# Number Grid Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results

Amanda Gaber

March 2006

Mathematics Coursework

Number Grid

Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results.  I will start off by working this out on a 2x2 square in a 10 x 10 grid and then will investigate varying the widths and lengths of squares and rectangles.

Method: In order to simplify the process, the investigation has been divided into sections according to the size of squares in the grids.

2 x 2 Squares

To begin with, I used a 10 x 10 grid, looking at 2 x 2 squares:

Then I took 2 x 2 squares from this grid and multiplied the opposing corners to calculate the difference between the two products.

1 x 12 = 12

2 x 11 = 22

22 – 12 = 10

So the difference between the answers is 10. I then took another 2 x 2 box from the above 10 x 10 grid:

47 x 58 = 2726

48 x 57 = 2736

2736- 2726 = 10

The difference is 10 again.  Perhaps this means that because it is a 10 x 10 grid, that all the differences would be 10.  I would still like to further investigate this theory.

The grid below is once again a 2 x 2 box derived from the original 10 x 10 grid.

89 x 100 = 8900

90 x 99 = 8910

8910 – 8900 = 10

This once again confirms what I stated; that the difference between the products of cross-multiplied boxes will always equal 10 in a 10 x 10 grid.

I would like to determine if this is definitely correct, so I am going to do it again twice.

27 x 38 = 1026

28 x 37 = 1036

1036 – 1026 = 10.

43 x 54 = 2322

44 x 53 = 2332

2332 – 2322 = 10.

I can now confirm that my prediction is correct, and that all products from a 2 x 2 box in a 10 x 10 grid, when subtracted, equal 10.

I would like to prove that any 2 x 2 box in any sized grid, when multiplying the opposite corners and subtracting the product, will equal the size of the grid.  For example, the difference in the products in a 2 x 2 box within a 12 x 12 grid will always be 12!

I will now work out an equation that will allow me to work out the difference in any-sized grid.  I will use the term ‘n’ for the top left hand corner of the 2 x 2 box, and I will use the term ‘g’ for the size of the grid.  It is clearly evident that the top right hand corner is the top left hand corner plus one.  For the purpose of this exercise, we will call this n + 1. The bottom left hand corner is the top right hand corner plus the grid size, n+ g.  The bottom right hand corner is the bottom left hand corner plus one; n+ g+1.

These are the final expressions that are shown in the box below:

Now I need to create an algebraic formula that links the two calculations and their resulting product:

n(n + g + 1) = n2 + ng + n

n + 1(n + g) = n2 + ng + n + g

[n2 + ng + n +g] – [n2 + ng + n] = g

By looking at my final equation, I can say that my prediction was correct, that any 2 x 2 box in a 10 x 10 grid will always give the difference as being the size of the grid.

Now I want to see if this equation can work for any sized grid.  I will therefore investigate with an 11 x 11 grid. I predict that the difference between the products of a 2 x 2 box in the 11 x 11 grid will be 11.

1. 1 x 13 = 13

2 x 12 = 24

24 – 13 = 11

1. 19 x 31 = 589

20 x 30 = 600

600 – 589 = 11

1. 48 x 60 = 2880

49x 59 = 2891

2891 – 2800 = 11

1. 85 x 97 = 8245

86 x 96 = 8256

8256 – 8245 = 11

1. 92 x 104 = 9568

93 x 103 = 9579

9579 – 9568 = 11

Looking at these calculations above, it is clear that the prediction is correct and that ...

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

This is a well written piece of work with only a couple of minor errors. This piece of work shows a good application of some algebraic techniques. There are specific strengths and improvements suggested throughout.