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UKHostMths advises on maths coursework
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Find out how many pieces of coursework you will be assessed on and what percentage it is worth. Some syllabuses have coursework tasks marked by your teacher and moderated by other people, others have the coursework as a formal written exam with a specific length of time for you to do it. You may only have to submit 2 tasks; your teacher will give you other tasks to do as practice. If you’ve the opportunity, ask your teacher what you need to do to improve your coursework.
- don’t leave your work until the last minute
- plan ahead
- stick to the deadlines set by your teacher
- always try your best
Coursework could have the following sections:
- The problem
- Methods
- Results
- Extension
- Evaluation
- Further lines of enquiry
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Whatever the circumstances, try to follow these guidelines:
Whenever you are given a coursework task to do, make sure you understand what maths is actually involved. Most tasks are geared towards algebra, some are of a statistical nature and a few involve shape and space. Even shape and space tasks often involve some algebra or trigonometry.
Once you’ve the task, start with simpler cases and build up your ideas. Try to record results in table form; it is easier to spot patterns and eventually find an algebraic rule that will allow you to make a generalisation. If you can, look for exceptions to your rules. They don’t always exist, but they do in some cases.
To gain higher marks, you need to extend the original task; either by looking at more sophisticated cases, by introducing another variable or looking at a similar problem which is a bit different.
Always keep your rough working notes. When you write up your coursework task, start by setting out the problem as it was given to you.
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"A cube has all of its faces painted red. The cube is cut into 243 smaller cubes. Investigate how many cubes have 0, 1, 2, 3 faces painted."
State that your are going to look at simpler cases and show your working out in detail for all of them.
"I am going to look at a cube cut into 27 smaller cubes first."
Be consistent in your use of letters and symbols – this shows you are able to express yourself mathematically.
"I am going to use x to stand for the number of cubes along each edge."
After you’ve round about 5 or 6 results, draw a table and show any values you’ve used or worked out for each case. Now look for patterns. A good way to find algebraic rules is to look at the differences between values. I’m not going to use the example of the painted cube here because it is often used as a GCSE task so it wouldn’t be fair to give you any more help with it. To look at differences, I’m just going to use a sequence of values:
x |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Results |
4 | 11 | 18 | 25 | 32 | 39 | 46 | 53 | 60 |
1st difference |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
7 |
The first difference is 7. This tells us that we have to multiply x by 7. Next work out what we need to add or subtract to get the algebraic relationship:
7 x 1 = 7, 7 – 3 = 4 7 x 2 = 14, 14 – 3 = 11 7 x 3 = 21, 21 – 3 = 18
The rule is multiply by 7 and subtract 3, or x maps onto 7x – 3, y = 7x –3.
For algebra involving powers, it gets a bit more complicated:
x |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Results |
3 | 8 | 15 | 24 | 35 | 48 | 63 | 80 | 99 |
1st difference |
5 |
7 |
9 |
11 |
13 |
15 |
17 |
19 |
|
2nd difference |
2 | 2 | 2 | 2 | 2 | 2 | 2 |
Because it is the second difference that is constant, the rule involves a quadratic (x²) and because the difference is 4 (2 x 2) it involves 2 times x squared, so we need to look at the squares of x and see what is added or subtracted:
x |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
x² |
1 |
4 |
9 |
16 |
25 |
36 |
49 |
64 |
81 |
Results |
3 |
8 |
15 |
24 |
35 |
48 |
63 |
80 |
99 |
Difference |
2 |
4 |
6 |
8 |
10 |
12 |
14 |
16 |
18 |
Factors of difference |
1x2 |
2x2 |
3x2 |
4x2 |
5x2 |
6x2 |
7x2 |
8x2 |
9x2 |
The formula is: x² + 2x.
Another way of finding the rule if it has a first constant difference is to draw a straight line (linear graph) and find the gradient – this is the value you multiply x by and the y-intercept (where the line cuts the y-axis) this is the value to add at the end of the equation or rule. Linear graphs are written in the format: y = mx +c.
Make sure you explain everything in as much detail as you can. Show all your working out. Now you need to extend your task by introducing another dimension. You could change the shape you are investigating or the rules, e.g. with Frogs:
"You have m white frogs and n black frogs. There is one gap between the frogs. You have to swap all of the white frogs with all of the black frogs. You can only move the frogs in one direction. They can only slide into an empty place or jump over one frog of the opposite colour. Investigate how many moves it takes."
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You could change the layout of the board the frogs are on or introduce gaps in extra places.
Finally you need to evaluate your work, say what you have done, why you chose to do it that way, look for possible further developments or other lines.
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