Lessons taught; Lessons learnt

I run a weekly maths club for a group of primary children in the area, and part of this involves creating or altering resources so that they can be picked up and understood by ten year olds. The first resource from the club that I'm contributing to this blog is my template and instructions for creating a trihexaflexagon.

While there are no shortage of flexagon sites, descriptions and videos around (and I particularly like the Murderous Maths one, as I've seen Kjartan perform, and he was very entertaining), there does seem to be a gap in the market for a simple blank template, together with a simple one-page instruction guide to putting them together.

Source: Inkscape SVG and PDF.

Source: Inkscape SVG and PDF.

I'm happy to work on producing similar instructions for the more complicated flexagons -- leave a comment if you'd be interested.

Inspired by this excellent post on the 'Point of Inflection' blog, here's an interactive cylinder, which will let students explore the relationship between radius, height, and volume. The post linked gives some great throughts on the benefits of using interactive examples like this in classes.

Incidentally, I've just noticed while creating this post that the new version of Geogebra allows you to embed a simple Geogebra applet completely in HTML, without having to upload a separate .ggb file. A wonderful advance, but it does make it impossible to save the applet to your own computer. I would love to link to the .ggb file here, but the new version of Wordpress seems to have implemented some odd 'security guidelines' for uploads that I need to hunt down and disable!

It’s been a busy term, and it’s not over yet, but I promised that I would post something this week. So, a question for you to explore:

The second term of a geometric sequence is 2. What’s the sum to infinity?

If your first thought is ‘it could be anything’, then you’re not alone, but you’re also not correct!

If you get an answer algebraically, can you give a geometric interpretation?

I am at the Maths Association conference, being held in Robinson College, Cambridge. Over the next few days I’ll be writing reports and reflections on the sessions I attend.

Session title:

Embedding Rich Tasks into the Curriculum

Session speaker:

Charlie Gilderdale

What was meant to happen:

We have been working with a number of schools trialing how problems on the NRICH website can be integrated into the curriculum. Find out how this is being done and see how our mapping documents can help your school. Of course, we will do some problem solving along the way.

What actually happened:

I have been to sessions run by Charlie Gilderdale before, and they have all been great fun. He is one of the driving forces behind NRICH, a wonderful resource of over 5000 maths problems, investigations and activities (many with teacher notes and solutions). They have also recently started to build up documents which map their activities into the UK maths curriculum. This was meant to be the focus of the session, but happily we spent most of the time working on an excellent set of problems which would be a great way to lead into some circle theorems.

Charlie started by showing us the Virtual Geoboard available on the NRICH site. If you’ve never seen a Geoboard in real life, it’s just a board with pins sticking out. You can then use rubber bands to create various shapes. The virtual geoboard lets you do this… virtually.

He then handed out a few pages of grids with 9-pin circular geoboards to everyone, and asked a simple question: What angles can we find, if the only things we are allowed to draw are segments joining two points on the geoboard? In particular, can we find angles of 10 degrees, 20 degrees, all the way up to 360?

This triangle, for example, gives us 40 degree and 70 degree angles:

We were asked to assume that we had primary-level knowledge about angles. So, we weren’t allowed to use any circle theorems, but were assumed to know that the angles in a triangle add to 180, the angles around a point add to 360, that vertically opposite angles are equal, etc.

Over around 15 minutes, we managed to find all the required angles except 150 and 170 degrees — can you fill in these gaps?

We were then asked to draw as many quadrilaterals as possible using these 9-pin Geoboards (without using the central vertex). Here is one example:

This lead to a number of interesting discussions:

• When are two quadrilaterals different, and when are they they same? We settled on saying that two would be the same if we could cut one out and place it exactly over the other (in other words, rotations and reflections don’t make new shapes).
• How can we try to systematically find examples?
• How do we record the examples that we find?
• How we could convince ourselves, and others, that we had found all the quadrilaterals?

Having found these quadrilaterals we were then asked to calculate their angles. Again we only used late-primary maths. We found a number of ways to calculate the angles, and particularly noticed how symmetry helped to make the calculations easier.

Charlie then pointed out that the results obtained from this activity could be used to get students to notice that the opposite angles in a quadrilateral added to 180. Would this happen if we changed the number of dots around the circumference? He handed out some 12- 15- and 18- pin Geoboards, and asked us to check, and then to try and develop a proof.

We soon found a very nice visual proof that the opposite angles added to 180 when the centre of the circle was ‘inside’ the quadrilateral, and were asked to try to develop a similar visual proof for the other case. He noted that this could be a great way to introduce this circle theorem to children, without having to immediate descend into the murky world of algebra, but also without copping out and just getting students to measure angles with protractors.

(Incidentally, there are a variety of other Geoboard-based investigations on the NRICH site. To find them just search for geoboard.)

In the last couple of minutes of the session we moved onto the mapping documents, which can be accessed from this section of the NRICH site. They are mostly useful for teachers in English schools, but could also be useful for teachers outside the UK, to help navigate around the multitude of problems on the site.

In summary, then, a fun hour, working through an investigation that was accessible, enjoyable, and has extensions leading in a number of different directions.

Thoughts:

It is not hyperbole to say that NRICH has developed into one of the best free maths teaching resources on the internet. Every month they present a selection of their problems, arranged around a particular theme, and at a variety of difficulty levels. They encourage students to write in with their solutions, which may get added to the site — several of my students have submitted solutions, although none of them have been featured so far.

The addition of the mapping documents has made it even more useful for those of us in the UK, because we no longer have the danger of spending a very enjoyable couple of hours on the site without actually coming out with anything we can use in a lesson!

I really liked the particular activity that was introduced in this session, and am looking forward to trying it out with the Primary Maths Club that I’m running this year.

As promised, here is an animated Geogebra visualisation. Click ‘Start’, and six moving lines will appear.

• What behaviours do the lines exhibit?
• What relationships are there between the different lines?
• How are these lines defined?

Geogebra applet (enable Java to see it).

(Source: movinglines.ggb and movinglines.html.)

Click on ‘Show Values’ and three values will appear — all six of these lines are generated from these three values. If you click on ‘Stop’, and move the sliders, then the lines will be automatically updated to reflect those new values.

• Can you make any conjectures now about how the lines are defined?
• What values would we need to make one (or more) pair of the lines
  • perpendicular?
  • parallel?
  • vertical?
  • horizontal?
• Can we generate any line we wish by setting the values appropriately?

I managed to captivate both a lower school class and an A-level class with this visualisation for quite a while last year, and the A-level class made good progress in making and justifying conjectures about what was going on.

Technical Note

One of the few downsides of Geogebra compared to other dynamic geometry software, is that it contains no in-built facility for animation. Thus, to create any animations using Geogebra, you have to create the animations manually in Javascript. It’s not particularly difficult to do, particularly if you know some Javascript, as you can see if you view the source of the .html file above. Putting the ability to animate inside Geogebra itself is, though, my one big wish for future versions.