1. MATH10000 Mathematical Workshop http://www.maths.manchester.ac.uk/undergraduate/ ugstudies/units/level1/MATH10000/ Dr Louise Walker Newman 1.24 louise.walker@manchester.ac.uk http://www.maths.manchester.ac.uk/undergraduate/ ugstudies/units/level1/MATH10000/ louise.walker@manchester.ac.uk

2. MATH10000 Mathematical Workshop Projects Individual and group work Project reports Presentations MATLAB 100% coursework

3. Timetable Semester 1 Week 1Introduction to the Workshop Week 2Project 1 - Cryptography Week 3Project 2 - Conic Sections Week 4Project 2 - Conic Sections Week 5Mathematical word-processing Week 6Mid-semester break Week 7Introduction to MATLAB Week 8Project 3 - Numerical Methods Week 9Project 3 - Numerical Methods Week 10 Project 4 - Determinants Week 11Project 4 - Determinants Week 12MATLAB assessment

4. Assessment Each project report will be assessed. There will be a mark for the correctness of the mathematics, a mark for the quality and clarity of presentation and a group mark (for group projects). Group presentation and word-processing exercise also assessed. The Workshop is worth 20 credits over both semesters.

5. Writing Mathematics maths is often poorly communicated who are you writing for? write in sentences use a suitable balance of words and symbols use diagrams and examples http://www.stat.ualberta.ca/~wiens/purdue1_write.pdf

6. Thinking Mathematically Entry Attack Review

7. Entry: Read and understand Use examples and diagrams Look for patterns

8. Attack: Generalise from specific examples Make conjectures Use logical arguments to prove conjectures Don’t worry about getting stuck Convince yourself, convince a friend, convince an enemy

9. Review: Checking your working Have you covered all cases? Can you extend your arguments to other cases?

10. Stuck? Being stuck can be a good thing Don’t give up Have you seen something like this before? Go back to your examples Explain your problem to someone else. Summarise your ideas

11. Which whole numbers can be written as a sum of at least two consecutive whole numbers? 1no9 = 2+3+4 or 4+5 2no10 = 1+2+3+4 3 = 1+211 = 5+6 4 = no12 = 3+4+5 5 = 2+313 = 6+7 6 = 1+2+314 = 2+3+4+5 7 = 3+415 = 1+2+3+4+5 8 no16 no…etc

12. Conjecture 1 – all odd numbers can be written as the sum of 2 consecutive numbers Conjecture 2 – all numbers that are not a power of 2 can be written as a sum of consecutive numbers

13. Proof of conjecture 1 Let n be an odd whole number. Then n = 2k+1 for some whole number k. We can write n= k + (k+1), the sum of two consecutive numbers as required. QED

14. Proof of Conjecture 2 If n is not a power of 2 then it has an odd divisor. Suppose n is divisible by 3, then n = 3k for some whole number k. We can write n = (k-1) + k + (k+1) as required. Can you generalise to n = 5k, n=7k, n = 11k etc ?