Lessons taught; Lessons learnt

(Update: This post is featured in the 39th Carnival of Mathematics. Check it out!)

Last summer I developed a bit of an obsession with hexagons. More specifically, with the patterns you can produce from tiling multiple copies of a single, simply decorated hexagonal tile:

I cut out and laminated around 100 of these tiles, and spent several happy hours moving them around into new patterns, trying different types of symmetry, and experimenting with different layouts.

If you’d like to follow me down a similar path, here’s a sheet with six of these hexagon tiles to print and cut out: hexagon-tiles-122.pdf (svg source), made using the free vector drawing program Inkscape.

Background

Although I wasn’t aware of precursors at the time, I later found this shape in several places: it is one of the tiling generators you can buy from the ATM, and it’s one of the tile shapes in the game of Tantrix. I recommend you browse the ATM’s store if you are a maths teacher — many excellent things await!

I also recall reading a very interesting web page which found this exact pattern being used as a paving slab in Spain, and developed quite an in depth mathematical discussion, but sadly this page now seems to have disappeared into the æther.

Although less common than they used to be, hexagons were often used for tiling and paving. I particularly like this hexagon paving tile designed by Gaudi, and used all over Barcelona.

Returning to the particular hexagon above, we get the following when we tile it:

The fun comes when you rotate some of the hexagons. Even with just six patterned hexagons you can find a range of different interesting patterns… but it’s much more fun to explore with a large pile of them!

In My Classroom

At the start of last year, the tiles appeared on my classroom walls in the place now occupied by the Pascal’s Triangle wall display. As with Pascal’s Triangle, the tiling wasn’t static, but slowly changed form and shape over time, and several interesting questions were raised by some of my groups (several of which couldn’t believe at first that all the different patterns were generated by a single type of tile).

Eventually the hexagons migrated over to the other side of the classroom, and took the form revealed in the very first post I made on this site:

Read more for: more examples of patterns you can produce from these tiles; some questions for you to explore; my thoughts on the mathematical content of this ‘pattern space’; and source files for all the diagrams.

I suggest you pause here, print out the hexagons, and have a play before continuing.

Some Patterns

Here are eight patterns made with the hexagon tile, on a 6-by-6 grid:

(click on the images for a better view). These by no means exhaust the patterns you can generate with this tile.

Questions Raised by the Patterns

I believe that these tilings form a rich and non-trivial environment for pattern spotting, and the generation of mathematical questions and conjectures in the classroom. Here, for example, are some questions which occurred to me as I was exploring the ‘pattern space’:

• Several of these patterns have rotational symmetry of order three. What other rotational or reflectional symmetries are possible? What about if we allow ‘holes’ (i.e. we don’t have to place hexagons in every grid position)?
• The last of the 6-by-6 patterns demonstrates glide reflection. There are seven possible frieze patterns; can we generate examples of all of them?
• Various motifs recur in different patterns: a small circle using three hexagons; an oval using four hexagons; several braids; a large circle and a trefoil using six hexagons. What other closed loops can we make using only a small number of tiles? Also, are there any forbidden lengths, with no examples of loops of that size?
• Fix a small grid size (for example, a 2-by-3 rectangle). How many distinct patterns can we make?
  The meaning of ‘distinct’ here is something that would need to be negotiated with the students. If we take a pattern and rotate or reflect it, should we count that as a new pattern?

I think all these questions could be generated, explored, and altered quite productively by students at a range of levels. They also give students an opportunity to demonstrate several key reasoning skills, such as systematic exhaustion: how can we be sure that we’ve found all the possible patterns of a given grid size, or loops of a given length?

Complexity

In some sense, all the questions in the previous section were combinatorial. We had to count the number of ways that something happened, or generate examples.

Here’s a deceptively simple question which leads into an investigation of another sort:

Which of the patterns above is the most complex?

We might perhaps say that simplest patterns are those which we can describe in the shortest amount of space. The very first pattern, for example, could be described by saying ‘all the tiles are this way up’. How could we describe how to generate some of the other patterns?

Given this view of complexity, what do complex patterns look like?

What do simple patterns look like?

This seemingly naive and simple view of complexity, when formalised, leads to algorithmic information theory, and in particular to the concept of Kolmogorov complexity. This also has connections to an immense body of work in computer science, including compression algorithms.

Transformations

Moving away from complexity, let’s now consider what happens when we start with a pattern, and want to alter (transform) it in some way.

It is obvious that we can transform one pattern into any other pattern by rotating each tile in turn — but what happens when we impose constraints on the ‘moves’ we are allowed to make? For example, suppose we had a rectangular grid of tiles, and were only allowed to rotate the tiles a row/column at a time.

Starting from the basic pattern we saw right at the start:

we could rotate the second ‘column’ one step anticlockwise:

and then the third row:

and then the fourth column:

and then the second row:

Could we generate every possible pattern this way?

This is an interesting question, but probably a little difficult to approach at the secondary school level. So, let’s be even more restrictive:

Suppose we rotate all the tiles at the same time. What happens?

To be more specific, let’s start with this pattern:

What do we get if we rotate all the tiles one ’step’ clockwise?

Now is an excellent time to print out some hexagons and find out!

What about if we rotate again? And again?

What happens with different starting patterns?

What is preserved by this transformation, and what is not? If one pattern repeats every three hexagons for example, then the other one will as well. But what about symmetry? What about the ‘loops’ formed by the lines?

Notice that we are exploring, in an accessible way, several advanced concepts: applying a transformation to something; preservation of features; iteration of transformations among others.

Where’s The Maths?

Usually, in school mathematics, we only consider functions which transform numbers into other numbers — even transformations such as rotation, reflection and enlargement are almost never talked about as functions which can be combined, or reversed, or iterated. This naturally makes it harder for students to ’see the maths’ in situations which don’t directly involve numbers.

One way to help students to become comfortable with these ‘non-traditional’ areas could be to improve the emphasis, throughout their school careers, on key concepts and questions, like iteration (what happens if we do something many times?), inversion (how do we undo what we just did?) and iso- & homo-morphism (what has changed, and what has stayed the same?). These are applicable at all levels of mathematics:

What happens if we add the same number lots of times?

How do I undo a multiplication?

What properties of my triangle stay the same when I enlarge it?

What happens when I differentiate a polynomial lots of times?

How do I undo exponentiation?

What properties of number stay the same if we allow the square root of -1 to be a number, and which have to change?

The same types of question recur throughout the mathematical development of a pupil.

Conclusion

Even something as simple as a hexagonal tile can lead to some really deep mathematical questions, and thinking about them can develop skills which are applicable in many more traditional areas of mathematics.

It also lets you use the pun in the title — and we don’t get the opportunity for puns often in maths!

[As promised, here are the source files for many of the above diagrams.]

“What are those numbers on the wall, sir?”

“An interesting question. What do you think they are?”

This isn’t a single taught lesson, but something which impacted many of my lessons for at least a week. It’s the story of one of the first pieces of decoration I put up in my new classroom. A wall-display… to be precise, this one:

(as with most images on this site, click for full size)

Background

Pascal’s Triangle has an enormous number of properties, and can be the starting point for a large number of investigations. I think it should be mandatory for all maths classrooms to feature it in a prominent place!

The Story of the Wall-Display

The above image isn’t how the display started, though. When I first put the (individually cut out and laminated) numbers on my wall, there were no questions, no header, and several rows missing. All the questions which subsequently appeared were ones which had been asked by students, mainly during the ‘dead time’ at the start of lessons, while waiting for a critical mass to arrive.

Before the questions arrived on the board, I gave the numbers their heading, but only after every group I teach had had the chance to see the numbers ‘naked’. These groups vary both in age and ability, but the initial questions asked by the students were all very much the same (and notice that I say ‘asked by the students’ rather than ‘asked of the students’).

Most of the sets figured out the generating relationship very quickly, but the colours took a little extra time, particularly as the focus of the lessons being taught that week was never on the display itself. Several of the students I discussed it with quickly saw that the white numbers were those divisible by three, but were unsure what the other colours meant. For several, it was their first exposure to the ideas which will lead into modular arithmetic, which is not actually in any UK GCSE or A-level syllabus.

Although it took a significant amount of time to cut out and laminate each number individually, I feel it was worth it for the increase in flexibility you get: you can easily remove several numbers, or a row, and ask students to put them back; you can remove a diagonal column, and lay it out somewhere else for inspection, so that students can see (for example) the trianglular numbers without getting distracted by the rest of the triangle; you can slowly build up the triangle over several days, so that the display isn’t seen as something static and fixed from the very start.

The display remained on my wall for the rest of the year for several reasons — one being that the numbers are actually useful in several different contexts. In statistics, for example, they give the number of ways of choosing k elements from an n-element set, and in pure mathematics they form the coefficients needed in the Binomial theorem. It also looks pretty!

The Quest for Ownership

By the time this display arrived, I had had the following poster on my walls for several weeks:

As you can see, it contains essentially the same information as my wall display, although in a slightly less colourful format. It actually goes much further, revealing a large number of patterns in the Triangle which my students never guessed at. Despite this, not a single student demonstrated any interest in Pascal’s Triangle until my display appeared… and I certainly didn’t get the impression that anyone in any of my sets had read anything on this poster.

To me, this emphasises the importance of getting students engaged in the material by feeling that they are making a contribution to something dynamic and developing. The static, over detailed poster with all the information already there ellicited no interest, while the simple, changing, colourful display (which didn’t intially have a title imposted on it) was very successful at engaging a wide variety of pupils.

Future Development

Next year, I would love to use this wall display as the basis of a mathematical investigation with one of my groups. It might lead to combinatorics, or to fractals (via the Serpinski gasket), or to the Stirling numbers, or to elementary number theory… I need to spend some time over the summer thinking about how best to structure the activity.

I also feel there is a need to put this in some historical context, both European (who was Pascal, and what else did he do?) and international: Indian commentaries on poems, Chinese diagrams with rod numerals, etc. This historical context is something that I will be spending much more time on with all the areas I teach next year (expect a post on this in the future!).

Sources

Here are the files I used to create the wall display:

• Pascal’s Triangle PDF [Zipped SVG]
• List of Questions on Pascal’s Triangle PDF [SVG]

Both of these were originally created using the excellent free vector drawing program Inkscape, which I plan to write about in the future. Inkscape’s native format is .SVG, and I have uploaded in this format for anyone wishing to edit the files.

Through the magic of embedded media, you can also browse them: