When I left my first teaching school (the one I’m back working at again now) I decided to make a piece of art as a leaving gift for my colleagues in the maths team. I’d seen an image on Pinterest of a mural made from different coloured paper tetrahedra, and wanted to try it for myself. However, being a maths teacher, I was intrigued by the possibilities of tessellating smaller tetrahedra within the gaps left by bigger tetrahedra....

Now I know that A-sized paper has some pretty cool properties, so, using the photocopier I reduced an A4 tetrahedron net down through the various A-sizes - and, yes, I made some lovely discoveries (which made perfect sense once I started thinking about it)!

Each iteration of A-sized paper decreases in area by half, (so the side lengths decrease by a factor of 1/(sqrt2)): A5 is half the area of A4, A6 half the area of A5, and so on. So the base of an A6 tetrahedron is a quarter the area of an A4 base, meaning that four A6 tetrahedron bases can be packaged in the gap left by an A4 tetrahedron. I played around and found some other combinations (see some examples below).

I printed and cut out all my nets, assembled my different sized tetrahedra (step-by-step instructions to follow below) and played around with lots of different arrangements. When finally happy, I stuck them down onto a ready-made canvas.

A week ago my ex-colleague Ali got in touch because she wanted to try the activity with her class. I talked her through the instructions and her students went on to produce the wonderful artwork below :)

Here are the step-by-step instructions if you would like to try this activity with your students.

I've just finished a five week stint running our school's Y7 and Y8 STEM Club - and what fun it's been!  I chose the theme of 'fractallations' - a fractal-tessellation hybrid that would allow me to include lots of my favourite things, including, you guessed it, some paper folding  :)

We started by exploring what characterises a 'fractal' pattern (in visual terms) and then investigated and drew Sierpinski Triangles using this excellent resource from the STEM Learning centre. Then, in week 2, we looked at tessellations, and learnt how to fold paper polygons with which to create some tessellations of our own (below).  For this I used my 'Patchwork Paper Patterns' presentation containing Liz Meenan's instructions - available on the Mathematical Art Lessons page. 

With the basics now in hand, we could start folding Christmas fractals...

Inspiration for this activity struck whilst I was attending the recent ATM/MA 'Problem Solving with Paper Folding' course delivered by Fran Watson of @nrichmaths.  We were introduced to a very clever method for folding an equilateral triangle from A4 paper that I'd not come across before.  I'd been wanting to make Koch Snowflakes with the students, and now I had my medium! 

However, things got very small, very quickly.  Next time I might suggest they start with a much larger central triangle tessellated from four, or even nine, A4-sized ones, and make GIANT fractal snowflakes!

Reflecting on this activity afterwards it struck me that it had potential to form the basis of a length and area scale factors investigation.  In addition, older students could be set the challenge of calculating the (finite) overall area of the snowflake.

No mathematical Christmas can be considered complete without Matt Parker's (@standupmaths) ThinkMaths' Fractal Christmas tree!  This fab resource, available here, is a great test of teamwork and problem solving skills, and I was so impressed with how the students tackled it.  Deciding to call themselves 'Christmas engineers', they organised themselves into teams, set up production lines and got completely stuck into the challenge. No pics of their tree yet, but this is one I made with students last year.

We didn't have time to make these during STEM Club in the end, so I'm planning to make them with my Y12 further mathematicians after their end of term assessment instead (they are a very spoilt bunch).  Full instructions and resources are available on the Fractal Foundation website here.  Alternatively Emma Morgan (@em0rgan) has put together this useful walk-through video.

Her students' cards look fantastic!

Happy festive fractal folding :)

I've come across a few mathsy-arty Halloween-themed ideas over the last week or two, and include here three that would make lovely classroom activities.

An Origami Bat

Geometric Halloween masks

These geometric paper masks by @Wintercroft (below), available from www.wintercroft.com, have often caught my eye on Twitter, and when I found out that there was a sixth form Halloween Fancy Dress Day coming up on the 31st, I finally had an excuse to try them out!  I treated my Y12 Further Maths group to a mask-making session after their end of term assessment.  No photos of their handiwork yet, as students have taken them home to finish - I'm very much looking forward to seeing their end results on Monday.

These masks are not strictly-speaking origami, but are instead constructed from printable 'nets' or templates.  Although the masks are sold for personal use only, Wintercroft are keen to promote craft in educational settings, and will allow the printing of class sets for one-off projects, so long as the digital files are not saved or stored on shared computers, or otherwise shared or distributed.

The masks can be made from cereal boxes or other recycled card, but I printed the templates directly onto white 160 gsm card, which was just about sturdy enough.  We used the slightly more time-consuming, but more robust, tabbed assembly method and stuck the pieces together using a glue stick.  I made the bat half-mask (above, top left) and found it very straight-forward to assemble.  I shall spray paint it black, and swoop around campus on Monday :)

The templates come with full instructions, but there is also a video guide.  And the maths?  As well as encouraging patience, perseverance and attention to detail, this would be a good activity to link to polygons, nets and polyhedra - how many different polygonal faces can your students spot and name?  And what about the challenge of designing their own geometric mask?!

Tessellated Halloween skulls

Chris Watson (@tessellationART), a digital tessellation artist, has produced a wonderful short video that shows the design process behind one of his recent pieces, a skulls triptych inspired by Mexican 'Day of the Dead' imagery.  As well as incorporating tessellations, the skulls also include a repeating fractal element where the skull appears again in the eye socket, and again, disappearing off into infinity.  This is a really nice resource for stimulating class discussion on lots of levels.

Do feedback in the comments below, or on Twitter, if you try any of the activities.  I'd love to see photos of your students' work.  And Happy Halloween!