On a full page in the back of your exercise book draw a perfectly regular pentagon.


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A Mathematics Lesson Starter Of The Day

Topics: Starter | Construction | Geometry | Shape

  • Mark Richer, Brockworth Enterprise School, Gloc.
  • I like the construction task.
    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.
  • Mark Richer, Brockworth Enterprise Scool
  • 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.
    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?
  • Mr Mark Richer, Brockworth Enterprise School
  • Are comments from teachers on the 'Starter of the Day' website vetted?
    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.
  • Transum,
  • 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 to but they do make an interesting read!

How did you use this starter? Can you suggest how teachers could present or develop this resource? Do you have any comments? It is always useful to receive feedback and helps make this free resource even more useful for Maths teachers anywhere in the world.
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Note to teacher: This activity could be used to revise finding the size of angles in regular polygons followed by a pencil, ruler and protractor construction. Pupils might be surprised to learn that there is also a nice origami solution!


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Laptops In Lessons

Teacher, do your students have access to computers such as tablets, iPads or Laptops?  This page was really designed for projection on a whiteboard but if you really want the students to have access to it here is a concise URL for a version of this page without the comments:

However it would be better to assign one of the student interactive activities below.

Laptops In Lessons

It is quite easy to draw a perfect pentagon using the repeat command in Logo. Follow the link below for an online version.

It is also possible to fold a kite from an A4 sheet of paper. Follow the link below for more information.

Student Activity


Curriculum Reference

See the National Curriculum page for links to related online activities and resources.


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