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

Opposite Corners

Extracts from this document...

Introduction

Opposite Corners


This piece of coursework uses a grid of numbers and a x by x box drawn around a set of numbers. I am trying to see if there is a pattern between the differences of the products.

Aim: My task is to investigate the differences of the products of the diagonally opposite corners of a rectangle, drawn on a 10x10 grid, with the squares numbered off 1 to 100.

I will aim to investigate the differences for rectangles and squares of different lengths, widths.

I plan to use algebra to prove any rules I discover which I will hopefully find by analysing my results that I will display in tables. I will test any rules, patterns and theories I find by using predictions and examples.

...read more.

Middle

n        n+1

n+10        n+11

n(n+11)= n²+11n

(n+10)(n+1)=n²+10n+n

n²+11n+10

n²+11n

10

To find a formula for any square on a 10x10 grid I created this table from the parts above:

Squares

Difference

Diff. /10

Square root of Diff/10

2x2

10

1

1

3x3

40

4

2

4x4

90

9

3

5x5

160

16

4

Looking at the last column I noticed that the numbers were 1 less than the dimension size in the 1st column.

Working backwards I took 1 dimension(d) then subtracted 1, squared it then multiplied by 10 to give this formula:

(d-1)²x10

I then proved my formula worked by testing it with a 3x3 box.

(3-1)²x10

2²x10

4x10=40

Rectangles:

...read more.

Conclusion

n        n+1        n+2

n+10        n+11        n+12

n(n+12)=n²+12n

(n+10)(n+2)=n²+12n+20

n²+12n

n²+12n+20

        20

I created this table for rectangles:

Rectangle

Difference

Diff /10

2x3

20

2

2x4

30

3

3x4

60

6

3x5

80

8

I worked out from the table and other evidence the formula:

(w-1)(l-1)x10

w=width        l=length

Grid Size:

I started to try and find a formula for any grid size for a square and rectangle.

I saw that in both formulae there was a x10 at the end. I thought that this may be the grid size.

I tested my prediction with a 3x3 box and a 2x3 rectangle on a 20x20 grid.

3x3-        (3-1)²x20                                        2x3-        (2-1)(3-1)x20

        2²X20                                                1x2x20

        4x20=80                                                2x20=40

Actual Differences:

3x3-        1x43=43                                        2x3-        1x42=42

3x41=1232x41=82

8040

...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)

    57 58 59 60 62 63 64 65 66 67 68 69 70 72 73 74 75 76 77 78 79 80 82 83 84 85 86 87 88 89 90 92 93 94 95 96 97 98 99 100 In a 9�9 square: 2 � 100 = 200 10

  2. Marked by a teacher

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

    3 star(s)

    = n2 + 19n + n n + 1(n + 19) = n2 + 19n + n + 19 ? [n2 + 19n + n +19] - [n2 + 19n + n] = 19 My prediction of what the difference would be was correct.

  1. Marked by a teacher

    In this coursework, I intend to investigate the differences of the products of the ...

    3 star(s)

    has a length of 3 squares and see if the same patterns apply and investigate whether the above rules are true for rectangles of any shape. I believe I shall require only one example of each and will proceed on this principle.

  2. Marked by a teacher

    I am going to investigate taking a square of numbers from a grid, multiplying ...

    3 star(s)

    1 x 22 = 22 4 x 19 = 76 76 - 22 = 54 The difference is 54, so the equation is correct and can be used to work out the difference for any 4x4 square in any square grid.

  1. Number Grid Coursework

    is the formula for an area relating to the box. When the grid width was 10, as in the last section, the formula for the difference of the two products was 10(p - 1)(q - 1). Being 10, I guessed that the '10' referred to the grid width.

  2. Investigate the differences between products in a controlled sized grid.

    5 represents the top left hand number in the box. This is to prove the equation is correct and that the difference is 90. x�+33x x�+33x+90 5�+(33*5) 5�+(33*5)+90 =190 =280 280-290= 90 This proves that the equation does work

  1. My coursework task is to investigate why, in a number grid square of 1-100, ...

    Will it be 10 or 40 or something else? (4x31)- (1x34) = 90 (64x91)- (61x94) = 90 (70x97)- (67x100) = 90 (37x64)- (34x67) = 90 As we can see the result is clearly 90, no matter what section of the number square we use.

  2. Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted.

    the second, third and fourth column in so that I could easily see if there were any relationships between the numbers. There are not many obvious patterns visible from this table and not many clues to a formula. If I draw a graph I may then be able to work

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