# I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid

by orangealexteh (student)

Opposite Corners

Opposite Corners

* Throughout this piece of work, the horizontal side will be referred to as the length, whilst the vertical side will be referred to as width.

Introduction/Aim

I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid. I am going to investigate different rectangles, of different areas, lengths, widths and positioning on a grid. After finding a pattern, I will try and prove that it will work for other cases by using algebra and making an algebraic formula. As I go along, I will have to record any ideas I have or patterns I see. Afterwards, I will go on to investigate how this rule may differs on a different sized grid.

2x2 Square, 10x10 Grid

This is a 10x10 grid. On it (outlines in red) is a 2x2 square.

Firstly, I’m going to see what the difference between the products of the corners is (D):

55x64=3520

54x65= 3510

3520-3510=10, D=10

Now: What if the same sized rectangle was placed in a different area of the grid?

29x38=1102

28x39=1092

1102-1092=10, D=10

The difference is still the same; I’m going to do one more example to confirm this:

2x11=22

1x12=12

22-12=10, D=10

From this I am assuming that the difference of the corners product (D) is the same wherever the square/rectangle is place. I will later try and prove this mathematically.

2x‘n’ Rectangles, 10x10 Grid

Assuming that the difference is the same regardless of the rectangles position, I will only need to do one example for each rectangle:

2x3

25x33= 825

23x35=805

825-805=20, D=20

2x4

80x87=6960

77x90=6930

6960-6930=30, D=30

From these I theorize that the difference is always a multiple of 10 (and goes up in 10).

Using this, I predict that 2x5 rectangles will have a difference of 40:

2x5

87x93=8091

83x97=8051

8091-8051=40, D=40

This has proven my prediction correct.

Table of Results 1(and further predictions)

} +10

} +10

} +10

} +10 (we can assume this, looking at the pattern of the difference increasing in increments of 10)

} +10

Possible Rules

One possible rule is taking one away from the length and multiplying by 10 will give the difference. This, written algebraically is:  10(L-1). However, this may not work with a width      that isn’t 2 [I’ll look into this further on].

One formula that takes into account the width is: 5W(L-1). This can be written in words as the Length minus one, times by five ...

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

**** This is a well thought out and constructed algebraic investigation. There is a good use of a general formula to generalise the relationships discovered. Specific strengths and improvements have been suggested throughout.