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If you are a teacher then please also visit my new site: intermathematics.com for over 2000+ pdf pages of resources for teaching IB maths!

Modelling tides: What is the effect of a full moon?

Let’s have a look at the effect of the moon on the tides in Phuket.  The Phuket tide table above shows the height of the tide (meters) on given days in March, with the hours along the top.  So if we choose March 1st (full moon) we get the following graph:

Phuket tide at full moon:

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If I use the standard sine regression on Desmos I get the following:

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This doesn’t look a very useful graph – but the R squared value is very close to one – so what’s gone wrong?  Well, Desmos has done what we asked it to do – found a sine curve that goes through the points, it’s just that it’s chosen a b value of close to 120 – meaning that the curve has a very small period.  So to prevent Desmos doing this, we need to fix the period first.   If we are in radians the we use the formula period = 2pi / b.  Therefore looking at the original graph we can see that this period is around 12.  Therefore we have:

period = 2pi/b

12 = 2pi/b

b = 2pi/12 or pi/6.

Plotting this new graph gives something that looks a lot nicer:

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Phuket tide at new moon:

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Analysis:

Both graphs show a very close fit to the original data – though both under-value the tide at 2300.  We can see that the full moon has indeed had an effect on the amplitude of the sine curves – with the amplitude of 1.21m for the full moon and only 1.03m for the new moon.

Further study:

We could then see if this relationship holds throughout the year – is there a general formula to explain the moons effect on the amplitude?  We could also see how we have to modify the sine wave to capture the tidal height over an entire week or month.  Can we capture it with a single equation (perhaps a damped sine wave?) or is it only possible as a piecewise function?  We could also use some calculus to find the maximum and minimum points.

There is a very nice pdf which goes into more detail on the maths behind modeling tides here.  There we go – a nice simple investigation which can be expanded in a number of directions.

Essential resources for IB students:

1) Revision Village

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Revision Village has been put together to help IB students with topic revision both for during the course and for the end of Year 12 school exams and Year 13 final exams.  I would strongly recommend students use this as a resource during the course (not just for final revision in Y13!) There are specific resources for HL and SL students for both Analysis and Applications.

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There is a comprehensive Questionbank takes you to a breakdown of each main subject area (e.g. Algebra, Calculus etc) and then provides a large bank 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 a large number of ready made quizzes, exams and predicted papers.   These all have worked solutions and allow you to focus on specific topics or start general revision.  This also has some excellent challenging questions for those students aiming for 6s and 7s.

Essential Resources for IB Teachers

1) Intermathematics.com

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If you are a teacher then please also visit my new site.  This has been designed specifically for teachers of mathematics at international schools.  The content now includes over 2000 pages of pdf content for the entire SL and HL Analysis syllabus and also the SL Applications syllabus.  Some of the content includes:

  1. Original pdf worksheets (with full worked solutions) designed to cover all the syllabus topics.  These make great homework sheets or in class worksheets – and are each designed to last between 40 minutes and 1 hour.
  2. Original Paper 3 investigations (with full worked solutions) to develop investigative techniques and support both the exploration and the Paper 3 examination.
  3. Over 150 pages of Coursework Guides to introduce students to the essentials behind getting an excellent mark on their exploration coursework.
  4. A large number of enrichment activities such as treasure hunts, quizzes, investigations, Desmos explorations, Python coding and more – to engage IB learners in the course.

There is also a lot more.  I think this could save teachers 200+ hours of preparation time in delivering an IB maths course – so it should be well worth exploring!

Essential Resources for both IB teachers and IB students

1) Exploration Guides and Paper 3 Resources

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I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission.  Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator!  I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams.  The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.