# 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

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Introduction

Opposite Corners

Introduction

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.

Hypothesis

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

Plan

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.

Methodology

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.

Middle

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.

Conclusion

(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.

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

Evaluation

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.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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## Here's what a teacher thought of this essay

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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