## 1 December 2014

### 5 Maths Gems #16

It's December! Time for students and teachers to start getting overly excited about the upcoming Christmas holidays. There's still three more weeks of teaching to get through though, and most schools do mock GCSEs in December or January, so things are as busy as ever. Let's look at some of the ideas that have been shared on Twitter this week. I've got lots of nice geometry for you today.

1. Frozen Fractals
I'm not a fan of Disney princess films, but I have daughters now so I've been forced to watch Frozen many many times. I actually quite like it. Being a mathematician, my favourite part is when Elsa sings about 'frozen fractals' in the song Let it Go. I had no idea what the word fractal meant until a few years ago so it seems odd to hear it in a children's film. It got me thinking - is there any value in teaching our students about fractals at secondary school? I think there is.

It's impossible to cover absolutely every branch of mathematics at school, but it's worth spending an hour on fractals - simply because they're interesting. You could start by explaining what fractals are. If you're not sure yourself, then I suggest you read What are Fractals and why should I care? by George Dallas. This is an incredibly clear and accessible article. The best way to understand fractals is to draw one - the article describes how to draw a Von Koch curve.

Give your students some dotted or triangle graph paper and they could follow the instructions in this worksheet to draw a 'Koch snowflake'. While they're creating their fractals, show them this lovely microscopic timelapse video of real snowflakes (thanks to @misseyfs for sharing this).

Because I'm writing this post on 1st December I'm thinking of Christmassy snowflakes, but we can look at other fractals too. The Sierpinski Triangle is also a nice introduction to understanding fractals. The awesome website fractalfoundation.org has an excellent worksheet (a 'fractivity') in which each of your students make their own Sierpinski Triangle - they then stick them together to make a 'class fractal'. Show the video on the page Fractal Trianglethon to wow your students with giant Sierpinski Triangles!
Alternatively your students could create Fractal Cutouts like the one pictured below.
And this Mathematical Christmas Tree is as fractally as they come.

If you want to further explore fractals, your students might be interested in this Numberphile video 'Measuring Coastline'.

Whether you make snowflakes, triangles or cutouts, the point is that students come away from the lesson understanding what a fractal is. It may not be on the syllabus but it's interesting mathematics nonetheless. And your students will be able to tell their friends that they understand the mathematical reference in Frozen.

2. Angles
I love this angles question from @j_lanier. I'd use it in a GCSE revision lesson. Students have to apply a number of rules to find all four angles - it's a good opportunity for them to practise articulating their reasoning.

3. Pythagorean Triples
I was looking at recent posts on my favourite blog, the ever-brilliant Median by Don Steward, when I noticed a link to a blog I'd not seen before - Pythagorean Triple Tasks by Hannah Jones. There's some lovely activities here.
By the way, speaking of Don Steward, did you know he also has a blog for practice resources? These are as well designed as his rich tasks. There's loads of good questions for GCSE revision, which you might find useful this week if your Year 11s have mocks coming up. I particularly like these grids of 'practice makes perfect' questions. Also, I spotted this fantastic problem in his A & A* questions - a lovely starter for my Year 12s:
4. Semi-circle
Did you see Ed Southall's awesome semi-circle problem? I'd use this as an extension activity in a Pythagoras lesson, but there's loads of methods for solving it - others have used similar triangles, trigonometry and coordinate grids. It's both accessible and challenging for students from Key Stage 3 up to A level.
5. Christmassy Maths
I started to write a seasonal maths resources page but lots of other lovely bloggers have done something similar. So rather than reinvent the wheel, I'll point you in the direction of:
There are hundreds of excellent Christmas-themed maths resources. I'd like to recommend a few in addition to those in the posts above. There are two from Teachit Maths that I particularly like - the 12 days of Christmas budget activity which involves problem solving using percentages, and Mistletoe and Lines which is good practice of straight line graphs. I also like these resources from Oxford University Press. Most Christmas resources are aimed at Key Stage 3 students, but your Year 11s might like this Christmas Shading Graphical Inequalities by frickard on TES. And how about this lovely Christmas tree logic problem?
 Logic problem from ageofpuzzles.com

That's it for this week. Please don't forget to keep tweeting or emailing me photos of misconceptions or common mistakes in your students' work. Thanks!