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Geometric Construction

Discussion in 'Mathematics' started by Returning, Jan 28, 2011.

  1. I have a moderately bright group of year 9 students who usually pick up any new topics fairly comfortably. All fine until we got to geometric construction (perpendicular and angle bisectors, perpendicular to and at a point). Somehow, they just did not grasp this.

    I need to come back to the topic and try again but I cannot work out what they don't understand. Any ideas. What makes this topic harder than everything else we've looked at?
  2. I have a moderately bright group of year 9 students who usually pick up any new topics fairly comfortably. All fine until we got to geometric construction (perpendicular and angle bisectors, perpendicular to and at a point). Somehow, they just did not grasp this.

    I need to come back to the topic and try again but I cannot work out what they don't understand. Any ideas. What makes this topic harder than everything else we've looked at?
  3. PaulDG

    PaulDG Occasional commenter

    I think the hardest thing for them to understand is that rulers (for measuring) & protractor aren't allowed - they don't understand why there's that restriction so it all feels like a silly false task to them.
  4. I used to spend some time talking about rhombuses. Marvellous shape the rhombus, because the diagonals bisect the four angles and each other at right angles. So really, if you can construct a rhombus then you're laughing. All you need to worry about is where it goes - which can be tricky. Then (and this is probably obvious, sorry) deal with the constructions one at a time, starting with either one of the bisectors before moving on to the other, then deal with the 'at a point' ones. It might drag a bit doing the same construction several times but its all too easy to get them muddled. You can make the exercises slightly more interesting by throwing in some loci for each one. Then move on to mixed exercises and loci requiring combinations of constructions.
    Hope this is of some use

  6. David Getling

    David Getling Senior commenter

    Some (a lot?) teachers simply tell students the rules and the algorithms. Are you sure you have explained carefully exactly why each construction works?
  7. googolplex

    googolplex Occasional commenter

    Have you tried approaching it from the other end, using something like geogebra to show what constructions can do: circumcircles, etc.?
  8. bombaysapphire

    bombaysapphire Star commenter

    I always spend a bit of time making sure that they understand the vocabulary. Do they know what perpendicular means? Do they know what bisect means? Can they draw a pair of perpendicular lines on a grid?
  9. Do they realise how precise this all has to be? Like the difference between root2 and 1.41 , one is the pure exact answer and one is roughly correct.
    Same for construction. There is an exact way to do it, and a rough way to do it. Mathematicians are interested in the exact way to doit. And maybe mention how no one has found a trisection?

    Or is it that they aren't good at construction? I like to start with curve stictchingto see how coordinated they are, and build up a sense of being precise. Then constructing a hexagon (and maybe curvestitching inside that if they are happy). Sometimes they have just never used a compass before and couldn't draw a decent circle if they had to.
  10. David Getling

    David Getling Senior commenter

    Please NO!!!

    Maybe mention that it has been proven that, in general, trisection is impossible. The world already has far too many would-be trisectors and cube doublers.
  11. Nazard

    Nazard New commenter

    I prefer pipipi's way of doing it! If pupils believe that perhaps it will be possible for them to achieve the first trisection and then spend ages exploring different constructions for themselves, this sounds brilliant. They will do so much maths. If they come up with what they think is a solution then it will be great for them to explore with their teacher why it is flawed, learning mathematical skills on the way, or if they fail to find a solution they can then be directed to a proof of why it is impossible. What a great learning opportunity!
  12. I love this topic! In pairs,on an A3 sheet of paper,
    1) get the pupils to construct different sized equilateral triangles.
    2) get them to construct a 60 degree angle on its own
    3) show them how to construct a 90 degree angle (using perpendicular bisector)
    4) get the pupils to do some on their own
    5) show them how to bisect a 90 degree angle to give a 45 degree angle
    6) ask them how they might construct angles of 120 (60 plus 60) or 135 (90 +45) degrees.
    7) let them experiment to see what angles they can come up with
    Whilst the better ones are being very creative, you can help the ones who are struggling with the basics.
    If any pair can come up with a unique angle, award a small prize/ merit/ certificate, if you want to.
    By doing it on A3 paper, the students won't worry as much about making a mess in their books.
    It can be a very 'buzzy' lesson/series of lessons.
    I would leave dropping a perpendicular from, or to, a point until they are confident with the above.

  13. I approach the topic in a similar way to salsamaths and also use a webcam supported by a stand from science so I can show the construction on the IWB using ruler and compass.

    The terminology can be a problem for some pupils and so we do some work on that as well.

  14. David Getling

    David Getling Senior commenter

    Do you really think introducing Field Theory to young students is realistic? I assume you do realize that this is what's needed to prove the impossibility of trisection.
  15. i just asked my y9 son - no, he hasn't had this explained - thanks for mentioning it - congruent triangles lesson coming up tomorrow afternoon [​IMG]
  16. Nazard

    Nazard New commenter

    Nice idea!
    What I meant was that a "proof" that pupils come up with for themselves will clearly have a flaw and the exploration of this flaw can throw up some interesting mathematics. Giving an 'appeal to authority' sort of explanation at the very beginning ("mathematicians have proved it doesn't work, so don't bother exploring it for yourself") would deny the pupils this opportunity.
  17. David Getling

    David Getling Senior commenter

    Fair point. But I wonder whether there would be enough time to do a decent job of it. From what I've seen, very little time is usually allocated for geometric construction.
  18. which is a real shame - i do it as a gat morning with y5/6's - part of a 'maths and art' series and it's very popular - regardless of the maths behind it -particularly with bright under-achievers
    if it were approached with enthusiasm and 'here's somthing a bit different and good fun' it could be a high point of most y9's

  19. David Getling

    David Getling Senior commenter

    Good point Flora. I've got a little book on the design of Islamic patterns, and I think there are a few books on this subject. So there's an opportunity to draw together several strands of the curriculum: maths, art and social studies.
  20. Thanks for the many good ideas. I did try to reply yesterday but my reply never went up for some reason; technical glitch on the phone maybe. Anyway.....
    To answer some of the earlier points. Yes, I did make sure that they understood the meaning of perpendicular, bisector etc; that bit was fine. We also spent some time discussing accuracy etc. We talked about why the constructions work etc. But the shutters came down when we started on the practical and from then it just went from bad to worse.
    I had been given information that left me thinking that I should get through perpendicular and angle bisector and some loci in a one hour lesson. In retrospect, I realise this was possibly "cloud cuckoo land" and think I just went to fast. How long would others spend on this? - some of the exploration ideas sound great, but time consuming!
    I am going to revisit the topic next week but take twice as long and spend at least some time learning to draw circles with a compass! Oddly the same group grasped the significance of 0 in sig figs in about 10 mins (ie diff between 4, 4.0 etc). Though whether they will remember it next lesson is anyone's guessed

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