When we think of paper snowflakes, we usually imagine the 'kirigami' type that involve first folding a sheet of paper into sixths, and then cutting bits out to create a papercut snowflake exhibiting perfect six-fold symmetry. (If you fancy trying some, there are some beautiful templates from First Palette here and some fun Star Wars and Harry Potter themed templates by Anthony Herrera, here). However, you can also make beautiful origami snowflakes from a single sheet of folded paper and no cutting, and they are wonderfully mathematical!

I have been planning an origami snowflake window display as part of my Christmas decorations this year, and this weekend got down to some folding. I chose to use tracing paper so that the internal structure and symmetry of the snowflakes would be visible when held up to the light - any translucent paper will do, including baking parchment or tissue paper.

In doing my research I came across several different styles, all starting from a paper hexagon. This is where the maths begins! The best method of cutting a paper hexagon for this type of snowflake is to start from an A4 (or letter size) rectangular sheet, as all the fold lines that remain form useful pre-creases for your final model.  The method itself is ingenious, and would make a great little geometric proof challenge for your students. Here is a video of me performing the method. Why does it produce a perfect hexagon?

Now to fold your snowflakes! Here are links to the instructions for the three most effective models that I found online. I have also included photos of the interim stages of the folding of my first attempts.

Diagram by Dennis Walker: http://www.oriwiki.com/origamidennis/diagrams/oridiag.htm

Video by Sara Adams, @happyfolding: https://www.youtube.com/watch?v=7m72m8L0xuA

Video by Riccardo Foschi: https://www.youtube.com/watch?v=x6UjDVLSqOk

Video by @senbazurueurope (in French, but clearly demonstrated): https://www.youtube.com/watch?v=EYMxVAlnnS0

I hope you enjoy folding these wonderful designs as much as I have. The layers upon layers of symmetry are incredibly pleasing to create, and there are lots of 'ooh' and 'aah' moments to enjoy as the cleverness of each design reveals itself. 

Let it snow! ❄️❄️❄️

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 :)

This weekend I taught two origami workshops on the theme of Christmas decorations. I thought I'd share the links to the projects here so that you can try them too.  In each case I have provided a link to a diagram or set of photo instructions, as well as a link to an instructional Youtube video.  The videos are really useful if you are new to interpreting origami diagrams.  I have numbered the folds in order of difficulty.

A big thank you to all the origamists and bloggers who have shared their ideas and designs for others to enjoy.

PS Maths fans will particularly appreciate the clever set of folds which produce a regular pentagon from a starting square and a single cut (video number 3 below).

1.        Christmas Wreath:
Diagram: http://www.homemade-gifts-made-easy.com/homemade-christmas-decorations.html
Video: https://www.youtube.com/watch?v=5eRvQndCJz8
 
2.        Lucky Stars (from 1:20 to 1:30 strips):
Diagram: http://www.origami-fun.com/support-files/origami-lucky-star-print.pdf
Video: https://www.youtube.com/watch?v=u4Woj9oP3AI
 
3.        5-Pointed Star (from a pentagon; see video):
Diagram: http://www.origami-resource-center.com/support-files/5pointedstar.pdf
Video: http://www.homemade-gifts-made-easy.com/5-pointed-origami-star.html
 
 4.        Froebel Stars and Wreaths (from approx. 1:25 strips):
Diagram: http://www.origami-resource-center.com/german-stars.html
Video: https://www.youtube.com/watch?v=h5NplkRIwcw
 
5.        Modular Gift Star (‘dollar star’):
Diagram: http://mvg-ori.nl/ori-l/models/ornament/moneystar.pdf
Video: https://www.youtube.com/watch?v=cj3dgRJ3mUs

1.        Simple Christmas Tree (with a Lucky Star on top):
Diagram: http://www.origami-fun.com/support-files/origami-pine-tree-print.pdf
 
2.        Fluted diamond:
Diagram: http://www.origami-resource-center.com/fluted-diamond.html
Video: https://www.youtube.com/watch?v=v-3U_vu_4JY
 
3.        Stellated Octahedron by Sam Cuilla:
Diagram: http://britton.disted.camosun.bc.ca/stellated_octahedron.pdf
Video: https://www.youtube.com/watch?v=dwUW5QOY_ZU
 
4.        Gyroscope by Lewis Simon:
Diagram: http://web.archive.org/web/20070319203345/http://www.fabricorigami.com/misc/pdffiles/misc55.pdf
Video: https://www.youtube.com/watch?v=OFelvhmFANs
 
5.        Christmas Cracker Gift Box:
Instructions and Video: http://www.paperkawaii.com/2014/12/07/origami-christmas-cracker-video-tutorial/

Happy 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!

This year I have started an exciting new role as STEM Faculty Lead Practitioner and KS5 teacher of maths at a stand-alone sixth form campus.  This has meant a whole new classroom to decorate (and how I love a display challenge!).  However, it being an A Level only classroom, I have had to completely rethink what displays might be suitable, and ditch some old favourites - alas, 'squares and cubes' is no more!

Over the summer I kept an eye out on Twitter for ideas and resources, and spent a couple of enjoyable afternoons in the last week of the holiday putting it all together.  With a big thank you to those who share their wonderful resources and ideas for free, here are some photos of the end result:

I thought it might be useful to provide links to all the displays that I collated from Twitter.   I'll take it wall by wall, from left to right...

The side wall:

The Greek Alphabet display is by Sarah Carter (@mathequalslove) and is one of a host of fantastic display resources on her website.  I think it adds a note of historical interest and background to some of the new mathematical symbols my students will encounter over the course of their sixth form studies.

The A Level Notation Periodic Table is a lovely twist on the well-known original by Mel Muldowney (@Just_Maths) and is by Clare Mazurkiewicz (@MrsMazzy).  It's an absolute must in an A Level classroom!  I've referred to it so many times already, and love the fact that it whets the appetite of those students new to A Level: look at all that exciting stuff ahead!  I used pre-cut display lettering to make the title underneath.

The 'Wall of Death' is a set of those 'Everytime you do this: ... an ... dies' (insert horrific-yet-alarmingly-common misconception and cute animal of choice) posters that you will have come across before.  Helpfully, @MathCurmudgeon has collated the full set on his blog, and, I think, added a few of his own.  Unfortunately I seem to be referring to these posters rather a lot at the moment (those poor, poor kittens!)!

Finally, above the whiteboard there's my version of the 'How to Learn Maths' flow chart which is available for download on my Classroom Display Ideas page.

The back wall:

This wall is dominated by large whiteboards, a resource that's new to me, and which I'm really enjoying seeing the students use.  Great for on the spot assessment, discussion and feedback.  However, there's still space above and below them for some bright displays...

Above the whiteboards I've put up my quotes from famous mathematicians and thinkers, which are available for download on my Classroom Display Ideas page.

Along the bottom I've used the wonderfully creative 'Maths Mr. Men' from Ed Southall (@solvemymaths), a poster set that's clever, fun and informative - ideal for an A Level classroom.  There are two lots of posters available, downloadable from Ed's excellent website.

The window wall:

I haven't had a classroom with a view for a while, so I decided I needed to frame my beautiful new view with some maths bunting.  Most of the bunting out there is GCSE-focused (see the links on my display page) but this lovely bunting from K Pitchford (@Ms_Kmp) is perfect and lends a note of cultural and artistic interest.

And the windowsill provides the perfect surface on which to display all my modular origami (in an attempt to entice more students along to my club!)  If you are interested in finding out more about this highly geometric form of origami, see my earlier blog post on the subject.

The front, teaching wall:

On the far left is my version of @mathequalslove's Growth Mindset display, available for download on my Classroom Display Ideas page.  I felt the message of this display was just as relevant at A Level, and have already referred to it a few times this term.

In the centre, at the top, I've put up my old 'mistakes quotes', also still relevant and also available for download on my Classroom Display Ideas page.

Underneath these there's a number line.  I ummed and ahhed at first about whether or not to put this up; it felt a bit too 'lower school'.  But then I thought back to how often I'd referred to my number line in sixth form lessons previously, and realised it was still important for teaching at this level.  I use it when talking about limits to infinity, domains of functions, the modulus function, solution sets for inequalities and so on. 

On the right of the IWB is a giant graphics calculator poster which was already in the classroom when I arrived.  Billy Adamson, (@Billyads_47) my head of department, and major 'GC' fan, had these printed for the team a while ago.  If you're interested in where he got these done, do contact him on Twitter.  I have to confess I've never mastered my GC, but will be teaching Stats 1 for the first time this year, so it'll soon be time to face my fears!

Above the right hand whiteboard I've put the 1-9 number facts poster set by Ed Southall. These are very intriguing.  For example, why is the number 1 'narcissistic'?  Why on earth is 5 'hungry'?  And why is 7 'odious'?!

With all the great display resources out there, there's no reason at all why A Level maths classrooms can't be bright, colourful and informative learning environments.  Feedback from my students has been really positive.

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.

Several people have asked for tips about getting started with geometric modular origami, so I'm hoping this post will be useful.  A quick caveat before I begin: I'm no expert, just an enthusiastic amateur with a great deal still to learn!  However, the wonderful thing about this form of origami is that beginners can produce fabulous pieces right from the start, as many complex models are built from relatively simple individual units. 

But first, what is modular origami and why fold it?  Essentially, modular origami is the use of two or more sheets of paper to create complex models, built up from several units, or modules.  Construction involves inserting the flaps of one module into the pocket of another.  If identical modules are used, the resulting flat or 3D shape contains repeating pattern and symmetry, and can hence be described as 'geometric' in form.  This is the kind of modular origami that I most enjoy folding.  As I mention on my Origami Gallery page, I find that the process of folding dozens of the same modular unit, over and over again, induces a meditative, flow-like state. This is then followed by the substantial, and often frustrating, challenge of constructing the given model.  And, as all maths lovers know, there's not much that's more satisfying than cracking a difficult puzzle!  Better still, you end up with a beautiful geometric form to enjoy.

So, a few starting points.  

Most origami instructions come in diagrammatic form, so you will need to be familiar with the system of symbols used.  These are relatively self-explanatory, and quick to pick up.  You can, of course, learn to fold modular origami models from the wealth of Youtube videos shared online, but I personally find that folding from original diagrams feels more authentic, and ultimately more rewarding.  Interpreting them is part of the challenge!  However, video instructions certainly have their uses, and often prove an invaluable aid to add some visual clarity when encountering particularly tricky folds. 

You'll also need some paper! ​ I have to confess that I usually fold from normal 80gsm coloured copier paper; it's cheap and readily available.  I do treat myself to 'proper' origami paper sometimes, though nothing special.  But to be honest, I find that modular pieces seem to hold together better with copier paper as it has a slightly grainier surface and is therefore a little less slippery.  Experts would probably be horrified!

Saying that, I have just purchased this bargain 500 sheet pack of coloured origami paper (on special offer at Amazon at the moment).  And recently I found two packs of the most gorgeous rainbow geometric paper (above), which I've yet to use: Rainbow Patterns paper and Geometric Origami paper.  

I also often use packs of craft or scrap-booking paper which can be picked up quite cheaply at shops like The Works, The Range or the Craft Superstore.  These come in some lovely designs, but as the paper is a bit thicker, and 'frays' a bit more easily under repeated folding, they aren't so well suited to particularly intricate or small-scale models.  ​

In a nutshell, though, pretty much anything goes!  More information about different types of paper can be found at Robert J Lang's blog, here.

It is also useful, although certainly not essential, to have access to a small guillotine or paper trimmer, as well as a small ruler, set square or even an old credit card to use as a 'folding bone'.  This will help achieve a good, crisp fold, especially important when folding modular units which will need to interlock firmly afterwards.  Accuracy is important: small inaccuracies in the individual units can cause problems during the assembly of the final model.

Next, you'll need a project to get started on!  A really useful starting point is this page from the Origami Resource Centre, which has links to instructions for many modular designs, handily organised (roughly speaking) by increasing complexity, and hence difficulty.  When you are feeling more confident, Meenakshi Mukerji's website has another page of useful links here.

​Another rich vein of modular origami ideas to tap is the wonderful world of Kusudama: Japanese flower or 'medicine balls'.  There are many traditional models to try, but there are also several new designers from across the world, whose designs seem to become ever more intricate.  Ekaterina Lukasheva (@kusudamame) and Maria Sinayskaya (@MariaSinayskaya) are two such origamists whose beautiful work is well worth exploring (see the list below). Ekaterina has recently published a new book with step-by-step instructions for making her designs: Modular Origami Kaleidoscope.

And finally, for reference and inspiration, here are the details of some modular origami A-listers (this is by no means intended as a definitive list):

• Tomoko Fuse:  the renowned master of modular origami and author of several books of designs
• Robert Neale:  alongside Mitsubo Sonobe, accredited with reinventing modular origami in the 1960s
• Thomas Hull: mathematical origamist; author of Project Origami: Activities for Exploring Mathematics
• David Mitchell: mathematical modular origamist and author of several books on the subject
• Meenakshi Mukerji: modular origamist and author of several books of designs
• Rona Gurkewitz and Bennett Arnstein: modular origamists and co-authors of several books of designs
• Maria Sinayskaya: kusudama and geometric modular origamist; see photos of her work here
• Ekaterina Lukasheva: kusudama and geometric modular origamist; see photos of her work here
• Krystyna Burczyk: mathematical origamist, specialising in 'twirligami'; see photos of her work here
• Francesco Guarnieri: geometric single sheet and modular origami; see photos of his work here
• Carmen Sprung: elegant geometric origami including stars and flowers; see photos of her work here
• Nick Robinson: @nickorigami, author of two Geometric Origami Kits
• Francesco Mancini: see photos of his work here​

There, that should be more than enough to get you started!  If ​you have any questions do contact me here or in the comments below.  And happy folding :)