fractals - kock snowflake

Fractals aren’t actually on the syllabus – but they do offer quite a good opportunity to look at limits, infinite sequences, complex numbers (eg Julia sets etc), the relationship between maths and art and so on.

This video is a fantastic introduction to fractals – looking at how the Koch snowflake has simultaneously a finite area and an infinite perimeter (interesting to link this to Gabriel’s Horn which has finite volume and infinite surface area – though this is not related to fractals):

Even more amazing, the Koch snowflake has a fractional dimension – more than 1 but less than 2.

PBS Nova have created a really detailed and interesting look at fractals and how they occur in real life:

To introduce students to fractals, you can also use the Sierpinski Triangle – which can be generated quite easily (instructions here)

and while you at it, the video of an ultra-detailed Mandelbrot zoom in is pretty impressive – a form of pictorial infinity:

There’s also the Dragon Curve – which is explained in a Youtube video. This allows students to see the beginnings of a fractal design from simply repeatedly folding a strip of paper.

From this you can start to look at both Julia and Mandelbrot sets. Dan Pearcy has posted a fantastic blogpost on the topic – which explains how the amazing fractal nature of these shapes are generated. There are also some amazing Julia Set generators and Mandelbrot generators on Geogebra.

All of this can lead onto the coastline paradox (or indeed, it might be a good place to start a lesson) – which asks can we ever actually measure the length of a coastline – because the more detail we go into, the longer the perimeter becomes.  A good link to ToK knowledge.

IB Revision

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If you’re already thinking about your coursework then it’s probably also time to start planning some revision, either for the end of Year 12 school exams or Year 13 final exams. There’s a really great website that I would strongly recommend students use – you choose your subject (HL/SL/Studies if your exam is in 2020 or Applications/Analysis if your exam is in 2021), and then have the following resources:

Screen Shot 2018-03-19 at 4.42.05 PM.pngThe Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and each area then has a number of graded questions. What I like about this is that you are given a difficulty rating, as well as a mark scheme and also a worked video tutorial.  Really useful!

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The Practice Exams section takes you to ready made exams on each topic – again with worked solutions.  This also has some harder exams for those students aiming for 6s and 7s and the Past IB Exams section takes you to full video worked solutions to every question on every past paper – and you can also get a prediction exam for the upcoming year.

I would really recommend everyone making use of this – there is a mixture of a lot of free content as well as premium content so have a look and see what you think.