## GeoGebra: Fixed Point Iteration: Rearrangement Method x=g(x)

Numerical method for finding roots of f(x) = 0, by rearranging to x = g(x) and iterating
xn+1 = g(xn)

Click the image to link to GeoGebraTube

Designed principally for use with the MEI-OCR C3 Coursework on Numerical Methods

## GeoGebra: Fixed Point Iteration: Newton-Raphson Method (Newton’s Method)

October 29th, 2013 1 comment

Newton-Raphson is an iterative numerical method for finding roots of equations.
xn+1 = x− f(xn)/f′(xn)

Click the image to link to GeoGebraTube

Designed principally for use with the MEI-OCR C3 Coursework on Numerical Methods

## Pythagoras’ Theorem works with Negative Length

March 31st, 2011 1 comment

When taking square roots of both sides of an equation, one should be careful not to turf out the negative result without first considering whether it has a true meaning.  When using Pythagoras’ Theorem, the last step is to take square roots. So, can we have a hypotenuse with length -5?

## The Traversal Problem (3 dimensions)

##### [This problem is related to the easier 2-dimensional “Diagonal Problem” – click here for a full solution to the 2d problem]

1. Imagine a cuboid made up of lots of little cubes.

2. Imagine a straight line (or “traversal”) connecting two opposite vertices of the cuboid

3. How many little cubes does the line pass through?

Note: Cubes are only counted if the traversal passes through them, so if your traversal goes exactly through a vertex or an edge separating cubes within the cuboid, then it is not considered to pass through the adjacent cubes.  In the 2×3×4 example shown, the traversal actually passes through an edge in the centre of the cuboid.

## The Diagonal Problem (2 dimensions)

##### [This problem is followed by the more challenging 3-dimensional Traversal Problem – click here to view that post]
1. Draw a rectangle on squared paper.
2. Draw a diagonal across your rectangle.
3. How many squares does it pass through?

Note: If your diagonal goes exactly through a point, like in the 2×4 example, then it is not considered to pass through either of the diagonally adjacent squares.

Problem: How many squares does a diagonal pass through in a 190×884 rectangle?b

## Instructions: Using an iframe to embed a GeoGebra worksheet in a WordPress blog post

1. From the WordPress Dashboard, install a plugin that enables iframes to be embedded into posts, for example Embed Iframe. (Simple instructions for installing plugins into WordPress may be found here) Read more…

## “Higher or Lower” – Part II

In Part I we calculated the probability of winning a single-suit version of the Higher of Lower card game.  The objective of this post is to find the probability of winning the full, unsimplified, 4-suit version of the game played in pubs across the land. Read more…

## “Higher or Lower” – Easy card game? Part I

I was recently roped into joining some friends for a pub quiz.  I hate pub quizzes.  For a start, I don’t watch soaps or football or take any interest in divorcing celebrities.  The sum total of my contribution is usually a question about the periodic table and another about some obscure ’90’s one-hit wonder.

At the end of this particular quiz there was a competition to win a pot of dosh.  Those optimistic enough to enter first had to be lucky enough to have their number drawn from a hat, and then attempt to win the ominous game of Higher or Lower, or “Play Your Cards Right” as it is sometimes known, via a crudely written flash program running on the host’s laptop.  If you win the game, you win the cash.  Easy?  Apparently not.  Two players failed in succession and then the host declared that the contents of the pot would roll over to next week.  Almost everyone in the pub had just lost a quid.  Would anyone ever win the money?  I didn’t know, but I was alarmed and it was my duty to discover the probability of winning this game.

## Money, Debt & Greed – A Brief Commentary

March 9th, 2011 1 comment

### On Friday I will be talking in our school chapel. Here is the penultimate draft of the monologue (the style does lend itself to being read out loud)

Good morning.

I want to talk to you today about our monetary system.  Most, perhaps all, of us here at Forest School have benefited from the monetary system.  Money affords us food, entertainment, transport, holidays; a place to live; stability.  Money creates incentives; it gives us jobs, careers and aspirations; and taxes allow huge investments like high-speed rail links and the Olympic park.

But does everyone benefit from our monetary system?

We’re born into a society of which money seems to be the driving force.  Where does money come from? Who gives us it to spend?