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Order of Rotational Symmetry

Discussion in 'Mathematics' started by Mac the Maths, Oct 30, 2003.

  1. Is the minimum order of rotational symmetry 2?

    Surely shapes with only the trivial case have order 1?

    Eg (by my understanding)

    F - has order one (or no rotational symmetry as some books would have it)

    A non-square rectangle - Order two, we count both the 180 rotation position and the trivial case or orignial position.

    What do other people teach? No rotational symmetry = order one or do you say minimum order of rotational symmetry is 2 and below that there is none.

    Anybody wishing to lay down geometrical law out there or is it one of the eternal moot points?

    Thanks,
    Mac.
     
  2. Is the minimum order of rotational symmetry 2?

    Surely shapes with only the trivial case have order 1?

    Eg (by my understanding)

    F - has order one (or no rotational symmetry as some books would have it)

    A non-square rectangle - Order two, we count both the 180 rotation position and the trivial case or orignial position.

    What do other people teach? No rotational symmetry = order one or do you say minimum order of rotational symmetry is 2 and below that there is none.

    Anybody wishing to lay down geometrical law out there or is it one of the eternal moot points?

    Thanks,
    Mac.
     
  3. I teach 'no order of symmetry' for eg. 'F' and 'order 2' for 'H'.

    I reason that F cannot be rotated to any other position so has no.... but H can and needs 2 moves to return to its original position hence order 2.
     
  4. I didn't see what they were getting at either when I first had to teach it. I assumed that you couldn't have Order 1 -glad that I might've guessed right ( if I haven't there'd be in the region of 1600 kids taught incorrectly : ) )
     
  5. If the minimum angle of rotational symmetry is 360/n, then the order of rotation is n. n=1 is quite acceptable.
     
  6. That's the way I've always taught it.

    'No rotational symmetry' is the same as 'order 1'
     
  7. i teach it by saying how many times does the shape achieve it's original 'look' in a 360 degree turn. That way you get order 1 for the trivial case.
     
  8. pencho

    pencho New commenter

    F has rotational symmetry order 1, surely. I remember on teaching practice a teaching telling me otherwise though.
     
  9. The order of a group is the number of elements it contains.
    Using the example of 'F', the rotational symmetry group of 'F' contains only 'e', the identity (so it is the trival group) thus the rotational symmetry group of 'F' has order 1.
    This is what I was taught when doing Groups and Geometry (yawn). Hope it helps.
     
  10. Thanks. It does help me because that is what I was taught in Y1 Algebra and Geometry but so many people (and books) say that you have Order 2 or nothing I was starting to doubt myself.

    Any gurus of the NNS out there? Should we teach them to say no rotational symmetry, order two, order three, ...,

    and then tell them that it is order one, order two, order three, ....,

    so that they get full marks on the KS3/KS4 paper or should I carry on teaching that EVERYTHING has at least Order 1?

    I love it when maths isn't always black and white - it really annoys the RE teachers!

    Thanks for the replies,
    Mac.
     
  11. Did anyone read this weeks TES Scotland?
    There is an article on p3 titled 'Failed by 'Victorian lessons''. Comments from the former head of the inspectorate Douglas Osler, "Many young people are turned off not by learning but by what they are asked to learn. They can't square what they do in school with the world they see and are often right."

    Is rotational symmetry useful for young people in the real world?

    I believe that 'no order of rotational symmetry' and 'rotational symmetry of order 1' are both acceptable and correct but what makes more sense to a 10 year old?
     
  12. I would imagine that Order One makes most sense to a ten year old because if one counts the 360 in order two why then wouldn't you if it was the only rotation?

    I did not read the article in TES Scotland - in fact most of the time I avoid reading the TES, I find much of it quite depressing - but I do not think that one has to teach "supermarket maths" (i.e. where the real-world relevance is immediate to the children) to make maths lessons engaging, fun, enjoyable, interactive or non-Victorian.

    Is your arguement that we should teach only basic arithmetic and very basic algebra? The point of a comprehensive (or at least a broad curriculum) is that children have a wide range of experiences and a firm foundation from which THEY can choose what to do in adult life.

    The joy of much mathematics is inherant in itself not necessarily in its application (although this too can be fun and rewarding).

    Hope you all have a good second half of this term,
    Mac.

     
  13. "..... arguement ...."

    Definitely not but certainly basic english. (Sorry!)

    "supermarket maths" - definitely not either but while we are shopping, are you advocating a one shoe fits all maths curriculum?
     
  14. Argument - sorry. I don't mind people pointing out typing/spelling errors - it's not one I usually make but if it was one I often got wrong I'd thank you for pointing it out.

    With reference to your second point, I would not advocate teaching all of the school mathematics curriculum to all children but I am keen that students get an appropriately wide and deep diet of maths up to a certain age; I would think that at ten one still deserves the chance to be considered a late bloomer and should, therefore, experience as much maths as they can cope with (and that time allows for).

    Should we teach transformations? Well, that's a debate for which I neither have the time or energy.

    Mac - who already feels like there wasn't a half term!
     
  15. frustum

    frustum Established commenter

    I may not often write down the order of rotational symmetry of a shape in everyday life, but I certainly use the concept, and it's difficult to teach the concept without the language.
     
  16. How very interesting.

    Firstly I was taught that the order of rotational symmetry is n where n is defined from the smallest angle (360/n) that a shape can be rotated through to fit exactly back upon itself.

    By this definition shapes with no rotational symmetry have order 1.
    I went through university studying maths without this ever being mentioned again.

    It never came up again until I taught A- levels in a sixth form college. By happenstance, one year I had a most obnoxious group of students who complained about everything and nothing. Believing they had caught me out they duly complained when I taught them no rotational symmetry is order 1.

    I had an almost hysterical head of dept (also head of faculty) screaming at me about this. I simply said, "Really sorry I always thought it was one". We duly checked the syllabus. According to the syllabus no rotational symmetry was to be taught order zero.

    I understood the logic of this argument to be that it has no symmetry therefore it should be designated order zero. As a graduate maths and computing I am familiar with the idea of 2 state definitions from both factorial definition and recursive programming with requires a ground resolvent.

    So nowadays I teach the children that it is zero but I still personally think that its not.

    I notice a lot of angst in the messages on this board. People seem to get quite up tight about getting ?right?. Underlying this is the notion that there must be ONE right answer. In fact I?ll start a new thread about it.
     
  17. I am an English teacher who has to teach Maths and Science. I have always used the concept of order zero because that was what I was taught!

    I can see from this thread that there isn't a definitive answer to this but us there concensus among exam boards? When I spend that brief period every year glossing over rotational symmetry, what would be the safest bet from the point of GCSE exams?
     
  18. well margesimpson the safest bet is teach what it states on your syllabus.

    If it doesn't clearly state which, you should contact the exam board and get written confirmation.
     
  19. I've never seen a textbook which says "order 0" for a shape with no rotational symmetry, but I have seen "none", which might be interpreted as the same.

    Zero wouldn't seem to make sense, because then what would order 1 mean? If you had to draw shapes with orders of rotational symmetry 0, 1, 2 and 3 I can't see what you'd draw for 1.

    So 1 must be the minimum.
     
  20. According to the New South Wales Mathematics Syllabus,

    '... if an object only matches itself once, it is NOT considered to have rotational symmetry.'

    Does this mean order 1 doesn't exist for NSW?
     

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