It’s not easy finding the time nowadays with so much going on (not least with the new puppy – Sir Isaac Newton – to play with) but it’s great to sit down with a cup of tea and write a blog post. I’m hoping to write another post next week but this post is mainly about some of the shape stuff I’ve been doing this term. Hopefully you’ll find at least one of the ideas interesting and I hope that a few people will be able to add to these ideas or provide new ideas that I can try out next year.

1) A great question when you’re at the pub with friends…

Which is bigger, the length of a pint glass or the circumference of the top of the glass? I bet most people will say the length. Of course, in a maths lesson, I don’t need to use the word “pint” to get my students hooked on a perplexing question. I used my water flask instead…

It turns out in this case that the circumference of the top is slightly longer than the length. Once students have a full understanding of where the circumference formula comes from, they’re amazed to check the results using mathematics (it’s quicker to measure the diameter and calculate over cutting out a piece of string and directly measuring which is where the maths comes in). In future, I might open this up to an investigation into different prisms. E.g. **If the top cross section wasn’t a circle, would this still happen? If so when and why?**

2) Surface Area and Structure: Why this shape at the supermarket?

This is the “separating thingy” that separates people’s shopping at the supermarket. We had a great discussion on why they would use this particular shape over any other object. Why this over a cuboid? Nice intro to minimising surface area whilst maintaining the structural importance of the shape.

3) Ipads, Exterior Angles and Hopscotch – Discovery

After doing some work with interior angles, I thought I’d get the students discovering the integral fact about exterior angles of polygons through the use of the iPad app, Hopscotch. Hopscotch is essentially the new Logo. It is a beginner’s coding app in which you can use “ready-made” commands to make a code.

I made this worksheet to go along with the activity.

4) Understanding Area Conversions and Population Density

In a study conducted by the European Commission in 2011, Europeans were asked what they thought the World’s 10 biggest problems are. Obvious stuff like climate change, the economic crisis and terrorism were up there in the top 5. One of the top 10 was the exponentially increasing population of the human species. Hence, I really like doing stuff with population density because it’s relevant and connects to so many areas of mathematics (namely ratio, proportion, standardised measures, area, converting between metric units of area). To introduce this, I like to start with getting the students to internalise how big 1 m² is. How many students can fit inside 1 m²? With the square against a wall, I was quite amazed that 11 students managed to get in there!

5) Understanding 3-D Shapes and Prisms before Volume

If students don’t know the difference between prisms and other shapes then they’ll simply start memorizing formulae for volumes of different shapes with little understanding. Hence, to start this topic I simply got the 3-D shape sets out and asked them in groups of three to organise the shapes in any way they thought was most logical. After 5 minutes, we had a little class discussion about what they’d done and then I encouraged them to try to categorise the shapes into 3 groups. Low and behold, all but 1 group sorted into Prisms, Pyramids and Spheres. This allowed us to move on to properly define prisms using the applet I made a while ago.

5) Islamic Artwork and Transformations Project – Discovery

I decided to get Year 7/Grade 6 discovering transformations on the co-ordinate grid using the hook of Islamic Artwork this year. Since one of my “secondary objectives” is to get students familiar with Geogebra, it was a great way to get them discovering, using and effectively communicating transformations.

I started with an introductory lesson (type “Islamic Artwork” into google to find some interesting pics) which got them hooked into thinking about the artwork from a mathematical perspective. We talked about rotational symmetry and then I provided a summarised historical perspective using information from Wikipedia:

Islamic Art began around the 7th century. It was designed and made by people who lived within the Islamic territory that was inhabited by or ruled by Islamic populations. It is thus a very difficult art to define because it covers many lands and various peoples over some 1400 years.

There are often repeating elements in Islamic art which is used to symbolize the infinite nature of God. Mistakes in repetitions are sometimes on purpose to show humility by artists who believe only God can produce perfection.

With it being an introductory lesson, I then allowed for the all important “tinker phase” of introducing new tech/software – you certainly cannot expect much structured learning of mathematics during this important phase(although of course it’s massive for discovery)! They had to perform a rotation, enlargement and reflection without any initial class discussion. Obviously, some of them have almost no idea what’s going on at first but they get there eventually. If anyone found out a good way of using the software, they were encouraged to write it down on sugar paper for the whole class to share solutions. The all important part of this project is how you bring in meaningful mathematical learning without students becoming so memorized by the technology and artwork. That’s where effective classroom pedagogy takes over in planning, class discussions and quality questioning. Furthermore, making the students write ever step down as instructions (using correct mathematical terminology) focuses them more on the mathematics rather than just the artwork (60% grade for instructions, 40% artwork marked with Art teacher). The great thing about the discovery element is that students who understand quickly and finish their work can investigate translations, negative enlargements and dilations. Since I don’t have any of the students’ work available electronically, take a look at a previous post on this.

6) What do tessellations have to do with the Salt Flats in Bolivia? Previous post.

So there’s a load of shape stuff. If anyone has any ideas to improve on these or any extra ideas on anything to do with shape, please share.

interesting ideas – I’ll have a look at the hopscotch app. I have to admit that until recently I ploughed through shape and space without much enthusiasm – I didn’t find it an especially engaging topic either for the students or to teach. A couple of months ago I bought a couple of books on non-euclidean geometry however – which has really sparked my interest in the topic. I’m thinking about how to put together some kind of introduction to non-Euclidean geometry accessible for secondary students – it’s a lot more interesting than the textbook maths of straight lines and circle theorems so seems a shame to exclude students from. I wrote a blog piece on the Riemann sphere a while back – and I think that this idea should be just about comprehensible to higher level sets. Equally a discussion about situations when the parallel postulate fail could be interesting…..

Hi Andrew, great comment! I love how you turned it around and commented on some aspects of shape which aren’t part of any standard school curriculum. I teach the MYP with no external assessment but I’m often fearful of branching out too much onto the types of topics you mention. I read your post and I’m glad that I did because I struggled in Geogebra with defining a point on the surface of a sphere and now have some new tools to stab away at that problem – I’ve never thought about mapping a point on the plane to a point on a sphere in that way before (interesting to see an infinite line mapping to a circle! I’m starting to think about how it would work for other 3-D shapes now – cylinder for example? I pretty much always make reference to the angle sum of a triangle on a sphere but very rarely go beyond that. Your comment has certainly given some food for thought!

One of the cool applications on Geogebra is the circular inversion tool – the topic of mapping points to other points is a little difficult for students to understand without visual help. The circular inversion maps a point P to a point P’ according to the formula:

OP.OP’ = r^2. Where r is the radius of a circle.

Circles map to circles, infinite lines through part of the circle go to circles, segments to part circles, and other shapes to all sorts of things! In fact, I might write something on this because I’ve just been playing around with it now. It’s pretty cool. The circular inversion button is the “Reflect Object in Circle” one – and just needs a circle and a point/shape.