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traffic simulation

Simulations -Traffic Jams and Asteroid Impacts

This is a really good online Java app which has been designed by a German mathematician to study the mathematics behind traffic flow.  Why do traffic jams form?  How does the speed limit or traffic lights or the number of lorries on the road affect road conditions?   You can run a number of different simulations – looking at ring road traffic, lane closures and how robust the system is by applying an unexpected perturbation (like an erratic driver).

There is a lot of scope for investigation – with some prompts on the site.  For example, just looking at one variable – the speed limit – what happens in the lane closure model?  Interestingly, with a homogenous speed of 80 km/h there is no traffic congestion – but if the speed is increased to 140km/h then large congestion builds up quickly as cars are unable to change lanes.   This is why reduced speed limits  are applied on motorways during lane closures.

Another investigation is looking at how the style of driving affects the models.  You can change the politeness of the drivers – do they change lanes recklessly?  How many perturbations (erratic incidents) do you need to add to the simulation to cause a traffic jam?

This is a really good example of mathematics used in a real life context – and also provides some good opportunities for a computer based investigation looking at the altering one parameter at a time to note the consequences.


Another good simulation is on the Impact: Earth page.  This allows you to investigate the consequences of various asteroid impacts on Earth – choosing from different parameters such as diameter, velocity, density and angle of impact.  It then shows a detailed breakdown of thee consequences – such as crater size and energy released.   You can also model some famous impacts from history and see their effects.   Lots of scope for mathematical modelling – and also for links with physics.  Also possible discussion re the logarithmic Richter scale – why is this useful?

Student Handout

Asteroid Impact – Why is this important?
Comets and asteroids impact with Earth all the time – but most are so small that we don’t even notice. On a cosmic scale however, the Earth has seen some massive impacts – which were they to happen again today could wipe out civilisation as we know it.

The website Impact Earth allows us to model what would happen if a comet or asteroid hit us again. Jay Melosh professor of Physics and Earth Science says that we can expect “fairly large” impact events about every century. The last major one was in Tunguska Siberia in 1908 – which flattened an estimated 80 million trees over an area of 2000 square km. The force unleashed has been compared to around 1000 Hiroshima nuclear bombs. Luckily this impact was in one of the remotest places on Earth – had the impact been near a large city the effects could be catastrophic.

Jay says that, ”The biggest threat in our near future is the asteroid Apophis, which has a small chance of striking the Earth in 2036. It is about one-third of a mile in diameter.”

Task 1: Watch the above video on a large asteroid impact – make some notes.

Task 2:Research about Apophis – including the dimensions and likely speed of the asteroid and probability of collision. Use this data to enter into the Impact Earth simulation and predict the damage that this asteroid could do.

Task 3: Investigate the Tunguska Event. When did it happen? What was its diameter? Likely speed? Use the data to model this collision on the Impact Earth Simulation. Additional: What are the possible theories about Tunguska? Was it a comet? Asteroid? Death Ray?

Task 4: Conduct your own investigation on the Impact Earth Website into what factors affect the size of craters left by impacts. To do this you need to change one variable and keep all the the other variables constant.  The most interesting one to explore is the angle of impact.  Keep everything else the same and see what happens to the crater size as the angle changes from 10 degrees to 90 degrees.  What angle would you expect to cause the most damage?  Were you correct?  Plot the results as a graph.

If you enjoyed this post you might also like:

Champagne Supernovas and the Birth of the Universe – some amazing photos from space.

Fractals, Mandelbrot and the Koch Snowflake – using maths to model infinite patterns.

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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.

Website Stats


IB Maths Exploration Guide

IB Maths Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes:

  1. Investigation essentials,
  2. Marking criteria guidance,
  3. 70 hand picked interesting topics
  4. Useful websites for use in the exploration,
  5. A student checklist for top marks
  6. Avoiding common student mistakes
  7. A selection of detailed exploration ideas
  8. Advice on using Geogebra, Desmos and Tracker.

Available to download here.

IB Revision Notes

IB Revision Notes

Full revision notes for both SL Analysis (60 pages) and HL Analysis (112 pages).  Beautifully written by an experienced IB Mathematics teacher, and of an exceptionally high quality.  Fully updated for the new syllabus.  A must for all Analysis students!

Available to download here.

IB HL Paper 3 Practice Questions (120 page pdf)

IB HL Paper 3 Practice Questions 

Seventeen full investigation questions – each one designed to last around 1 hour, and totaling around 40 pages and 600 marks worth of content.  There is also a fully typed up mark scheme.  Together this is around 120 pages of content.

Available to download here.

IB Exploration Modelling and Statistics Guide

IB Exploration Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations – focusing in on non-calculator methods in order to show good understanding. Includes:

  1. Pearson’s Product: Height and arm span
  2. How to calculate standard deviation by hand
  3. Binomial investigation: ESP powers
  4. Paired t tests and 2 sample t tests: Reaction times
  5. Chi Squared: Efficiency of vaccines
  6. Spearman’s rank: Taste preference of cola
  7. Linear regression and log linearization.
  8. Quadratic regression and cubic regression.
  9. Exponential and trigonometric regression.

Available to download here.

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