10) * (y – 1)

X = the number of squares across

Y = the number of squares down

An example of this is a 3 * 3 square would equal

(10*3 – 10) * (3-1)

=  20 * 2

= 40

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

If we work this out by multiplying the opposite corners and subtracted the sums we get

3 * 21 – 1 *23

= 63 – 23

= 40

This proves my formula works.

...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

Here's what a teacher thought of this essay


The general pattern for a 10 x 10 grid is identified. To improve this investigation more algebraic manipulation is needed to verify the identified pattern. There should be multiplication of double brackets and the identification of an nth term. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/04/2013

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

    opposite corners

    5 star(s)

    find the general rule straight away for any rectangle on any grid. The letters I will use are: L: the length of the rectangle must be bigger than 1 and equal to or smaller than the grid size. H: the height of the rectangle.

  2. Marked by a teacher

    Opposite Corners. In this coursework, to find a formula from a set of numbers ...

    4 star(s)

    Also for a grid arranged in 13 columns, the difference for a square size, 10�10, is to be considered. Solutions: Below is a 20 by 20 grid. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

  1. Marked by a teacher

    I am going to investigate the difference between the products of the numbers in ...

    4 star(s)

    A+L A+2L ? A+G(W-1) A+1+G(W-1) A+2+G(W-1) ? A+(L-1)+G(W-1) These new changes can now be put into the proof. Proof (A+(L-1)) x (A+G(W-1)) = A2+(GA(W-1))+ A(L-1) + G(L-1)(W-1) A x A+(L-1)+G(W-1) = A2+A(L-1)+(GA(W-1)) A2+(GA(W-1))+ A(L-1) + G(L-1)(W-1) ? (A2+A(L-1)+(GA(W-1)))= A2+(GA(W-1))+ A(L-1) + G(L-1)(W-1) - A2-(GA(W-1)) A2+(GA(W-1))+ A(L-1) + G(L-1)(W-1) - A2-A(L-1)-(GA(W-1))= G(L-1)(W-1) + A2-A2 +(GA(W-1))-(GA(W-1) + A(L-1)-A(L-1)

  2. Investigate the number of winning lines in the game Connect 4.

    I should achieve a formula that will tell me the number of winning lines in all possible 3 directions on any sized board for Connect 4. 2 (w-3) (h-3) + h(w-3) + w(h-3) = 2wh - 6w - 6h +18 + wh - 3h +wh - 3w = 4wh -

  1. Number Grids Investigation Coursework

    = a2 + 33a + 90 - a2 - 33a = (a2 - a2) + (33a - 33a) + 90 = 90 Therefore I have proved that the difference between the products of opposite corners in 4 x 4 squares must always equal 90 because the algebraic expression for the

  2. Number Stairs Maths Investigation

    blocks (for example, on a 3 by 3 number grid the block in the top right hand corner will be 3w and on a 8 by 8 number grid the block in the top right hand corner will be 8w). Upon elaboration of the number grid on Page 2 (Fig.

  1. The Magic of Vedic Mathematics.

    The RHS of the answer should always contain 2 digits. Square of a number having all digits 1. Suppose you want to find the square of 11111, Check the number of 1s in the number, i.e. 5. So write 12345 and then write in the reverse order, 4321.

  2. Number Stairs

    Stair Total of this 3-Sstair is 62 24 14 15 4 5 6 4+5+6+14+15+24=68 The Stair Total of this 3-Sstair is 68 25 15 16 5 6 7 5+6+7+15+16+25=74 The Stair Total of this 3-Sstair is 74 26 16 17 6 7 8 6+7+8+9+16+17+26=80 The Stair Total of this 3-Sstair

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