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.

Egg Box Update!

Note - it is best to have this link playing in the background while reading this post. Open it in another tab and enjoy. http://www.youtube.com/watch?v=2wdWGtTQ7hA


The first set of egg boxes are complete and I have photos!

I'm going to reflect on the project as a whole in another post, but for the time being I thought I'd assume the role of artistic director of the gallery and give you a guided tour. I've put in a selection of designs from top and middle sets. I haven't got a picture of any work from the lower set groups, but I have seen a couple of really nicely decorated cubes and cuboids from them.

(N.B. My tongue is now firmly in my cheek)




1. Untitled
On the left here we have what is considered to be the pre-eminent piece in the collection from two of our gifted and talented artists. The talented duo who created it declined to name the work, but at the gallery we have chosen to refer to it as "Possibly the head of a panda princess?"  A great deal of skill went into the production of this piece and the artists should be commended for their ingenuity in creating the spherical shape. They were also highly commended for their excellent research skills which enabled them to look up the formulas for the surface area of each section.

 

2. Bright Yellow Rabbit with Oversized Teeth and Cavernous Eyes

This is the work of another one of our most distinguished artistic duos. The work encourages us to reflect on the contrast between the jaunty yellow colour that dominates the head of the rabbit and the deep black colour used for the creature's eyes. Again, this pair of artists were highly successful in calculating the surface area of each section of the work, not forgetting the ears.

3. Chateau de l'Oeuf (unfinished) 

This ambitious work was sadly unfinished as the artists had to run to their next lesson, but one can get a sense for the scope of their design by this partially completed exhibit. This exemplifies a trend in some of the middle set groups towards creating "buildings" for their eggs. Judges were particularly impressed by their use of the formula 3/4 x pi x radius squared, for the cone shaped turrets, which were made using 3/4 of a circle.

4. The Eggcitement of Easter
This egg-box design, showing a rabbit in open-mouthed delight as it is about to consume a chocolate egg, encapsulates the joy of the Easter season. Although they had never been taught how to calculate the correct dimensions when designing the net of the cylinder, this group should be particularly appluaded for their "can-do" attitude and problem solving approach. When discussing how to work out the length of the rectangular part of the net, the group realised that it had to be the same as the circumference of the circle, but they had not been taught a formula to work this out. Instead, they took a piece of string and used that to measure the circumference of the circle. Ingenious!

5. The Nodding Rabbit

 This design demonstrates that sometimes simpler is better. The group used 2 cubes, attached with a piece of card at the back. Viewers in the gallery are encouraged to interact with this exhibit by tapping the head of the rabbit, which bobs up and down. Although not particularly ambitious from a mathematical point of view, this is a much valued contribution to the gallery and will be used as a model to inspire future generations of artists, especially those who are less confident.

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6. Sheep
As artistic director, this is my personal favourite in the gallery. Again, the acutal net is not particularly mathematically adventurous (it is a cuboid) but who could fail to love an egg box covered in cotton wool and made to look like a sheep? 



7. Egg-Barn

Our guided tour concludes with a very accomplished piece of work from two artists who have made fantastic progress this year. Not content with a simple cuboid shape, the pair decided to construct a triangular prism for the roof of their Egg Barn. They realised that the length of the triangle had to be the same as the length of the corresponding side of the rectangles in the net. With some assistance from their artistic director, they were guided to experiement with using compasses to construct inter-secting arcs to find the third vertex of each triangle.Once they had mastered the technique, the artists became experts in their field and showed another group how to do it.

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Musings on the pedagogical uses of laughter....

One of the most important things in teaching is building a relationship with your students. I always think things are going well with a class if they can laugh at/with me in a good natured way AND I can laugh at them (just a little bit of course).

Both of my current year 11 classes are pretty good at this.One class recently described me as "like a really funny Mum", which I took as a compliment, even though I'm only 10 years older than them and couldn't possibly be their mother. A student in the other class recently said "Miss, you should do that thing where teachers go on TV and teach difficult kids. I'd definitely watch you." I thought this was quite a big compliment, so I made the mistake of asking why. "Oh because you're really funny when you're cross".  Ah. That swiftly deflated my ego!*

Laughing at someone else is easy, but laughing at yourself takes real character. Even as adults, not everyone can do it. I have recently realised that some of the friends and family I admire most are people who can laugh at themselves and don't take themselves too seriously.


I'm pleased to say that most of my y11s seem to have this character trait. One of the most able students that I teach (and one of the most confident) stared at a distance time graph recently and shouted across the room "Miss this scale is all wrong. It makes no sense. It goes one thousand, one thousand and thirty, one thousand one hundred, one thousand one hundred and thirty. What is it doing?" Rather than give her the answer, I just burst out laughing. "You'll find it funny when you realised what you've done" I said. "Read the question again" (It was a distance time graph, so the scale actually read 10.00, 10.30, 11.00, 11.30). Similarly, in the other y11 class I asked the girls what the letter D stood for in 2D and 3D. "I know!" shouted one student "Dime... Dim..... Dementia?" To which I burst out laughing as well (along with the rest of the class). Again, in a great show of character, she took it on the chin and laughed along with the others.

We ask our students to cope with criticism every day. They are constantly being told how to improve their work, or being asked to correct their mistakes and think about which topics they need to revise. It can't be easy. In my first term doing a history degree at Oxford I felt completely out of my depth and I knew I was handing in rubbish work. I was so embarrassed that I couldn't bear to read the pages of comments my tutor wrote on every essay. If I had done, I would probably have improved much more quickly. As it was, it took about 3 months before a different tutor took me to task and verbally went through how I could make my writing better. She described the next week's essay as "a transformation".

By encouraging students to laugh at themselves when they have made a mistake, we can help them see that making mistakes is part of learning. But to have that sort of atmosphere in the classroom, we need to let them laugh at us too. Using humour in the classroom is a risky strategy, but I think that when it pays off, it's hugely worthwhile.

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* To re-assure readers that I'm not a complete idiot in the classroom, I am hardly ever genuinely "cross" with this class, because they are really nice. Instead, I tend to say things like "if you haven't brought your calculator I'm going to give you a dirty look" and I pull a face at them. 

Easter Egg Box Project

I don't remember a huge amount about maths lessons when I was at school. I remember enjoying the SMP system in year 7 where you worked through lots of little books and essentially taught yourself. I really liked that. I didn't like the GCSE coursework where you had to count words in sentences and then work out the standard deviation of the sentence lengths. As a lover of literature I despaired to see To Kill A Mockingbird reduced to such soul-destroying analysis.

However, some lessons did capture my imagination and I remember particularly enjoying a series of lessons where we had to design an easter egg box. Luckily for me, I was allowed to work with my hugely talented best friend who was both a mathematical high achiever and ridiculously creative. She came up with the idea of designing a net that looked like a hen's head.

It was pretty impressive. It consisted of an octagonal prism and a cone all in one net, which we decorated to look like a hen's head and beak. I've searched on the internet for something similar and this is the closest image I can get to it, but it doesn't really do it justice.......

So I wonder what our y7s and y8s will come up with when we give them the same project in a week or so's time.

Here are the resources on the TES: http://www.tes.co.uk/teaching-resource/Design-Your-Own-Easter-Egg-Box-Project-6323990/

If you are looking for a project to do before easter, I think it should be a good one. It wasn't difficult to put together and so far I think it's ticking all the boxes (ho, ho! I love a good pun)

Which boxes has it ticked? Well it......

*hasn't taken long to plan
*helps to deliver quite a lot of national curriculum content
*allows students to be creative
*has a cross-curricular element
*gives us the chance to assess objectives without testing students to death
*shows how maths applies to real life situations
*AND is open-ended enough for all students from level 3 to level 8.

The resources on the TES include a powerpoint  and a student assessment sheet. I haven't uploaded the nets that we are going to use because we've taken most of them from www.senteacher.org. We also drew some ourselves on sqaured paper, so that the level 2/3 students can count squares.

If we get any particularly impressive designs, I'll post them up on the blog too.

darts project resources

A quick update! Here are the darts project resources.

http://www.tes.co.uk/teaching-resource/Darts-Project-number-and-geometry-project-6323154/event/22/

Enjoy.

Codebreaking resources

The resources for this lesson

http://www.guardian.co.uk/teacher-network/teacher-blog/2013/mar/04/codebreaking-second-world-war-maths-history

Are available here:

http://www.tes.co.uk/teaching-resource/WW2-cross-curricular-codebreaking-project-6323054/requestAction/add//