• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
  1. 1
  2. 2
  3. 3
  4. 4
  5. 5
  6. 6
  7. 7
  8. 8
  9. 9
  10. 10
  • Level: GCSE
  • Subject: Maths
  • Word count: 1516

In this coursework, I intend 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

Extracts from this document...


Opposite Corners


In this coursework, I intend 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 that are aligned differently, of different lengths and widths.


This coursework will aim to establish if there is a pattern from which we could derive a formula from the numbers in the grid.


The plan is to use Algebra to analyse the results which will be logged into tables. Subsequently I will test any rules, patterns and theories I find by using predictions and examples.


To start with I will break up my work into sections, my first is:

2x3 Rectangles

I will concentrate on one particular aspect at a time.

What is the difference between the products of the corners?
Is the difference the same for a rectangle drawn anywhere in the grid?


I now have several questions:

Is the difference different when the rectangle is aligned so that its shortest sides are at the top and bottom?


What is the same between the two alignments?

What About Other Sizes of Rectangles?

I will now try rectangles, all in the 2 x X series, with different lengths.

...read more.


I believe I shall require only one example of each and will proceed on this principle.

3 x 3


3 x 4


3 x 5



What Do I Notice?

        The “L-1 x 5H” does not apply.                   But L-1 x 20 does.

I feel confidant that L-1 is correct but I still need to find how H can be used to calculate what to multiply L-1 by.

For 2 x X                L-1 x 10

For 3 x X                L-1 x 20

Could the universal rule be:                 D = L-1 x 10 x (H-1)?                        

Subtract one from the length, multiply by 10 and then multiply by the height subtracted by one?

Testing the Rule

4 x X

I will now look at 4 x X and see if my new found rule truly encompasses all rectangles.

4 x 4


4 x 5


The difference is advancing by 30.  4 x 6 will have a difference of 150.

4 x 6



(table calculated with a calculator)

Could the  x 10 in my rule be related to width of my number grid, which is ten squares wide?

To test this idea I will try my rule on a different sized grid.


...read more.



(X + L - 1) x (X + (G(H - 1))) = XxX + X (G + X(H - 1)) + L-1 X + L-1 (G  (H - 1))_

 X x  (X + (G(H - 1))) + L - 1) = XxX + X(G + X(H - 1)) + X L-1 _________________

L-1 (G (H - 1))

To Test This Expression

L-1 (G  (H-1))

Let us substitute the value of X for 1 in a 2x2 rectangle on a 10x10 grid and check that it calculates the square values correctly.

image26.png   So far so good

And now I will use the above expression to find the difference (which should be 10)

L-1 (G (H-1))

2-1 (10 (2-1))                        

1 x (10 x 1)

1x 10 = 10

It seems fine so far.

I will now check it with a 3 x 4 rectangle on a 10x10 grid.  I know the difference should be 60 from previous workings.

L-1 (G (H-1))

4-1 x (10 (3-1))

3    x (10 (2))

3    x 20 = 60

The expression is true!

L-1 (G (H-1)) is the universal rule with the brackets in different positions:

L-1 (G (H-1)) = L-1 x (Grid Width) x (H-1) = Difference


I believe I have completed my aims and succeeded in attaining the goals I set myself.  I have found the rule to work out the difference for any rectangle of any size on any grid.  I have done this through the use of algebraic methods.

I have taken this course work as far as I can in the time that I have been allotted. I am happy with the work that I have done and can not think of anything I could have carried out better or improved upon.

...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

3 star(s)

This is a reasonably well structured investigation. It uses a wide variety of experimental examples to support a discovered pattern. To improve this more algebraic expressions to represent the differences need to be included. Specific strengths and improvements have been suggested throughout.

Marked by teacher Cornelia Bruce 18/07/2013

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)

    For instance I could instead of having; 1, 2, 3, 4, 5... I could have; 2, 4, 6, 8, 10... as I have in the last few investigations I will only use algebra as numbers as well are to time consuming.

  2. Marked by a teacher

    Opposite Corners of a Square on a Number Grid

    3 star(s)

    by 10 Square root 2*2 1 1 3*3 4 2 4*4 9 3 5*5 16 4 6*6 25 5 Forming an Expression: The number before the term number is squared and then multiplied by 10 to reach the result. So this means the new expression would be written like this: 10(n-1)

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

    Therefore, we have 5L -24 present in the formula. I will use this formula to prove its correctness and to look further for additional differences. Number In Sequence Formula Formula Equation Results L-Sum 1 5L -24 (5 x 19) - 24 71 71 2 5L -24 (5 x 20)

  2. Number Grid Investigation.

    3 x 3 = 40 4 x 4 = ? = 40 + 30 (p.d.) + 20 = 90. The difference between each p.d. increases by 20 on the previous one each time, (see increase T.2.) Therefore, this can be shown as M+20, where M is the difference previously. E.g.

  1. GCSE Maths Sequences Coursework

    Spatial Justification Now that I have found that my formulae are correct by making predictions then proving them, I will now use diagrams to spatially justify my formula and explain why these formulae work. I will use two different stages of the sequence in each justification to ensure this is true in all cases.

  2. Algebra Investigation - Grid Square and Cube Relationships

    = n2+33n Stage B: Bottom left number x Top right number = (n+30)(n+3)= n2+3n+30n+90 = n2+33n+90 Stage B - Stage A: (n2+33n+90)-(n2+33n) = 90 When finding the general formula for any number (n), both answers begin with the equation n2+33n, which signifies that they can be manipulated easily.

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

    9w - 9h +18 (Formula for winning lines on any sized board) To test the formula, I will do the same as I did before. I will predict the number of winning lines on a 5x4 board for a game of Connect 4 however this time I will include the

  2. Number Grids Investigation Coursework

    I will now try some more examples of this to make sure this is the case in all 3 x 3 squares in this number grid. 75 76 77 85 86 87 95 96 97 (top right x bottom left)

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