Saturday, November 7, 2015

[S1_20151107PAXA] Angle between two Aussie Fifty Cent Coins

Question

Introduction
This question appeared as Question 7 of this year's Australian VCE Further Mathematics Examination.  It has caused an uproar among some students who sat for the examination, hurling abuse at “50 cent”.  For a moment I was wondering: why should the eponymous American rapper face the music for an exam question he never set?  Hmmm...
Anyway, in Singapore, this type of problem belongs to just the plain old Lower Secondary Mathematics syllabus for the topic of angles and polygons, usually taught in secondary 1 (equivalent to about grade 7 or 8).  There is no “additional” or “further” mathematics in our lower secondary curriculum.  Students who do well can opt to take Additional Mathematics.  In the Integrated Programme schools, they may call their mathematics curricula by different names.
I would say this question would be of intermediate difficulty level in Singapore.  It is not so straightforward, but there is a quick solution, given the right insight.  How can we obtain the right insights?  This can be done by making observations and splitting a problem into smaller problems!  [Heuristics H04, H10, H11]

Important Principles
Let us do some recap.  A polygon is any closed shape consisting of a number (at least three) of straight edges.  An exterior angle of a polygon is obtained by producing (extending) an edge in one direction – it is the angle between this extended edge and the next nearby edge.  It is a fact that
 the sum of all exterior angles of any polygon is always 360°
This can be proven in various ways, but I think the best way to see this intuitively is to imagine that you are an ant on the polygon.  Starting from any vertex, go along the edges and every time you walk on a new edge, you turn by an amount equivalent to the exterior angle.  By the time you have come back round to your starting point, you would have turned a total of  360°[H01]
By its construction, the exterior angle and its interior angle always add up to 180°.  For a regular polygon  with  n  sides, all its sides (edges) are equal and all its interior angles are equal and so all its exterior angles are also equal.  Then it is immediately obvious that
 each exterior angle of a regular n-sided polygon is 360° ¸ n

Solution
We construct a vertical line segment in the middle and focus on, say, the right half of the angle (shown highlighted in magenta).  [H09, draw an “imaginary” line]
This half-angle is in fact an exterior angle of the 50 cent coin (a 12-sided regular polygon) and so it measures 360° ¸ 12 = 30°.  Thus the value of  q  is double that, i.e.  60°[H09]  Done!

Remarks
If you have three such 50-cent coins, you can actually put them together and you see a triangular hole in the centre.  [H01]  This is actually an equilateral triangle and each angle would be 180° ¸ 3 = 60°.   I am sure even a ten-year-old kid can do this!

H01. Act it out
H02. Use a diagram / model
H04. Look for pattern(s)
H05. Work backwards
H09. Restate the problem in another way
H11. Solve part of the problem

Suitable Levels
Singapore Lower Secondary Mathematics (Sec 1 » Grade 7/8)
Victorian Certificate of Education Further Mathematics (Australia)
* other syllabuses that involve polygons and angles
* whoever is interested, even 10 year-old kids