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!

I'm delighted to have been asked to take part in Emily Grosvenor's blog tour for her new children's book, 'Tessalation!'.  Emily (@emilygrosvenor) has always had a particular interest in tessellations and was instrumental in organising the first World Tessellation Day on 17th June this year (a wonderful celebration; check out the hashtag #WorldTessellationDay).  Her book, funded via a successful Kickstarter campaign, describes the adventures of girl named Tessa, who enjoys hiding herself in the tessellating patterns she finds in the world around her.  The illustrations by Maima Widya Adiputri are exquisite, and the book provides the perfect way to introduce young children to the mathematics of pattern-making by tapping into their inherent curiosity about the natural world.  

Emily's work got me thinking about the many ways we can use the appealing aesthetics of tessellations to engage the students in our mathematics classrooms.  The book itself contains clear step-by-step instructions for making tessellating tiles or 'tesserae' from a starting square.  These instructions would be suitable for use with students of all ages (see the central image below) and the method is explained in more detail in this post on the Kids Math Teacher blog.

Many other bloggers on the tour have contributed their ideas for exploring tessellations with children.  For secondary students, Brent Yorgey's post presents a clear introduction to the geometrical reasoning behind the mathematics of tessellations.  And the post by John Golden (@mathhombre) contains links to his comprehensive resources page covering many aspects of this rich topic.  ​

One of the lesser known ways of working with tessellations is through the medium of paper folding (a favourite pursuit of mine) and I have come across two great resources to support teachers in working with this technique in the classroom.  

The first is this fascinating publication from Liz Meenan, which explains how to create various Islamic-influenced tiling patterns (see below) from squares, equilateral triangles, kites and hexagons folded from A-sized paper.  Liz has kindly allowed me to use her instructions on this presentation and handout.

The second is this fantastic resource from mathematical origamist David Mitchell, found on his website origamiheaven.com.  The whole website is worth exploring, especially if you are interested in mathematical or modular origami.  The instructions for folding origami tiles from 'silver rectangles' (A-sized paper is a good approximation) are of particularly high quality, and the easiest ones to fold would be accessible to students from early primary upwards.  For example, it would never have crossed my mind that the trapezia created from a simple, single fold would tessellate in such a variety of ways!
​

David's detailed instructions also discuss the angles, geometry and symmetries of the various tiles, along with investigations of area and perimeter (including work with surds), which make this activity suitable for students across the full age range.  Truly worth a look.   

I do hope you've been inspired to explore some of these tessellation resources in your classroom.  As always, please do share your experiences on Twitter, or in the comments below.  

If you'd like a copy of Emily's book, click here.  And you can find an electronic copy here.