• 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
  16. 16
    16
  17. 17
    17
  18. 18
    18
  19. 19
    19
  20. 20
    20
  21. 21
    21
  22. 22
    22
  23. 23
    23
  24. 24
    24
  25. 25
    25
  26. 26
    26
  27. 27
    27
  28. 28
    28
  29. 29
    29
  30. 30
    30
  31. 31
    31
  32. 32
    32
  33. 33
    33
  34. 34
    34
  35. 35
    35
  36. 36
    36
  37. 37
    37
  • Level: GCSE
  • Subject: Maths
  • Word count: 2571

My coursework task is to investigate why, in a number grid square of 1-100, when a section of two by two squares is extracted and the two opposite squares are multiplied and then subtracted the result is always 10.

Extracts from this document...

Introduction

Math Coursework

My coursework task is to investigate why, in a number grid square of 1-100, when a section of two by two squares is extracted and the two opposite squares are multiplied and then subtracted the result is always 10.

I will also be testing and studying whether it is true for three by three, four by four, five by five e.t.c number squares. I shall also be studying what will happen if I change the size of the grid square upon which I am extracting the numbers from.

E.g.

image49.png

image00.pngimage01.png

image10.png

image50.png

image20.png

                     2x2 squareimage29.png

image68.pngimage37.png

image43.png

                                           3x3 square        

2 by 2 Analyses

image79.png

                                                     (2x11) - (1x12) = 10image44.png

image89.png

                (35x44) - (34x45) = 10image44.png

image100.png

                                  (3x12) - (2x13) = 10image02.png

image111.png

                (99x90) - (89x100) = 10image02.png

As we can see the results clearly show that no matter what selection of 2x2 square we use the result will always  be 10.

We can show how and why the result is always 10 by using Algebra, (representing numbers by using letters).

image132.pngimage122.png

   =

This is how we can express numbers using letters. As with numbers we can also put letters into a formula. This is how it would look:

(P+10)(P+1)- P (P+11) = 10

image03.png

Testing the Formula

Now we can put the formula to the test by using it with numbers:

(23x32)- (22x33) =10

image51.pngimage140.png

           =

As we can see the formula has proven to work well and the result is once again 10 as the formula suggests.

...read more.

Middle

²X10 is 360.

image90.png

(8x62)(2x68) = 360

image91.png

                                                                             (P+60) (P+6) – P (P+66)

image19.png

I will now try an 8x8 grid square … I predict using the table that the result will be 490, because 7²x10, is 490.

image92.png

        (29x92)(22x99) = 490

image21.png

As the other results have proven the rule that the number, -1, multiplied by ten and squared gives the result. I can safety say that I will try a 9x9 and 10x10 and I shall predict the result.

9x9 the result will be 640, as 10²x8= 640

image22.png

image93.png

(10x82)-(2x90)= 640

image23.png

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

This is the ten by ten grid square, I predict that when the opposite corners are multiplied and then subtracted from each other that the result will be 810, because judging by my results table so far I have formed a theory that it will be 10x9², is 810.

I can now test the theory that states:     P²+99P+810 -P² - 99P = 810          

(10x91)- (1x100) = 810

The result clearly shows us that the formula expressing the theory is correct.

image24.pngimage94.png

image25.png

image27.pngimage26.pngimage27.pngimage26.png

Now that I have tried and tested the results within a 1-100 grid square or 10x10, I am now going to try it with a 9x9 grid square like this:

image95.png

image28.png

image30.png

image96.png

image97.png

I will be extracting the square of 2x2, 3x3, 4x4 and so on.  Like shown above and subtracting the product of the two numbers multiplied together.

2 by 2 analyses

                                                                                 (2x10)-(1x11)=9image98.png

image31.png

image99.png

                                                  (81x89)-(80-89)=9image31.png

image101.png

         (74x82)-(73x83)=9

image31.png

        (43x51)-(42x52)=9image31.pngimage102.png

...read more.

Conclusion

always  be 8. Even if it is extracted from the bottom right or left or top left or right hand corners.

We can show how and why the result is always 8 by using Algebra, (representing numbers by using letters).

image149.pngimage150.png

             =

This is how we can express numbers using letters. As with numbers we can also put letters into a formula. This is how it would look:

(P+9)(P+1)- P (P+8) = 8

image45.png

image52.png

        (3x17)-(1x19)= 32

image53.png

        (23x37)-(21x39)= 32

image54.png

           (21x35)-(19x37) = 32

image55.png

             (4x18)-(2x20)=32

As we can see the results clearly show that no matter what selection of 3x3 square we use the result will always  be 32. Even if it is extracted from the bottom right or left or top left or right hand corners.

We can show how and why the result is always 32 by using Algebra, (representing numbers by using letters).

image56.pngimage57.png

             =

This is how we can express numbers using letters. As with numbers we can also put letters into a formula. This is how it would look:

(P+18)(P+2)- P (P+16) = 32

image46.png

As we can see there is a formula beginning to evolve as there was with the squares. So now we can draw a results table explaining and displaying our results.

image58.png

image47.pngimage48.png

 I have now proved, using algebra, that when the two corner numbers of any size square or rectangle are extracted from a number grid of any size, the results will always be, the size of the square (n-1)².

...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

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

    4 star(s)

    224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273

  2. Marked by a teacher

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

    4 star(s)

    The exact same thing can be done for any sized grid. The only thing that needs adapting is the value the number increases when moving down. For example, on a 13x13 grid, it would increase by 13 when you go down.

  1. Marked by a teacher

    Mathematics Coursework: problem solving tasks

    3 star(s)

    Formulas found for each type of spacer No. of rows + 1 tile +2 tiles +3 tiles +4 tiles +5 Tiles +6 tiles Complete Column 1 T 0 2 4 6 8 10 = 2n - 2 1 L 4 4 4 4 4 4 = 4 1 + 0 0 0 0 0 0 = 0 1

  2. Number Grid Coursework

    Product 2 (TR x BL) Difference (P'duct 2 - P'duct 1) 1 56 306 250 13 884 1134 250 21 1596 1846 250 32 2784 3034 250 43 4214 4464 250 e) Here are the results of the 5 calculations for 7x7 Box on Width 10 Grid: Top-Left Number Product 1 (TL x BR)

  1. Algebra Investigation - Grid Square and Cube Relationships

    = n2+nw+5hn-6n Stage B: Bottom left number x Top right number = (n+5h-5)(n+w-1) = n2+nw-n+5hn+5hw-5h-5n-5w+5 = n2+nw+5hn-6n+5hw-5h-5w+5 Stage B - Stage A: (n2+nw+5hn-6n+5hw-5h-5w+5)-(n2+nw+5hn-6n) = 5hw-5h-5w+5 When finding the general formula for any number (n), any height (h), and any width (w)

  2. Maths - number grid

    r + s Results Multiple of 12 3 x 2 24 12x2 12x2x1 2x3 24 12x2 12x1x2 5x3 96 12x8 12x4x2 6x4 180 12x15 12x5x3 7x4 216 12x18 12x6x3 8x5 336 12x28 12x7x4 My final part of my investigation was looking at the exact same size of rectangles as in Chapter Two except using my new 12x12 number grid.

  1. Maths coursework. For my extension piece I decided to investigate stairs that ascend along ...

    As said before, just as these individual values in each square would always be the same no matter where this 3-stair shape is translated around any size grid, the formula Un = 6n + 4g + 4 for working out the stair total for any given stair number - where

  2. algebra coursework

    Z+3 Z+30 Z+33 Z = top left number = 7 (in this case) Z+3 = 7 + 3 = 10 (top right number) Z+30 = 7 + 30 = 37 (bottom left number) Z+33 = 7+33 = 40 (bottom right number) Z (Z+33) = Z� + 33 Z (Z+3) (Z+30)

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