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This is an example of how an investigation into area optimisation could progress.  The problem is this:

A farmer has 40m of fencing.  What is the maximum area he can enclose?

Case 1:  The rectangle:

Reflection – the rectangle turns out to be a square, with sides 10m by 10m.  Therefore the area enclosed is 100 metres squared.

Case 2:  The circle:

Reflection:  The area enclosed is greater than that of the square – this time we have around 127 metres squared enclosed.

Case 3: The isosceles triangle:

Reflection – our isosceles triangle turns out to be an equilateral triangle, and it only encloses an area of around 77 metres squared.

Case 4, the n sided regular polygon

Reflection:  Given that we found the cases for a 3 sided and 4 sided shape gave us the regular shapes, it made sense to look for the n-sided regular polygon case.  If we try to plot the graph of the area against n we can see that for n ≥3 the graph has no maximum but gets gets closer to an asymptote.  By looking at the limit of this area (using Wolfram Alpha) as n gets large we can see that the limiting case is the circle. This makes sense as regular polygons become closer to circles the more sides they have.

Proof of the limit using L’Hospital’s Rule

  

Here we can prove that the limit is indeed 400/pi by using L’Hospital’s rule.  We have to use it twice and also use a trig identity for sin(2x) – but pleasingly it agrees with Wolfram Alpha.

So, a simple example of how an investigation can develop – from a simple case, getting progressively more complex and finishing with some HL Calculus Option mathematics.

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.

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Exploration Guide

A comprehensive 63 page pdf guide to help you get excellent marks on your maths investigation. Includes advice on how to choose a topic, over 70 hand-picked topic ideas, detailed marking criteria guidance on how to maximize your marks, common students mistakes and how to avoid them, in-depth topic examples, great technology advice to enhance your exploration, some examples of beautiful maths and much more! [Will be emailed within the same day as ordered].

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Modelling and Statistics Guide

A 60 page pdf guide full of advice to help with modelling and statistics explorations. Contains an in-depth Pearson's Product investigation, and investigations using Chi Squared, Spearman's rank, t-tests, binomial and more. Also includes detailed regression techniques (linear, quadratic, cubic, trigonometric, exponential, linearization using log scales. [Will be emailed within the same day as ordered].

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Online IB Tuition

Online IB Tuition

Need help with IB tuition? Spires Tutoring are the world leaders in highly qualified IB tuition across all IB subjects. They only accept 4% of tutors who apply to teach.  This means you only get genuinely top-quality IB teachers.

Revision Village

Revision Village

Revision Village is the best IB maths online resource out there for students to prepare for their exams.  They have questionbanks, practice exams, video tutorials and much more!

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