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I am searching for ideas to construct Project Based Learning type that involves Modular arithmetics with eventually geometry and could fit in High School level

What could a nice research question for a project that could explored which requires the use of modular arithmetics?

Any suggestion is welcome.

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Rotations of regular polygons.

Start with an equilateral triangle. Label the vertices one, two, three. Define a rotation to be 60° counterclockwise, for example (or you could go clockwise). Note that the locations of the vertices have shifted. Consider this a new “state“ or “configuration”.

Now, what “configuration” does one end up with after n rotations?

Then, extend this to any n-sided polygon. A real life application would be modeling the movement of molecules. Gallian’s abstract algebra book shows a cross-section of a 28-sided HIV virus, for example. Or, how about rendering graphics in video games? Modeling movements of satellites and space shuttles in outer space?

A more accessible example would be a clock, or days of the week.

Incidentally, you could then use mod 12 or mod 7 to discuss notes on musical scales.

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  • $\begingroup$ Thx. Very interesting ideas. I can see how it could be applied to rotation of figures as you explained. Same also with clock. the 28-sided HIV virus, graphic in video games, and movement of satellites => I have been checking NASA website, and google with little success. I ll keep searching. $\endgroup$
    – gegu
    Apr 1, 2018 at 3:21
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I think the idea of check digits is pretty compelling. A bunch of examples of where check digits are used can be found here, including things like government ID numbers in various countries, UPC and ISBN codes, credit cards, etc. The basic idea is that you're given a number, say an ISBN number on a book. You'd like to know if the number you've been given is a valid ISBN number. What you do is calculate a certain (weighted) sum of all the numbers in the ISBN except the check digit (the last digit in the ISBN number, it takes on 11 values from 0 to X) and reduce the total mod 11. If the result is not equal to the check digit, then you were not given a valid ISBN number. If they're equal it might be inconclusive. What's more, in many cases, the procedure can also identify what kind of error occurred and where.

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