Transcript Week1presentation

MATH10000
Mathematical Workshop
http://www.maths.manchester.ac.uk/undergraduate/
ugstudies/units/level1/MATH10000/
Dr Louise Walker
Newman 1.24
[email protected]
MATH10000
Mathematical Workshop
• Projects
• Individual and group work
• Project reports
• Presentations
• MATLAB
• 100% coursework
Timetable Semester 1
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week 10
Week 11
Week 12
Introduction to the Workshop
Project 1 - Cryptography
Project 2 - Conic Sections
Project 2 - Conic Sections
Mathematical word-processing
Mid-semester break
Introduction to MATLAB
Project 3 - Numerical Methods
Project 3 - Numerical Methods
Project 4 - Determinants
Project 4 - Determinants
MATLAB assessment
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.
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
Thinking Mathematically
• Entry
• Attack
• Review
Entry:
• Read and understand
• Use examples and diagrams
• Look for patterns
Attack:
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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
Review:
• Checking your working
• Have you covered all cases?
• Can you extend your arguments to other
cases?
Stuck?
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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
Which whole numbers can be written as
a sum of at least two consecutive whole
numbers?
1 no
2 no
3 = 1+2
4 = no
5 = 2+3
6 = 1+2+3
7 = 3+4
8 no
9 = 2+3+4 or 4+5
10 = 1+2+3+4
11 = 5+6
12 = 3+4+5
13 = 6+7
14 = 2+3+4+5
15 = 1+2+3+4+5
16
no
…etc
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
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
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 ?