On a full page in the back of your exercise book draw a perfectly regular pentagon.
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"I like the construction task. Mark Richer, Brockworth Enterprise School, Gloc.
"Following my comment of 9 Dec where I refuted the Origami method of producing a regular pentagon from a sheet of A4 paper, and stated that I did not believe one could be constructed without a protractor, I was delighted to view on your web-page a graphical demonstration of a non-protractor method which showed how a pentagon could be constructed. Mark Richer, Brockworth Enterprise Scool
"Are comments from teachers on the 'Starter of the Day' website vetted? Mr Mark Richer, Brockworth Enterprise School
"Mark, thanks so much for your comments. We are just about to leave for the Christmas holidays but when we return will remove the word 'perfect' from the origami film. We will happily take down any of your previous comments if you request us too but they do make an interesting read!" Transum,
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However, the origami method does not produce a regular pentagon, but one which is very close.
The angle formed by the first two folds is in fact 109.47 degrees and not 108. This can be established through the fact that the ratio of sides of A4 paper (or A3, A2 etc) is always 1 to root 2, and similar triangles. Hence the tangent of half of the internal angle is root 2.
I became suspicious through understanding that it is impossible to construct a regular pentagon without a protractor."
Sunday, December 09, 2007
My instinct tells me that this is not possible. I have checked out the method shown, using traditional Pythagorean and Trigonometrical methods, using a standard calculator giving results to 8 or 9 d.p., I found lengths and angles to be within 40 parts per million of the theoretical values, which is well within the bounds of rounding errors in the complex series of calculator processes.
This leaves me in two minds.
1. Do I eat ‘humble pie’ and accept that my assumption was wrong? No angle-relationships or other mathematical methods known to me can prove that this construction method is valid.
2. Just like the Origami method which I know is flawed, could it be that some clever person has come up with another contrived construction method which ‘produces the goods’ in practice, but is flawed in theory?
"
Tuesday, December 11, 2007
Is there a mechanism by which mathematical statements which are clearly wrong can be deleted so that both pupils and teachers are not misguided?
I of course refer to the suggestion that a regular pentagon can be constructed.
I am concerned that for as long as this page shows an incorrect method (or even two), we are not informing learners, but arming them with invalid mathematical ideas. "
Friday, December 14, 2007
Saturday, December 15, 2007
Excellent, I would like to see more like this
Good, achieved the results I required
Satisfactory
Didn't really capture the interest of the students
Not for me! I wouldn't use this type of activity.
This starter has scored a mean of 3.7 out of 5 based on 23 votes.
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The internal angles of a regular pentagon are 108o and the external angles are 72o. The following diagram shows a way to constrict a pentagon without a protractor:
Did you know that you can fold a regular pentagon from a sheet of A4 size paper?