**Task 1: Research**

Research the Golden Ratio and write 3 pages which will pass the College’s plagiarism checker explaining what you have found. You may like to investigate the relationship to the Fibonacci series, occurrence of the Golden Ratio in biology, art, music. Good places to start your research include Ted Talks or Numberphile. Your research should link with the syllabus through the use of surds and the solution of quadratic equations.

**Task 2: Application**

Sensible Ann starts work at 20 and hopes to retire at 60. She contributes £6000 per year to her pension for the first 10 years of her career and then stops making contributions. Assuming that her savings grow at 7% per year, how much will she have by the time she retires?

Steady Bob also works from 20 to 60 but he puts off starting his pension until he is 30 years old and then saves £6000 per year. How much will he have by the time of his retirement?

Flash Charlie leaves it until he is 50 years old to start saving for his pension. If he saves £6000 per year, how much will his pension be worth? How much must he save each year if he is to have the same total as Ann by the time he is 60?

Annuity rates for a healthy person at age 60 are approximately 4%. i.e. if you have a pension pot of £100 000, the pension company would take that and promise to pay you 4% x £100 000 = £4000 per year for the rest of your life. How much could Ann, Bob and Charlie each expect to receive?

Clever Dave views his pension in a different way. He has decided to start by targeting a pension of £30 000 per year and work back. How much would he need to have in his pension pot at age 60 to achieve this?

Eric invests 1000 per year for the full 40 years of his career. Calculate the value of his pension pot. By how much would this have been reduced if he’d delayed his pension by 5 years? Vary the rate of interest and see how this affects the value of his pension. Record your results in a graph. Eric’s pension provider charges a 1% fee each year. How much of a reduction would this make to the eventual value of his pension pot?

Write up all of your answers neatly, showing your methods clearly and explaining the steps in your working.

**Task 3: Application**

For this question use the following rates;

- National Insurance is charged at 0% for earnings up to £8 000 per year, 12% for everything between £8 000 and £43 000 and 2% for everything over £43 000.
- Income Tax is charged at 0% for income up to £11 000, 20% for everything from £11 000 to £43 000 and 40% for everything over £43 000.
- The minimum wage is £7.20
- Pensions contributions are deducted at 9% of earnings.

Ann works for 15 hours per week at minimum wage. How much does she earn per year (52 weeks) before deductions? How much does she earn after deductions? What is her average hourly rate after deductions?

Bob works full time (40 hours per week). He earns the £25000 per year. How much does he earn after deductions? What is his average hourly rate after deductions?

Charlotte is a senior manager. She works 60 hours per week and earns £60 000 per year. How much does she earn after deductions? What is her average hourly rate after deductions?

Each of the three people above are offered a promotion which comes with a pay rise of £3000 per year (before deductions). The promotion involves doing 2 extra hours of work per week. What is the real rate of pay offered after deductions for each of the three people above?

Instead of a pay rise, the company decides to offer free spa membership to senior managers, worth £1 000 per year. What is the equivalent increase in Charlotte’s salary which would be required to provide a £1 000 increase in post-deduction pay?

Write up all of your answers neatly, showing your methods clearly and explaining the steps in your working. You may wish to work with pen and calculator or you may wish to use a spreadsheet.

**Task 4: Research**

Research prime numbers and write 3 pages explaining what you have found.

You may like to investigate the formula for prime numbers, the uses of prime numbers, Gauss’s prime number theorem, Chebyshev’s theorem, Mersenne primes, Germain primes (what was her relationship with Gauss during the Napoleonic wars?). Good places to start your research include Ted Talks or Numberphile. Your research should link with the syllabus through the use of surds and the solution of quadratic equations.

**Task 5: Research**

Research Fermat’s last Theorem and write 3 pages which will pass the College’s plagiarism checker explaining what you have found.

You may like to investigate who Fermat was, where did he live, when was he alive. What was the Wolfskehl prize and what is the story behind it. Who solved Fermat’s last theorem – what can you find out about him? Good places to start your research include Ted Talks or Numberphile. Your research should link with the syllabus through the use of surds and the solution of quadratic equations.

**Task 6: Research**

What is the relationship between the length and the width of an A4 sheet of paper? Why is this important? What does it imply about the relationship between lengths and widths of A3 or A5 paper. Can you show *mathematically* why this is so? Your answer should link with the syllabus through the use of surds.

**Task 7: Research: **

Watch and write 2 sides explaining the key points regarding infinities, using the links below. More about Georg Cantor here. Infinity and paradoxes (philosophy) here . Some infinities are bigger than others here.

**Task 8: Application**

a) The table below shows the number of possible half time scores in a football match, given that the full time score is as shown Complete the table to show the number of ways of reaching full time scores of 4-0, 5-0 & any number ‘m’ –0.

The table below shows another possible set of full time scores. Complete the table.

Can you find a formula for the number of half time scores that could lead to a full time score of m-n where m and n both represent any number.

**Task 9: Behavioural Economics and Psychology**

Listen to this short recording of Richard Thaler, Nobel Prize winner for Behavioural economics.

By using a tree diagram, work out the probability of making a loss in his coin-toss game, assuming that you play the game a) once b) twice c) three times d) 4 times. Either by continuing your tree diagram or using your knowledge of Binomial probability, extend your table to show the probability of making a loss for 5, 6, 7, 8, 9, 10 games.

Assuming that you play the game 10 times, draw a table showing the probability distribution for the range of possible prizes.

What upper and lower quartile values for your probable winnings after 10 games?

If you play the game 100 times what is the chance that, at the end of 100 games, you have lost money?

**Task 10: Statistics in the Movies**

In the 2013 film Rush, about the 1976 formula 1 championship, Niki Lauder’s character claims that ‘every time he goes out to race there is a 20% chance that he could die’.

If this claim is true, what would have been the chance of surviving a) 2 races, b) 3 races, c) 4 races?

There were 16 races in the 1976 season. What, according to the film, should have been the chance of surviving the whole season?

The season started with approximately 30 drivers. If the film’s claim were true, approximately how many drivers would have died in the first race? How many would have been left to compete in the sixteenth and final race of the season?

Bonus point; what is the grammatical error contained within the dialogue of this video clip?

**Task 11: The Importance of the Serve in Tennis**

First start by simplifying the rules of tennis to ignore the possibility of deuce. Thus, you can win a game 40-0, 40-15 & 40-30.

A score of 40-15 may have come from the following sequence: 0-15, 15-15, 30-15, 40-15. How many other sequences could lead to 40-15?

How many sequences could lead to 40-30.

If the server’s advantage means that s/he has a 75% chance of winning a point, what is the chance of winning the game 40-0? What is the chance of the non-server winning the game 0-40?

Use a tree diagram to find the probability that the server wins the game.

Repeat the calculations a few times to find how the probability of winning a game varies, if you change the server’s advantage?

*Sixth Form Extension:*

Can you find a formula for the probability of winning the game if the server’s advantage is p? Plot the graph of the probability of winning the game versus the probability of winning the point. What does the gradient of this graph tell you about coaching strategy?

How does the analysis change if you include the possibility of deuce? Before you can answer this, you will need to have studied infinite geometric series.

**Task 12: Who Do You Think You Are?**

Watch this Youtube clip of the Eastenders actor, Danny Dyer, finding out that he is descended from Edward III, King of England. Is this a likely or unlikely discovery?

**Working Downwards from Edward III**

Edward lived from 1312 until 1377.

By the late 1500s we have a reasonably accurate estimate of the population of England; 3.8million. Make an estimate for the number of generations of Edward’s family to have been born between his birth and the late 1500s?

We know that he had 245 Great-Great-Grandchildren and, from that point onwards, the number of his descendants was increasing by a factor of about 3 per generation. How many descendants would he, therefore, have had by the late 1500s?

What percentage of the English population was therefore descended from Edward III at this point in time?

**Working Upward from You**

How many direct ancestors do you have in your parents’ generation? How many in the generation of your grandparents, your great-grandparents? What is the pattern?

Approximately how many years is it since the late 1500s to the date of your birth? How many generations of your family do you estimate this to be? How many ancestors do you, therefore, have by the time you’ve reached in the late 1500s?

**Combining the two estimates**

Using the number of ancestors worked out above and the probability you found for an individual to have been descended from Edward III, calculate the number of your own ancestors who would have been descended from Edward III.

Comment upon any strengths or flaws in this form of analysis.