• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 2891

Investigating when pairs of diagonal corners are multiplied and subtracted from each other.

Extracts from this document...

Introduction

Investigating when pairs of diagonal corners are multiplied and subtracted from each other. Introduction: In this coursework I shall be investigating when pairs of diagonal corners are multiplied and subtracted from each other. To do this I shall take square grids and then select 3 square boxes from each grid and multiply the opposite corners then find the difference, then I will change the box size, to try and find a pattern or formula. Then I will change the grid size and then once again take boxes of different sizes and multiply the opposite corners and find the difference. I will try and find a formula for and square box on and size square grid. Then I shall investigate further by using rectangle boxes instead of square. 34 x 45 = 1530 35 x 44 = 1540 Difference = 10 68 x 79 = 5372 69 x 78 = 5382 Difference = 10 81 x 92 = 7452 82 x 91 = 7462 Difference = 10 In a 2 x 2 box on a 10 x 10 grid the difference is 10. Algebraic Method x x + 1 x + 10 x + 11 x(x + 11) = x� + 11x (x + 1)(x + 10) = x� + 10 + x +10x = x� + 11x + 10 x� + 11x - x� + 11x + 10 = 10 1 x 23 = 23 3 x 21 = 61 Difference = 40 36 x 58 = 2088 38 x 56 = 2128 Difference = 40 73 x 95 = 6935 75 x 93 = 6975 Difference = 40 In a 3 x 3 box on a 10 x 10 grid the difference is 40. Algebraic Method x x + 2 x + 20 x + 22 x(x + 22) = x� + 22x (x + 2)(x + 20) = x� + 20x + 2x + 40 = x� + 22x + 40 x� + 22x - x� + 22x + 40 = ...read more.

Middle

N = Any Number (b -1) = Box size - 1 Let x = (b - 1) n n + x n + 8x n + 9x n(n + 9x) = n� + 9xn (n + 1)(n + 8) = n� + 8xn + xn + 8x� = n� + 9xn + 8x� n� + 9xn - n� + 9xn + 8x� = 8x� 8x� = 8(b-1) � I will now do the same process on an 11 x 11 grid to see if there is a pattern. 14 x 26 = 364 15 x 25 = 375 Difference = 11 28 x 40 = 1120 29 x 39 = 1131 Difference = 11 76 x 88 = 6688 77 x 87 = 6699 Difference = 11 In a 2 x 2 box on an 11 x 11 grid the difference is 11. Algebraic Method x x + 1 x + 11 x + 12 x(x + 12) = x� + 12x (x + 1)(x + 11) = x� + 11x + x + 11 = x� + 12x + 11 x� + 12x - x� + 12x + 11 = 11 5 x 29 = 145 7 x 27 = 189 Difference = 44 42 x 66 = 2772 44 x 64 = 2816 Difference = 44 89 x 113 = 10057 91 x 111 = 10101 Difference = 44 In a 3 x 3 box on an 11 x 11 grid the difference is 44. Algebraic Method x x + 2 x + 22 x + 24 x(x + 24) = x� + 24x (x + 2)(x + 22) = x� + 22x + 2x + 44 = x� + 24x + 44 x� + 24x - x� + 24x + 44 = 44 4 x 40 = 160 7 x 37 = 259 Difference = 99 78 x 114 = 8892 81x 111 = 8991 Difference = 99 74 x 110 = 8140 77 x 107 = 8239 Difference = 99 In a 4 x 4 box on an 11 x 11 grid the difference is 99. ...read more.

Conclusion

= n� + 9xn + ny (n + y)(n + 9x) = n� + 9xn + ny + 9xy n� + 9xn + ny - n� + 9xn + ny + 9xy = 9xy 9xy = 9(L - 1) (W - 1) I have noticed a pattern in my formulas. The number that comes before the brackets is the same as the grid size. So I can change that number to G when G = Grid Size. So the formula for any box size on any grid size: G (L - 1)(W - 1) Proving Formula: n n+(W - 1) n + G(L - 1) n + G(L -1) + (W - 1) N = Any Number (L - 1) = Length - 1 (W - 1) = Width - 1 Let x = (L - 1) Let y = (W - 1) n n + y n + Gx n + Gx + y n(n + Gx + y) = n� + nGx + ny (n + y)(n + Gx) = n� + nGx + ny + Gxy n� + nGx + ny - n� + nGx + ny + Gxy = Gxy Gxy = G (W- 1)(L - 1) Conclusion Formula Description 10(b - 1) � This calculates the difference of any square box on a 10 x 10 grid. 9(b - 1) � This calculates the difference of any square box on a 9 x 9 grid. 8(b - 1) � This calculates the difference of any square box on an 8 x 8 grid. 11(b - 1) � This calculates the difference of any square box on an 11 x 11 grid. G(b - 1) � This calculates the difference of any square box on any grid. 10(L - 1)(W - 1) This calculates the difference of any rectangle box on a 10 x 10 grid. 9(L - 1)(W - 1) This calculates the difference of any rectangle box on a 9 x 9 grid. G(L - 1)(W - 1) This calculates the difference of any rectangle box on any grid. Limitations: ...read more.

The above preview is unformatted text

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

    4 star(s)

    Again I will start with algebra. z-number in the top left corner. z z+1 z+2 z+3 z(z+23)=z�+23z z+10 z+11 z+12 z+13 (z+3)(z+20)=z�+23z+60 z+20 z+21 z+22 z+23 (z�+23z+60)-(z�+23z)=60 Difference This proves that with any 3*4 rectangular box the difference is always 60.

  2. Marked by a teacher

    Opposite Corners

    3 star(s)

    I shall now move on and investigate a 3x3 Rectangle (square) and more rectangles in the same way. 3x3 Rectangle (square) 54 55 56 64 65 66 74 75 76 71 72 73 81 82 83 91 92 93 54x76=4104 71x93=6603 74x56=4144 91x73=6643 40 40 10 11 12 20 21

  1. Marked by a teacher

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

    3 star(s)

    grid = Difference of 19 100 x 100 grid = Difference of 100 31 x 31 grid = Difference of 31 58 x 58 grid = Difference of 58 1000x1000 grid = Difference of 1000 3 X 3 Squares Having studied 2 x 2 boxes within different sized grids, I

  2. Number Grid Coursework

    If (p - 1) is the variable representing the side of the square, then the area calculated from that is (p - 1)2, or (p - 1)(p - 1). Because each of these brackets represents the side of the square, when it becomes a rectangle, it is safe to assume that the area will be: (p - 1)(q - 1)

  1. Algebra Investigation - Grid Square and Cube Relationships

    Because the second answer has +ghs2w-ghs2+gs2w+gs2 at the end, it demonstrates that no matter what number is chosen to begin with (n), a difference of s2ghw-s2gh+s2gw+s2g will always be present. Difference Relationships in a 10x10x10 Cube Top Face (TF) 1 2 3 4 5 6 7 8 9 10 11

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

    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

  1. Investigation of diagonal difference.

    a 2x2 cutout anywhere on the 10x10 grid by implementing the use of simple algebra. I can call the top left number in the cutout n, the top right number n + 1, the bottom left number n + 10 and the bottom right n + 11, as this is

  2. Maths-Number Grid

    6 � 6 Grid 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

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