## Circles puzzle

Here’s a lovely puzzle I saw on Brilliant.org this week:

It’s a nice little workout. I did it entirely in my head and that is my challenge to you. Do it, go on. Do it now….

Scroll down for my answer….

Have you done it? You better have…..

I looked at this picture and my frat thought was that the blue and gold areas are congruent. Thus the entire picture has an area of 70. There are 4 overlaps, each has an area of 5, so the total area of 5 circles is 90. Leaving each circle having an area of 18.

This is a nice mental work out and I feel it could build proprtional reasoning skills in my students. I am hoping to try it on some next week.

Did you manage the puzzle? Did you do it a different way?

*This post was cross posted to better questions *here*.*

## Catchy Mnemonics

I find most memory aids a little silly. Why learn a rhyme about horses when you can just learn the trig ratios? Why learn a rhyme about the duke of York when you can just remember the order the colours come in?

However, I find that music is a good way of remembering things. For some reason music is good for us to remember words. I can, for instance, remember the words to a great deal of 90s pop songs even though I didn’t like them and never chose to listen to them because I heard them out places and on TV so often that they got lodged in my brain forever.

This is something I have seen work well in learning maths facts. Year on year I hear pupils sing “mean is average, mean is average…” etc in lessons to remember the averages. And I also hear a great many variations on the circle song.

Last year when I was teaching kinematics one of the students said “Sir,play the SUVAT song.” I’d not heard of the SUVAT song and he found it on you tube and we listened to it. It’s simple and it’s catchy and it really helped him and his class remember those equations. So on Tuesday I played it to one of my mechanics classes. By the end of the leson I’d heard three people sing it and it has been stuck in my head all week.

*What do you think about mnemonics? Do they have a place? Have you any songs or rhymes that you use to remember things or that you encourage students to use? And do they help?*

## Uniforms

*This post was originally published on Labour Teachers *here* on 9th September 2016.*

Uniforms are part and parcel of school life for the vast majority of us. They are often quite arbitrary and they differ from school to school. They are something that, for some reason, never stop being discussed.

They can be expensive, I’ve recently seen these costs as a parent for the first time and I understand them. But they aren’t a great deal more than other clothes.

So why is it they have hit the news again?

Well that’s because a school crackdown has caused outrage, as usual. What I imagine has happened is that the school has either brought in a new uniform requirement or, the more likely scenario, the school has decided to ensure that students follow the uniform policy.

It seems like basic common sense to me. If a school has a uniform policy, it should be enforced. If you attend a school with a uniform policy you should follow that policy. If your child attends a school with a uniform policy you should ensure they are following said policy.

It’s strange, I’ve worked in many jobs which have had many different dress codes. Some simple uniforms (a pub branded t shirt) some full uniforms (a branded suit and tie), some strict dress codes (suit and tie) and some more lax (shirt and tie). I’ve never thought to try and get round it.

I have, however, heard every excuse under the sun from students.

Following uniform policies is important. It’s the opening gambit. If you have a uniform policy and don’t enforce it you are saying to the world “our policies mean nothing” and inviting students to break the behaviour policy, the attendance policy etc etc etc.

## Circles and Triangles

Regular readers will know that I love a good puzzle. I love all maths problems, but ones which make me think and get me stuck a bit are by far my favourite. The other day Ed Southall (@solvemymaths) shared this little beauty that did just that:

I thought “Circles and a 3 4 5 triangle – what an awesome puzzle”, I reached for a pen an paper and drew out the puzzle.

I was at a bit of a loss to start with. I did some pythag to work some things out:

Eliminated y and did some algebra:

Wrote out what I knew:

And drew a diagram that didn’t help much:

I then added some additional lines to my original diagram:

Which made me see what I needed to do!

I redrew the important bits (using the knowledge that radii meet tangents at 90 degrees and that the line was 3.2 away from c but the center of the large circle was 2.5 away):

Then considered the left bit first:

Used Pythagoras’s theorem:

Then solved for x:

Then briefly git annoyed at myself because I’d already used x for something else.

I did the same with the other side to find the final radius.

A lovely puzzle using mainly Pythagoras’s theorem, circle theorems and algebra so one that is, in theory at least, accessible to GCSE students.

I hope you enjoyed this one as much as I did!

## A lovely old problem

Recently Ed Southall shared this problem from 1976:

I’m not entirely sure if it is from an A level or and O level paper. It covers topics that currently sit on the A level, but I think calculus was on the O level at some point. *Edit: it’s O level* I saw the question and couldn’t help but have a try at it.

First, I drew the diagram – of course:

I have the coordinates of P, and hence N so I needed to work out the coordinates of Q. To do this I differentiated to get the gradient of a tangent and followed to get the gradient of a tangent at P.

Next I found the equation, and hence the X intercept.

And then, because I’m am idiot, I decided to work out the Y coordinate I already knew and had used!

The word in brackets is duh…..

Now I had all three point.

It was a simple division to find the tangent ratio of the angle.

The next 2 parts were trivial:

And then I misread the question and assumed I’d been asked to find the shaded region (actually part d).

Because I decided calculators were probably not widely available in 1976 I did it without one:

I thought it was quite a lot of complicated simplifying, but then I saw part c and the nice answer it gives:

Which makes the simplifying in part d simpler:

*I thought this was a lovely question and I found it enjoyable to do. It tests a number of skills together and although it is scaffolded it still requires a little bit of thinking. I hope to see some nice big questions like this on the new specification.*

*Edit: The front cover of the paper:*

## Old school Venn

Recently I saw this picture from Ed Southall (@solvemymaths) and thought it interesting:

It is an O level question on Venn Diagrams from 1988. I had a go at it.

The Venn itself was easy enough to fill in and the forming and solving part followed nicely.

As did the rest.

Having gotten used to A level statistics this was relatively straightforward, it manages to test use and knowledge of Venns but doesn’t go as far as probabilities.

I like Venn diagrams and I think questions like this are a good start point to build on, students who can do this will find A level Venns much easier. I assume that this style of question may be what we can expect from the new style GCSE, and even if it’s not its certainly something I intend to use with my classes.

## A new term, and a new school

I’ve started a new job this year at a new school. This is the second time I’ve moved schools and I have to say it has been a much smoother transition than it was the last time.

This school is very close geographically to my last school and as such has a similarge make up of students.

I’ve now got to the point where I can remember most of the names of my students and we are working hard to put some real progress in the classroom.

It’s been a time of change all round really, my daughter started school this year too, and as such I have now become the parent of a school kid. That’s been weird all round but she’s enjoying it and I think we picked a really good school. My wife and I have been invited to attend a meeting there next week when the will tell us how they teach English and maths in reception. I’m interested to see what they say about it, particularly in mathematics!

Also this summer I finished my MA and I’m awaiting results for the dissertation. I think it went OK, but I won’t know until the brown envelope arrives with my feedback and grade. The dissertation was entitled “Investigating problem solving as a means to improving understanding in A level mathematics” – catchy I know. I enjoyed writing it and I may share a summary on here at a later date.

All in all its currently a time of change and that brings with it excitement and challenges.

How has your start to the new year been? How are your new classes? Have you started a new job? I’d love to hear about it in the comments or via social media.## Share this via:

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