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Many candidates will have to carry out coursework as part of their examination.
- State how you are going to try to investigate the task.
- If your method does not work, explain why and try another method.
- Write down any observations you can make from your tables, diagrams, graphs or formulae.
- Remember to check your statements before drawing conclusions.
- To obtain the higher grades, you must develop the task further.
- Do not write essays.
- Use short sentences and be precise.
(Note this is not a model answer, it is simply one method of approaching the question)
A firm has two schemes for the payment of travel expenses.
Scheme 1 pays p pence per mile for the first k mile travelled; and then q pence per mile for each mile travelled beyond the first k miles.
Scheme 2 pays r pence for each mile travelled.
1. Calculate the amount paid in travel expenses for a journey of 500 miles under each scheme when p = 33, q = 20, r = 25 and k = 75.
When a question is split into parts in this way, the first part will usually be fairly straight forward.
For 500 miles, Scheme 1 pays 33 pence per mile for the first 75 miles and then 20 pence per mile for each mile travelled beyond the first 75 miles.
In other words, (33 × 75) + (20 × 425) = (2475 + 8500)p = £109.75
Similarly, Scheme 2 pays £125.00 .
2. A company member travels a distance of 200 miles.
Write down expressions, involving p, q, r and k, which will give the amount paid for his travel expenses under each scheme.
Scheme 1: amount paid = p pence for the first k mile travelled [= pk] plus q pence for each mile travelled beyond the first k miles [= q(200-k) ].
So the amount paid = pk + q(200 - k)
Scheme 2: amount paid = 200r
3. The 'break-even point' is the distance for which the two schemes pay the same amount. The manager believes that there will always be a 'break-even point' for the two schemes.
(a) Set up expressions involving p, q, r and k which will work out the amount paid in travel expenses under each of schemes 1 and 2.
Where m is the number of miles travelled:
Scheme 1: (simply replace 200 in the above equation by m) amount paid = pk + q(m - k) = pk + mq - kq
Scheme 2: amount paid = mr
(b) Hence set up an equation which will determine the 'break-even point' for various choices of p, q, r and k.
The break even point occurs when the amount paid by the two schemes is equal, ie when
pk + mq - qk = mr
At this point you may like to select various values p, q, r and k and determine the break-even point. A graph might be handy (a graph of cost against number of miles travelled for the two schemes?).
4. Obtain a range of solutions for this equation, confirming or otherwise, with justification explanation or proof, whether or not there is always a 'break-even point'.
A good answer to this part of the question is needed to obtain the top marks (over 21 out of 24).
One possible way of approaching this question might be to determine when there will be a break-even point and when there won't. If, for example, k is negative, but (p - q) and (r - q) are both positive, the value of m where the two schemes pay the same amount of money will be negative,
since m = k(p - q) .
r - q
Unless a negative number of miles is travelled, the schemes will not pay the same amount of money, whatever value of m is picked.
-Pick some numbers which will show your claims and draw graphs to help show what you are trying to say.
-Try r = 25, q = 25, k = 75 and p = 33. Plot cost against number of miles travelled on a graph for each of the two schemes and you will find that there is no break-even point.
-Will there be a break-even point in any other case?
Forming equations gains credit. You could probably have gained 20 out of 24 for forming the equations that I have written in part 3. For other problems, spending time trying to find and explain the equation for a general pattern can be very rewarding.
Most of the awarding bodies offer two different methods for assessing the coursework tasks that are completed. These are:
- Your school can select the tasks to be completed, mark all of them and send a sample of marked scripts to the awarding body to be moderated.
- The awarding body will set the tasks and mark the work.
The detailed criteria for Using and Applying Mathematics and for Handling Data, which are common to all awarding bodies, are written by QCA and printed in the specification that you will be using. You should be able to obtain a copy of these detailed criteria from your teacher or the awarding body’s website.
Using and Applying Mathematics Strand 1: Making and monitoring decisions
4 marks - Gathering (in a systematic manner) enough results that are correct and enable you to write a generalisation about the given problem.
5 marks - Change one variable and undertake sufficient new work so you could make a further generalisation.
6 marks - Show a range of techniques to extend and develop the task further. For example if you had only been using simple linear equations such as y = 3x – 2 up to this point you could try to use
a graphical approach or simultaneous or quadratic equations to support this extended work. (This would link in with the requirement for 6 marks in the Communication strand where the consistent use of symbolism, i.e. algebra, is required).
7 marks - Attempt to co-ordinate three features in the work, perhaps by moving into 3 dimensions.
Using and Applying Mathematics Strand 3: Mathematical reasoning
3 marks - Show a progression from ‘making general statements’, i.e. a valid generalisation, derived from at least three of your results.
4 marks - Test your findings, formula or relationship by checking a further case (do not use the values you already used in deriving the formula or results).
5 marks Give a valid explanation as to why your generalisation works, referring to the shape of a grid, or size and structure of a shape.
6–8 marks - The progression continues up to 8 marks where a mathematically rigorous justification is expected.
Using and Applying Mathematics Strand 2: Communicating mathematically
4 marks - Present work in an orderly manner using two different methods, for example tables and diagrams, linking them together with a commentary.
5–6 marks - Show an increased use of algebra.
7–8 marks - Show a sophisticated use of algebraic techniques.
Handling Data Strand 1: Specify and plan
5–6 marks - Show clear aims and state a plan designed to meet these aims. The data used should be appropriate and the reason for any ampling should be explained.
7–8 marks - Demonstrate valid reasons for what you have done and explain any limitations, for example bias, that might arise.
Handling Data Strand 2: Collect, process and represent
5–6 marks - Show correct use of appropriate calculations using relevant data.
7–8 marks - Demonstrate evidence of higher level techniques applied accurately.
Handling Data Strand 3: Interpret and discuss
5–6 marks - Use summary statistics to make comparisons between sets of data and clearly relating your findings back to the original problem and evaluating the success, (or otherwise), of your strategy.
7–8 marks - Explain how you avoided bias and demonstrate the use, for example, of a pre-test or a pilot questionnaire.
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