Practical Volume Activities

Here's a quick post this evening about a sequence of lessons that I've done with y10 on volume.

I often find that y10 and y11 lessons fall foul of time pressure. There is so much content to cover, in such a limited amount of time, that there doesn't seem to be much room for creativity. In y7, y8 and y9 I'm always keen to make lessons as creative as possible but with the older year groups the prospect of exams always seems to loom large and we don't get to have as much fun.


I usually plan y10 lessons with a colleague of mine who teaches a similar, middle ability group and I always find it really useful to talk to her. Two brains are definitely better than one! When we saw volume on the scheme of work, we both agreed that we'd like to get some practical activities involved, and this is what we came up with.

In the first lesson we gave the students some nets and asked them to work out the surface area. We took the opportunity to recap how to work out the area of simple shapes (rectangles, triangles, circles) and some of them took on the challenge of working out more complex shapes (trapeziums, pentagons, hexagons).

In the next lesson they cut out and made the nets into 3D shapes. We then talked about classifying the 3D shapes and they got into the idea of prisms. Not content with the shapes they had made, the girls wanted to look for prisms in the rest of the room. Surprise surprise, there were quite a few dotted around the place (it's not like I was collecting examples or anything) including a lovely octagonal quality street tin that just happened to be sitting at the front.

It was a really worthwhile lesson, although I'm pretty sure the pace would have been considered too slow by Ofsted. It took a while to cut and stick all the shapes, but once they had made them, the girls clearly understood how all the faces fitted together and they had no problems appreicating that the volume of a prism was cross-sectional area x length.

We had another lesson on calculating the volume of more complex prisms including problems where they were given the volume and hade to work out a missing length or area. Then in today's lesson we looked at the cuboid challenge where you give students a single piece of paper and ask them to make an open-topped box with the greatest possible volume. I think this task is usually done as a nice introduction to calculus, or at least plotting the graph of a cubic equation (nrich, as always, have explanations of how it works if you're not sure http://nrich.maths.org/6399/solution ). But, much as I feel a bit pained to say it, I wasn't bothered about the algebra this time. I wanted students to get a feel for dimensions and to have a go at a practical version of trial and improvement. I also wanted to make a big deal of the second part of the challenge, where I gave them 4 multi-link cubes each and asked them to work out how many whole cubes they could fit in their boxes.

The picture above shows the efforts of the winning team of 4 students, and the photo on the right shows a section of working out in another student's book. I asked the girls to work out the volume of every box made by their team and only once they had finished that were they allowed some cubes so they could work out how many cubes would fit inside. At first they were unimpressed with the quota of 4 cubes each, but they quickly figured out what to do and even the weakest students in the class seemed pretty motivated.

There is much more you could explore with volume, but I feel pleased that even in a limited time we managed to cover a lot of content in a way that didn't feel pressured and allowed the students time to become really familiar with 3D shapes. I'm sure that making the shapes, physically measuring them and turning them round in their hands helped the girls to understand what they were doing far more than simply working from a textbook would have done.

Reader, I married him: Going with the flow on ratios

I overheard some year 11s today talking about one of their earlier lessons and getting quite agitated about it. “It’s just structure, structure, structure in those lessons” said one of them. “I think she plans for every second” said another.

At first I wanted to laugh. I have no idea who they were talking about (and have no desire to know either), but whoever it was sounded like an excellent teacher and I thought the students just didn’t know what was good for them. “Every second planned” they’d said. Crikey. It sounds like the work of a very organised person. Hats off to them.

But then I wondered; why were the girls so agitated about it? They are highly motivated students, the sort that have been turning up for after school revision every week since they were in year 9. They’re not afraid of working hard.

The answer seemed to come in the final comments I heard:

“She never lets us just go with the flow” said a different student. “Yeah,” said the first, “she should just let us get on with it”.

This little conversation links in to an issue that I come back to time and again in my teaching practice. How much independence should we allow/encourage? How much is too much? Do students do better in lessons where “every second is planned” or in lessons when teachers “let us get on with it” a bit more.  An interesting seam, but I think I’ll save mining that one for another day.

Following another thread, it made me think about the lesson I’d just taught. It was year 9 and we were doing ratios. I knew that I’d put together quite a boring lesson. Originally I’d wanted to do something really creative involving the ratio of the string lengths of musical notes, but I hadn’t sorted things out in time and I was stuck with a boring stack of textbook questions. So I started the lesson feeling a bit uninspired, and explained how to simplify ratios into the form 1:n.

The girls were not impressed. What was the point? You can’t get 0.64 of a person. Why not just simplify them in the normal way?

So I said the first thing that came into my head and it went a bit like this:

Well, when I was at university there was a bit of an issue about the ratio of boys compared to girls. My university was split into colleges and some colleges had very different ratios. One college might have had a ratio of 11:10, but another might have been more like 7:3.

They looked interested, so I ploughed on….

So in some colleges, the number of boys and girls was fairly equal, in others there were lots more boys compared to the girls. If you were choosing a college, you might want to know the ratio of boys compared to girls and pick somewhere based on that. It might be difficult to compare the ratios, because you can't tell straight away if 3:7 is better than 16: 21, but if you had them in the ratio 1:n you would always be comparing like with like. Ideally the ratio would be 1:1 or pretty close (I was rambling at this point) although to be honest….. my college wasn’t that equal (really should have stopped to think at this stage)…but I suppose it did have its advantages … I mean I’m marrying one of them so it worked out quite well for me

Cue raucous laughter and bright red face from me.

This definitely wasn’t a case of every second planned, but it certainly livened things up. It took a while for them to calm down and stop pointing out how red I’d gone, but they were clearly interested in the idea. So I ditched the textbook and made up some ratios on the board saying that they were the male/female ratios of different colleges. The students had to simplify them into the form 1: n, then explain which one they would like to go to.

Nearly all of them said the one with the ratio closest to 1:1, so we then ranked the answers in order from most fair to least fair ratio and had a useful discussion about ordering and comparing decimal numbers (is 1:1.6 equally as unfair as 1:0.6? Some of them thought it was at first).

I wish I’d thought of this idea before the lesson. I’ve since been online and looked at the male/female ratios on different courses at Oxford and the results are sadly unsurprising, but they are interesting and hugely relevant to a group of bright young things at an all girls school. The male/female ratio in history is 1:1.02 and in English it’s 1:1.7, but in maths it’s 1:0.44, in engineering it’s 1:0.3 and in physics its 1:0.2.

So definitely an idea worth using again, and despite the momentary embarrassment – I think it shows that it doesn't hurt to “go with the flow" occaisionally.