1 Introduction
Your first experience of mathematics was probably using numbers in everyday situations: for example, counting, measuring, and dealing with money. After that, you probably discovered that numbers could be used to work out all sorts of problems, and that looking at shapes and their properties had lots of practical applications too. Mathematics can be used to help solve many different problems – from everyday examples, such as working out your water bill or estimating the time you need to allow for a journey, to very complicated scientific and technological projects, such as sending astronauts to the moon. Although you will meet other ways of investigating problems during your studies, maths does play a very important part in tackling problems in many aspects of life. It is an essential tool for scientists and technologists, so learning about new mathematical ideas and how they can be used is very important for MST students.
A
ctivity Break Can you think of examples where you have used (or could use) maths in some everyday situations, such as travel, leisure, cookery or in your home?How do you think maths could be used in economics, architecture, politics, medicine or the study of the environment?
Here are a few examples to start you thinking.
Home World
• Planning my spending and budget.
• Scaling up recipes from four to six people
• Finding the quickest route between two places.
• Working out how much paint I need for the decorating.
• Finding a best buy.
• Working out the cost of a holiday from a brochure.
• Working out the inflation rate or tax rate.
• Predicting the outcome of an election.
• Deciding which is the most effective drug for an illness, or how much of a drug should be used.
• Investigating the safety aspects of a building.
• Predicting the effects of pollution.
• Working out the cheapest routes for supplying shops from warehouses.
How can mathematics help with these problems?
Most mathematical investigations can be described in terms of the cycle shown in Figure 4.1.
Figure 4.1
To see how the cycle works, consider the following problem.
You have agreed to organize a minibus trip to the Science Museum in London for your fellow students on the course. They would like to arrive there at about 11 a.m.
The ‘real-world’ problem is: ‘What time should you arrange to pick everyone up?’ The key question is: ‘How long will the journey take?’ The journey time will depend on traffic conditions, so it is difficult to be precise. However, a precise time is not needed; a rough idea of the time will do.
First, define the problem. From a map, the distance from your town to London is 150 miles. The problem is easier to solve mathematically if some
simplifications and assumptions are made.
1 Overall, the minibus travels at about 45 miles per hour.
2 Allow 30 minutes to park and walk to the museum.
Then, go into the ‘mathematical world’.
Using these facts and assumptions, the pick-up time can now be calculated.
The minibus can travel 45 miles in one hour.
Mathematical world Problem described by numbers,
equations and diagrams Use calculations and other
mathematical processes to solve the problem
Interpret the maths answer in terms of the
original problem
‘Real world’
Problem described in words
Define the problem Make simplifications
and assumptions Collect information
So, the time taken to travel 150 miles = 150 45 hours
= 313 hours
= 3 hours 20 minutes.
Hence, the time for the journey and parking = 3 hours 50 minutes.
Therefore, the pick-up time should be 7:10 a.m. (or 0710 hours).
Now, interpret the mathematical answer. From past experience, you know that some people in the group always arrive late. So, a sensible pick-up time would be 7 a.m. – allowing extra time for the latecomers to arrive. If some students protest that this is too early, you might have to work around the problem-solving cycle again, changing your assumptions about the parking time or the speed at which the minibus can travel. Note that the answer you get depends on the assumptions made and the data used. If these are wrong or inaccurate, the answer may not be reliable. For example, the traffic flow in London may make the average speed much slower than has been assumed here. So, leaving at 7 a.m. may not be early enough.
The cycle in Figure 4.1 can be used to find solutions to many different kinds of problem, although several trips around the cycle may be needed for complicated investigations, as assumptions are modified and the original problem redefined. Investigating problems in this way, however, does require certain skills and mathematical techniques. As an MST student, you will often use mathematics to investigate problems, so it is important that you develop the skills involved in studying new mathematical concepts and applying those ideas in practical situations. You will need to be able to:
X read mathematics and understand the language and notation
X choose and apply mathematical techniques, and decide on the next step to take
X decide whether an answer is reasonable or sufficient X sort out what to do when you get stuck
X reflect on what you have learned and on how it fits in with what you already know
X write down your own mathematics.
This chapter will help you with these skills. The next two sections – ‘Reading maths’ and ‘Practising maths’ – are particularly useful if you are learning maths as part of a course. They contain practical advice on understanding maths texts, using worked examples and tackling exercises. The final two sections – ‘Tackling mathematical problems’ and ‘Expressing yourself
mathematically’ – will be useful both as part of your studies and when you are using maths in a practical situation.
All the sections of this chapter use examples to illustrate ideas. If you would like extra help with the mathematical techniques involved, please refer to Maths Help towards the back of this book. The page references are marked in the text like this: see Maths Help on pages 00 to 00.