10 Summing up
In this course, you have looked at a variety of problems all of which involved using patterns or formulas and you have also extended some of your strategies for solving problems. One of the first steps in tackling any problem is to check that you understand both the problem and the information you have been given. This step can concentrate on what the question means. However, this can also involve looking up or checking on mathematical terms, notation or definitions as in Goldbach's conjecture. You may be able to sort this out yourself by referring back to your notes or books or, if it is an unfamiliar situation, it may be easier to discuss it with someone else. Understanding the notation is also important. The notation used for inequalities is often used to express ideas concisely and you may find it helpful to add this to your dictionary or to your notes for future use.
You have already seen how diagrams can be helpful and that applies to this course as well (for example a number line when dealing with inequalities).
Visualising a problem in a practical way, as in the currency exchange, can help you to work out what to do next as well. You may also like to try working through a problem with a few simple numbers first to get a feel of what's involved. This strategy was used in Goldbach's conjecture. However although we broke Goldbach's conjecture down into steps and found that the conjecture was true for all the examples we tried, this does not prove that the conjecture is always true. A pattern isn't proof!
Using formulas is an important skill, but it is important to check that the formula is appropriate for your particular problem (see the mobile phone problem) and also that any measurements are in the same units as those specified in the formula. It is easy to make mistakes in this way, so checking that the answer you obtain seems reasonable is very important too. You may like to make an estimate first as well. It is also important to check formulas you have derived yourself, particularly if you are using a spreadsheet. If you work out the numerical answer to the calculation first, you can then check that you get the same result when you use the formula in the spreadsheet. Although this doesn't guarantee that you have got the right formula (you may make the same mistake twice after all!), it can help to eliminate careless mistakes.