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Constructions

One topics I have never been a fan of teaching is constructions. I think that this is due to a few factors. Firstly, there is the practical nature of the lesson, you are making sure all students in the class have, essentially, a sharp tool that could be used to stab someone. I remember when I was at school a pair of compasses being used to stab a friend of mines leg and this is something I’m always wary of.

Secondly, the skill of constructing is one that I struggled to master myself. I was terrible at art, to the point where an art teacher kept me back after class in year 8 to ask why I was spoken about in the staffroom as the top of everyone else’s class but was firmly at the bottom of his. I explained that I just couldn’t do it, although it was something I really wished I could do. He was a lovely man and a good teacher and he offered to allow me to stay back every Monday after our lesson and have some one to one sessions. I was keen and did it, this lasted all through year 8 and although my art work never improved my homework grades did, as he now knew I was genuinely trying to get better. I have always assumed the reason I am poor at art is some unknown issue with my hand to eye coordination, and I have always blamed this same unknown reason for struggling sometimes with the technical skills involved in constructions. Since coming into teaching I have worked hard to improve at these skills, and I am certainly a lot lot better than I used to be, but I still feel I have a way to go to improve.

For these reasons I chose to go to Ed Southall’s (@solvemymaths) session “Yes, but constructions” at the recent #mathsconf19. Ed had some good advice about preparation and planning, but most of that was what I would already do:

Ensure you have plenty of paper, enough equipment that is in good working order, a visualiser etc.

Plan plenty of time for students to become fluent with using a pair of compasses before moving on.

He then moved on to showing us some geometric patterns he gets students to construct while becoming familiar with using the equipment. Some of these were ones I’d not considered and he showed us good talking points to pick out and some interesting polygons that arise. The one I liked best looked like this:

This is my attempt at it, I used different coloured bic pens in order to outline some of the shapes under the visualiser.

The lesson was successful, the class can now all use a pair of compasses and we managed to have some great discussions about how we knew that the shapes we had made were regular and other facts about them.

Next week we need to move on to looking at angle bisectors, perpendicular bisectors, equilateral triangles, and the such. I hope to get them constructing circumcircles of triangles, in circles of triangles and circles inscribed by squares etc.

Here are some more of my attempts at construction:

“Constructing an incircle” – I actually did this one in Ed’s session!

“A circumcircle” – I drew the triangle too big and the circle goes off the page. Interesting to note the centre is outside the triangle for this one.

“A circle inscribed within a square” – this is difficult. Constructing a square is difficult and that is only half way there if that. This is the closest I have got so far and two sides are not quite tangent.

“A flower” – nice practice using a pair of compasses and this flower took some bisectors too.

If you have any ideas for cool things I can construct, and that I can get my students to construct, please let me know in the comments or on social media.

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  1. benjamin
    July 12, 2019 at 6:35 pm

    If you need any other anecdotal data: My older son just finished geometry this year and constructions were one of his favorite parts. Also: I ended up doing Clarissa Grandi’s compass based art this year and it went really well (if you need an extension which is mostly for fun) https://mymathclub.blogspot.com/2019/06/63-tesselations.html

    • July 13, 2019 at 11:31 am

      Nice, thanks for the comments and the links. I will try it out.

  2. Gabriela Vulpe
    July 13, 2019 at 10:13 am

    Hi, I am planning to do a couple of lessons with year 7 set 1 on drawing circle patterns. First get used to compasses and practise drawing the diagram that you showed here, the one with the triangles and hexagons.
    Then I will give them copies of 3-4 patterns I found online and ask them to work out how to draw them (ie what radius is needed, where is the centre of each circle).
    Then draw them and colour them in!
    It is the first time I am doing this, hope it will work ☺️

    • July 13, 2019 at 11:32 am

      Sounds like a good plan. Let me know how it goes!

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