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  • Level: GCSE
  • Subject: Maths
  • Word count: 1645

Number Grid

Extracts from this document...

Introduction

Maths Coursework - Number Grid

Friday 13th July 2007

Number Grid Coursework

   For this piece of coursework, I will investigate the difference when 2x2, 3x3, 4x4, 5x5 and rectangle snapshots are taken from a 10x10 number grid and have their corners multiplied and the difference worked out. For the first part, I will use 2x2 snapshots.

2x2 Boxes

Box 1

2

3

12

13

 2x13=26

 36-26=10

3x12=36

Box 2

32

33

42

43

32x43=1376

 1386-1376=10

33x42=1386

Box 3

6

7

16

17

6x17=102

 112-102=10

7x16=112

Box 4

5

6

15

16

5x16=80

 90-80=10

6x15=90

   I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 10. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 2x2 snapshot, the difference I will find will be 10.

Box 5

25

26

35

36

25x36=900

 910-900=10

26x35=910

   My predictions that I made earlier about ‘Box 5’ have turned out to be correct as when I multiplied the numbers in the 2x2 box and worked out the difference, I was left with 10.

...read more.

Middle

59

60

67

68

69

70

77

78

79

80

47x80=3760

 3850-3760=90

50x77=3850

   My predictions that I made earlier about ‘Box 4’ have turned out to be correct as when I multiplied the numbers in the 4x4 box and worked out the difference, I was left with 90.

I should now try the same method but with boxes of 5x5 dimensions.

5x5 Boxes

Box 1

1

2

3

4

5

11

12

13

14

15

21

22

23

24

25

31

32

33

34

35

41

42

43

44

45

1x45=45

 205-45=160

5x41=160

Box 2

6

7

8

9

10

16

17

18

19

20

26

27

28

29

30

36

37

38

39

40

46

47

48

49

50

6x50=300

 460-300=160

10x46=460

      I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 160. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 5x5 snapshot, the difference I will find will be 160.


Box 3

23

24

25

26

27

33

34

35

36

37

43

44

45

46

47

53

54

55

56

57

63

64

65

66

67

23x67=1541

 1701-1541=160

27x63=1701

My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 5x5 box and worked out the difference, I was left with 160.

I should now try the same method but with rectangles.

Rectangles

Rectangle 1

1

2

3

4

11

12

13

14

21

22

23

24

1x24=24

 84-24=60

4x21=84

Rectangle 2

7

8

9

10

17

18

19

20

27

28

29

30

7x30=210

 270-210=60

10x17=270

      I have noticed the pattern here. Whenever the numbers are diagonally multiplied and then the difference is found, you always end up with 60. The hypothesis I am going to make is that if I was to work out the difference of another box taken from a 10x10 in a 4x3 snapshot, the difference I will find will be 60.

Rectangle 3

22

23

24

25

32

33

34

35

42

43

44

45

22x45=990

 1050-990=60

25x42=1050

My predictions that I made earlier about ‘Box 3’ have turned out to be correct as when I multiplied the numbers in the 4x3 box and worked out the difference, I was left with 60.

Algebra Variables

   For this specific investigation, I have decided to use algebra to represent and then make formulae for each snapshot I have taken from the 1-100 grid. I will use the letter H to represent the height of each snapshot and use the letter W for the width of each snapshot, no matter of size and dimensions. In this, S is the top left corner in each snapshot. I will now prove how 2x2 boxes always leave you with a difference of 10 through algebra.

Equation

S

S+1

S+10

S+10+1

...read more.

Conclusion

2

= S2-20W-11S+11WS+10W2+10

Next, I will show how to find the formula for any Rectangle.

Finding the Formula Which Can Be Used On Any Rectangle

S+10L-10xS+(W-1)=

S2+10LS-10S(W-1)S+10(L1)(W-1)

SxS+10(L-1)+(W-1)=

S2+10LS-10S(W-1)S+10(L-1)(W-1)-(S2+10LS-10S+(W-1)S)

=10(L-1)(W-1)image00.png

   I have now found the general rule to work out the formula for any rectangle on a 10x10 grid. The rule is 10(L-1)(W-1).


Expanding the Brackets

s(s+11)

S

+11

S

S2

+11

s2+11s

(s+10)(s+1)

s

+10

s

s2

+10s

+1

+s

+10

s2+11s+10-(s2+11s)

s2+11s+10-(s2-11s)image03.pngimage02.pngimage02.pngimage01.png

=+10image04.png

Algebra

2x2 Box

s

s +1

s+10

s+10+1

s(s+11)(s+1)-s(s+11)image05.png

3x3 Box

s

s+1

S+1+1

S+10

s+10+1

S+10+1+1

s+10+10

s+10+10+1

S+10+10+1+1

s(s+10+10+1+1)-(s+1+1)

  (s+10+10)

  s+20

4x4 Box

S

S+1

S+1+1

S+1+1+1

S+10

S+10+1

S+10+1+1

S+10+1+1+1

S+10+10

S+10+10+1

S+10+10+1+1

S+10+10+1+1+1

s+10+10+10

s+10+10+10+1

s+10+10+10+1+1

S+10+10+10+1+1+1

s(s+33)-(s+30)(s+3)

5x5 Box

s

s+1

s+1+1

s+1+1+1

s+1+1+1+1

s+10

S+10+1

S+10+1+1

S+10+1+1+1

S+10+1+1+1+1

S+10+10

S+10+10+1

S+10+10+1+1

S+10+10+1+1+1

S+10+10+1+1+1+1

S+10+10+10

S+10+10+10+1

S+10+10+10+1+1

S+10+10+10+1+1+1

S+10+10+10+1+1+1+1

s+10+10+10+10

s+10+10+10+10+1

s+10+10+10+10+1+1

s+10+10+10+10+1+1+1

s+10+10+10+10+1+1+1+1

s(s+44)-(s+40)(s+4)

DIFFERENCE IS 10 WHEN YOU GO DOWN A BOX.

Shiva Dhunna

11S/Ma2

...read more.

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