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Introduction

Louis Franks 10PC  10X2                                                                                                         10/12/01

## Opposite Corners

 Wx L Difference Increase Wx L Difference Increase 2 x  3 20 10 6 x  3 100 50 2 x  4 30 10 6 x  4 150 50 2 x  5 40 10 6 x  5 200 50 2 x  6 50 10 6 x  6 250 50 2 x  7 60 10

Middle

10

6 x  9

400

50

2 x  10

90

10

6 x  10

450

50

We are investigating the difference between the products of the numbers in the opposite corners of any rectangles that can be put on a 100 square.

2 x  3 Rectangles

 1 2 3 11 12 13

To keep things simple I have started with rectangles with a width of 2 squares. I kept the width to two squares and increased the length by one square.  (see results table above).  I discovered that the width increases by 10 every time the length increases by 1.

The difference can be worked out for all rectangles with a width of 2 squares by using several formulas:

1.(Length – 1 x 10 = Z)

3 – 1x 10 = 20 = Z

## Then

(Width x  Z ) – Z = difference of opposite corners

2 x  20 – 20 = 20

### OR

#### 2. L = Length,  W = Width

(L – 1)(10 (W-1)) = difference of opposite corners

Example:

(3 – 1) x (10 (2 – 1)) = 20

### OR

3.

 1 2 3 11 12 13

Conclusion

1. 6 – 1 x 10 = 50

Then

3 x 50 – 50 = 100

2. (6 – 1) x (10 (3 – 1) = 100

3.

##### 26

(y+20) (y + 5) = y+ 100 +20y+ 5y

= y + 100 +25y

y ( y + 25)

= y + 25y

(y + 100 +25y) – (y + 25y) = difference between product.

= 100

##### Conclusion

I have come to the conclusion that my three formulas work for all types of rectangles and squares.  There are several ways to achieve the end result for the difference of the opposite corners.

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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