# Geometry

Monday, 30. January 2012 13:33

We used Geogebra to allow the students to create geometrical functions, and see the links between geometry and algebra.

One activity used this starting point in Geogebra:

Parabola

and asked the students to plot all the points that were equidistant from the line and the point.  It is set up so that the students create y=x^2.

The students had a lesson on spherical geometry - showing them that the angles in a triangle do not always add up to 180, and a lecture on reflectional symmetry in euclidean geometry, spherical geometry and hyperbolic geometry.

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# Computer based statistics

Monday, 30. January 2012 12:15

This theme showed the power of computers in understanding statistics and in analysing huge amounts of data in an easy visual way.

The students were introduced to CAST, an online statistics textbook which uses interactive applets for explanations and practice and for student practice.

Students watched Hans Rosling's fascinating video showing 200 years of history in four minutes, and had the opportunity to use the Gapminder software for themselves.  Students explored worldwide trends, but were able to focus on Kenya using the trace function.  The software initiates mathematical discussions such as "What is a logarithmic scale, and why is it often used instead of a linear scale?"

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# Game Theory and Modelling

Monday, 30. January 2012 11:50

This theme was probably the most fun - from Prisoners Dilemna to Monty Hall to running outside in a group trying to mimic the behaviour of swarming birds; there was certainly lots of hands on action.

Students watched a TED talk by Steven Strogatz explaining the basic rules that can be used to model how each bird flies in a swarm.

1. Only aware of your nearest neighbour
2. Fly in the same direction as your neighbour
3. Maintain the same distance from your neighbour
4. Fly away if you see a predator

Easy to understand, hard to enact!

The main idea we were trying to get across was that maths is all about sets of rules.  Once you understand the rules you can do the maths.  We played card games every evening which reinforced this idea of being able to operate within rules.

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# Number systems

Sunday, 29. January 2012 13:14

This theme grew from the film "The Story of 1" narrated by Terry Jones of Monty Python fame, which we showed the students in the middle of the week.  We looked at counting in binary, helped by the Mathematica applets Binary Card Game (you may have seen this in a Christmas cracker!), Counting in Binary, and Binary Counting Sequence.

The story of Pythagoras and his denial of irrational numbers was discussed, and the existence of irrational numbers was proved using this Geogebra applet:

Pentagram

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# Number Theory

Sunday, 29. January 2012 9:47

During the maths camp last summer we used Simon Singh's excellent website The Black Chamber after introducing the students to the concept of codebreaking by splitting them into groups and asking them to create their own codes.  We looked at the frequency distributions of letters in English and Kiswahili and how you could use these to crack certain types of code.

We followed this with a session on modular arithmetic, leading to the use of primes in codebreaking, via inverses.

In the final two sessions we showed the students three problems  and asked the following questions:

Fermat’s Last Theorem (with n=3)

x3+y3=z3

and asked students if they could find any integer solutions.

Hendrik Lenstra sequence

\$latex x_{0}=1\$

Etc. etc.

Are the solutions always integers?

3n+1 Problem

Take any positive integer, if it is odd: multiply it by 3 and add 1, if it is even: half it.

Keep following these steps, stopping if you get to 1.

e.g. 5 – 16 – 8 – 4 – 2 – 1

Are there any numbers which don’t go to 1?

Students had time to explore these using pen and paper and then used Mathematica applets to explore further

We then used these problems to discuss the difference between an unsolved problem, a counter example and a proof.

See the 3n+1 it described here by Stephen Wolfram

Fermat's Last Theorem - see Simon Singh's site (the book is brilliant)

Lenstra ...

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# Current themes

Friday, 27. January 2012 19:30

In the inaugural maths camp in 2011 we decided to structure the week according to mathematical themes.  The logic was that the themes would run through the week, with roughly one lesson per day in each theme.  This wasn't made explicit to the students, but the idea was that the lessons and activities would build up in difficulty during the week and come together at the end.

Initially this blog will describe the themes we used to introduce our students to extra curricular mathematics in 2011, and then add to these themes and develop new ones through interaction for 2012.

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