Welcome to A – Level Maths! Year 12 Induction Welcome to A – Level Maths!

The Curious Set of Integers The integers 1.3.8. and 120 form a set with a remarkable property: The product of any two integers is one less than a perfect square. Your Challenge: Find a fifth number that can be added to the set without destroying this property. The fifth number is 0. However, if you examine this question in more detail, you will find that is has a very long and interesting history that goes back to Fermat and Euler.

Back to business (after all, we’re here to talk about your course) Key Topics: 2/3 of your course content is on core mathematics 1/6 of your course content is on statistics 1/6 of your course content is on mechanics

Core Mathematics Core maths involves working with: Algebraic expressions Quadratics Equations and inequalities Graphs and their transformations Binomial expansion Trigonometry (including identities and equations) Vectors Differentiation Integration Exponential functions and logarithms

Statistics Statistics involves: Data Collection Measures of location and spread Representations of data Correlation Probability Statistical distributions Hypothesis testing

Mechanics Mechanics involves: Modelling in mechanics Constant acceleration Forces and motion Variable acceleration

Expectations For every hour of class time, you should be studying 1 hour on your time! We are here to guide your learning, but you need to be independent and motivated learners Tell us when you don’t understand something - It’s important! Read up on the subject…. You need to read and learn about maths and it’s history

Would the volume change if x increased or decreased? What could be the possible max or min values of x? What is the maximum volume? If we adjust the size of the rectangle will the value of x change to maximize the volume? How would this all change if the shape was a square?

Summer Task Complete the Induction Homework booklet Research either a famous mathematician or theorem. Over two pages, type down what you’ve learned, including why the person (or theorem) is important and useful.

More Maths… Bronx v Brooklyn: A young man lives in Manhattan near a subway express station. He has two girl friends, one in the Bronx. To vist the girl in Brooklyn he takes a train on the downtown side of the platform; to visit the girl in the Bronx he takes a train on the uptown side of the same platform. Since he likes both girls equally well, he simply takes the first train that comes along. In this way he lets chance determine whether he rids to the Bronx or to Brooklyn. The young man reaches the subway platform at a random moment each Saturday afternoon. Brooklyn and Bronx trains arrive at the station equally often – every 10 minutes. Yet for some obscure reason he finds himself spending most of his time with the girl in Brooklyn: in fact on the average he goes there 9 times out of 10. Can you think of a good reason why the odds so heavily favour Brooklyn?

Solution The answer to this puzzle is a simple matter of train schedules. While the Brooklyn and Bronx trains arrive equally often – at 10 minute intervals – it happens that their schedules are such that the Bronx train always comes to this platform one minute after the Brooklyn train. Thus the Bronx train will be the first to arrive only if the young man happens to come to the subway platform one minute after the Brooklyn train. Thus the Bronx train will be the first to arrive only if the young man happens to come to the subway platform during this one minute interval. If he enters the station at any other time the Brooklyn train will come first. Since the man’s arrival is random, the odds are 9 to 1 for Brooklyn.