- From: Several students
- Date: April 1999
*Subject: Advice with GCSE coursework investigations*
I have got to do an investigation for my GCSE coursework. Can you give me any advice on how to get started and how to present my final write-up? |

*Firstly, please remember that Maths Help cannot do your coursework for you. It has to be your own
work. But we can give some general advice.*

Usually a typical GCSE investigation gets you started by giving you a situation and asking a specific question. These situations often consist of a geometrical diagram or a number pattern. For example, they might give you a square of dots and ask how many triangles you can make. Or they might give you a grid of numbers and ask you what the row totals are.

You should answer this question carefully and in full detail, using a clear diagram (or set of diagrams as appropriate).

They will probably then ask you the same question but based on a larger shape or a larger number grid. For example, they might ask firstly based on a 2-by-2 square, then on a 3-by-3 square, then on a 4-by-4 square.

Again, work it out in full detail.

Now it is over to you . . . You should "investigate further" by considering the next few cases yourself, based on how they have started you off. For example, you should consider the same question on a 5-by-5 square, a 6-by-6 square and so on.

Having worked out a few more cases yourself, you should now summarise your results in a
table. For example *(NOTE: This data is fictitious!!)*

Side of square (n) | 2 | 3 | 4 | 5 | 6 |

Number of Triangles | 6 | 12 | 20 | 30 | 42 |

What they are looking for is for you to identify the underlying pattern(s) in your table of
data. The question you should be asking yourself is:

**OK, I have worked this out for several different cases. Now can I find a rule which will
tell me the number of triangles in ANY size square?**

First try to explain the pattern in the sequence of numbers on the bottom row of the table. Explain in words first. How are the numbers increasing? What are the step lengths? Is there a pattern? What do you think the next number would be? And the next? Why?

To get extra credit, you should try to find a **formula for the n ^{th} term **in the
sequence.

Now you should use your rule (which you either explained in words or found a formula for)
to **predict** what the next result in the sequence would be. (In the fictitious example above,
you could predict the number of triangles in a 7-by-7 square.) Justify your prediction fully,
using the formula, your observed rule or the underlying geometry or pattern of the situation.

Finally, **confirm** your prediction, by actually drawing out the 7-by-7 square (or whatever
the next item in the sequence would be) and working our the value directly. If your prediction is
not correct, don't despair. Explain why you realise it is wrong, and try again.

This brings one "cycle" of the investigation to an end. To summarise:

- Accept the starting point as given, and work it out;
- Work out the next few cases yourself;
- Summarise your results in a table;
- Explain in words any underlying pattern in the sequence of the results;
- Try to find a formula for the n
^{th}term in the sequence; - Use your rule or formula to predict the next item in the sequence;
- Check your answer by working it out directly (from diagram, etc)

Now it is your turn to "make up the rules". Run through the investigation cycle at least one more time, choosing your own different starting point. For example, if the original investigation was about squares, repeat the process using rectangles, say. (Be clear - what kind of rectangles? 2-by-3, 3-by-4, 4-by-5, etc? Or 2-by-4, 3-by-6, 4-by-8, etc? Keep a consistent pattern.)

GENERAL POINTS:

There are some things it is always advisable to do in a coursework investigation:

- Do things in a logical order, one step at a time
- Illustrate as much as you can with clear diagrams.
- Explain everything, even if it appears obvious to you.
- Don't throw anything away - if you realise you have not got very far with a particular approach, include it anyway and explain why it didn't seem to work.
- Use a computer if you can. A spreadsheet can generate the number sequences, for example. Be sure to explain the commands you used.

PLEASE NOTE:

The above is very general, and for guidance only. Your teacher should be able to let you know
what your particular examination board expects. Most exam boards send schools copies of
examplar pieces of coursework, and guidelines on how they are marked. Ask to have a look at some of
these.

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