Many problems in mathematics can be considered in different ways. For instance a multiplication of decimal numbers can also be considered as a multiplication of fractions, which may make it easier. A graphical problem can be considered algebraically and vice-versa.

With geometrical problems, we can add lines to the problem that could help us. These could be parallel or perpendicular lines, lines of symmetry, lines joining points, or other useful lines.

We will now consider the same problem three different ways, to demonstrate that if you get “stuck” using one method it is always useful to consider alternative methods:

Worked Example

Work out the size of angle a in the below diagram:

Solution 1 to Worked Example

By adding an additional parallel line passing through a (as shown below), we can then spot alternate angles and sum these to solve the problem:

Solution 2 to Worked Example

To be a good mathematician you should learn as many methods as possible to tackle a problem, as the same method isn’t always the most efficient. So even if you understand perfectly the “solution 1”, you should strive to understand the other solutions.

Instead of adding a parallel line we could add a perpendicular line through “a” to make two triangles as shown below. We can then deduce the missing angles in the triangles and use these to calculate a, using a diagram as shown below:

Solution 3 to Worked Example

We can draw the alternative parallel line shown below and find the missing angles in the quadrilateral that it makes:

Do you understand all three methods? Do you have a preferred one? Why?

Important theory in mathematics can be proved many different ways using different methods. True mathematicians are interested in not only the answer to a problem, but also the different ways that they can find it. Over 300 proofs have been constructed of Pythagoras’ theorem, for instance, some of which are detailed here.

25 Questions of Increasing Difficulty

Worked Solutions to Questions