governor’s school for the sciences mathematics day 1
TRANSCRIPT
Governor’s School for the Sciences
MathematicsDay 1
Schedule
• Introductions• Hometown: Good, Bad or Ugly• MOTD• A Joke• Some Math!• Forms and Tests• Lab
Course Structure
• Instructors• Lecture, Work and Labs• Point System• Project• AfterClass
Where are you from?
Give us the Good, the Bad or the Ugly!
Tnmap.gif
MOTD: Abu al-Khwarizmi
• 780-850 AD• Hindu-Arabic
numerals• First to use 0• “Algorithm”
derived from his name
Why did Homeland Security arrest the mathematician?
Why did Homeland Security arrest the mathematician?
• They found out he was a member of the al Gebra network.
• They found a calculus book, a compass and chalk in his briefcase: weapons of math instruction.
All About Patterns
Patterns in number sequencesPatterns in letters/wordsPatterns in geometry
Mathematics is about finding and studying patterns.
Pattern “Black Box”
Nature gives us a pattern we need to create a matching box.
Simple PatternsWhat comes next?
• 1, 2, 3, 4, ___• 1, 2, 3, 5, ___• 1, 2, 4, 8, ___• 2, 4, 6, 8, ___• 1, 3, 2, 4, ___• 3, 5, 9, 17, ___
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, __• 1, 2, 4, 8, __• 2, 4, 6, 8, __• 1, 3, 2, 4, __• 3, 5, 9, 17, __
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_• 1, 2, 4, 8, __• 2, 4, 6, 8, __• 1, 3, 2, 4, __• 3, 5, 9, 17, __
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_• 1, 2, 4, 8, _16_• 2, 4, 6, 8, __• 1, 3, 2, 4, __• 3, 5, 9, 17, __
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_• 1, 2, 4, 8, _16_• 2, 4, 6, 8, _10_• 1, 3, 2, 4, __• 3, 5, 9, 17, __
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_• 1, 2, 4, 8, _16_• 2, 4, 6, 8, _10_• 1, 3, 2, 4, _3_• 3, 5, 9, 17, __
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_• 1, 2, 4, 8, _16_• 2, 4, 6, 8, _10_• 1, 3, 2, 4, _3_• 3, 5, 9, 17, _33_
Simple PatternsWhat comes next?
• 1, 2, 3, 4, _5_ • 1, 2, 3, 5, _8_• 1, 2, 4, 8, _16_• 2, 4, 6, 8, _10_• 1, 3, 2, 4, _3_• 3, 5, 9, 17, _33_
But any answer can
be justified!(That’s why the
SAT/ACT shouldn’t ask about sequences)
Easiest way to generate a pattern of numbers is to use a function f (call it the pattern generating function ) and compute
f(1), f(2), f(3), f(4), etc.
For example: if f(x) = 2x – 1, we get:
1, 3, 5, 7, 9, …
For f(x) = x2 – 3x + 2, we get
0, 0, 2, 6, 12, …
Now, suppose we have a sequence of numbers:
A(1), A(2), A(3), A(4), …
Is there a pattern generating function, I.e. a function f s.t. A(n) = f(n), for n = 1, 2, 3, …?
If so, how do we find it?
Since there are multiple choices, what’s the simplest answer?
Polynomial Interpolation
• General polynomial of degree n p(x) = a0 + a1x + a2x2 + … + anxn
• Finding p means finding a0, a1, a2, …, an
• After picking n there are 3 approaches:1. Vandermode Matrix2. Nested Form/Divided Differences3. MATLAB (computer lab)
Time for Boardwork!
Simple PatternsWhat’s the pattern generating function?
• 1, 2, 3, 4:
• 2, 4, 6, 8:
• 2, 5, 8, 11:
• 1, 3, 7, 13:
Simple PatternsWhat’s the pattern generating function?
• 1, 2, 3, 4: A(n) = n
• 2, 4, 6, 8: A(n) = 2n
• 2, 5, 8, 11: A(n) = 3n-1
• 1, 3, 7, 13: A(n) = n2 – n + 1
Test Time!
References
• Introduction to Scientific Computing by Charles Van Loan, Ch. 2 (Poly & Trig Interp), Ch. 7 (Least Squares) [Handout]
• mathworld.wolfram.com/topics/Interpolation.html (Pointers to other methods)
• Any book on ‘Numerical Analysis’ or ‘Numerical Methods’ should contain a chapter or 2 on interpolation.