## 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.*

## A Puzzling Heptagon

At somepoint Ed Southall (@solvemymaths) posted this heptagon puzzle:

I saved it in my phone to solve later, and forgot about it until the other day.

At first I looked at it and wasn’t sure where to start. I attacked the problem by sketching what I knew:

The heptagon is split along the base of the red shape into an isosceles trapezium and a pentagon. Because I know the interior angle of a heptagon (900/7) I know enough about this trapezium to work out the other angles (51 3/7).

This means I can deduce the angle BAE (77 1/7):

Because the heptagon is regular I know that AE = BE, and as such ABE = BAE. This means I can use the angle sum of a triangle and the sine rule to calculate each angle and each side of the triangle.

From here I could calculate the area of the triangle ABE either using absin(c)/2 or Heron’s Formula. I chose Heron’s Formula:

I know that triangles ABE and ADE are congruent so the area I’m looking for is double this area subtract the area of the overlap.

I considered the Pentagon above the line AE and how the triangles split it up into 4 triangles and a rhombus. I briefly considered the rhombus:

And quickly realised that using the rhombus properties and opposite angles I had enough information to calculate the area of the overlap:

First calculating the height, then the area:

Then I could find the area I was looking for:

*I thought this was a great puzzle, and it got me thinking about which of my students would be able to attack it. The skills needed are all skills that higher level GCSE students should have, and I think that some of my year ten class would give it a good go, but I also worry that some of my sixth formers may struggle with it. I think by exposing students to these problems early, and by sharing our own thought processes, we can start to build the resilience and mathematical thinking needed to succeed.*

*When I first attempted this problem it was late, and I made some daft errors transfering working from one line to another, and got a very wrong answer, this showed me that it’s easy to do, and we need to make sure we reiterate often the importance of checking our work to our students.*

*There was also the problem of rounding, as we were dealing with angles that didn’t give us nice exact trig ratios I had to round, through my working I rounded where it seemed sensible, but all these errors would have built up so I decided 1dp would be a good limit to round my final answer to, although now I feel nearest whole number would have been better.*

*I’m fairly sure that there is a much better and more concise way of solving this problem, but I currently can’t see it. If you do spot one, or solve it an even more long winded way, please let me know.*

## An interesting discussion

Yesterday I was teaching perimeter of compound shapes with my year 9s. After they had been solving problems based on shapes made up solely from squares and rectangles one if the pupils asked “Sir, will the perimeter always be even?” I thought this was a great point for discussion so I opened it up for the class.

They decided that for the type of shape we were looking at the perimeter would indeed always be even, as you had to cover every distance twice, meaning 2 was always a factor of the perimeter. I was impressed with their reasoning, but a little disappointed that one had to prompt non-integer side lengths. When I did they quickly dismissed this with “of course we knew that sir, but be were only considering whole numbers!”

I then asked them to consider is this applied to all shapes. They quickly concluded that triangles could have odd perimeters with integer side lengths, and circles, then they extrapolated to any shape with an odd number if sides. They concluded quickly that regular polygons with an even number of sides couldn’t have an odd perimeter if it had integer side lengths. Irregular polygons provided a more difficult challenge, and I left them to ponder it over the weekend!

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