The Counting Principle – 9-1 GCSE Content

In the interests of making my blog more reflective and useful as a T&L tool for myself (and others if they so wish!) I thought it may be nice to talk about some of the lessons I have taught and strategies I am using in the classroom. This is my first attempt at doing so! I should add here that this lesson, like many of my others, is a serious mish-mash of my own ideas with things other teachers have already put together. I have tried to credit everyone who’s ideas I have stolen but PLEASE let me know if something here is your idea and I haven’t – I would hate to not give you the credit you deserve.

I am teaching my first ever Year 11 class this year. It comes with all kinds of excitement and fear – it is a lot of responsibility! However, I’ve been lucky that my class are an absolute dream to work with – they are one of our top sets, they are polite and kind, and exceptionally hard working. This has made the fear les pronnounced, and the excitement kick in a little more. The 4 hours with them on my timetable are fast becoming my favourite for all the reasons above, but also because I have decided that rather than just revising the things they’ve been taught for 4 years already, I am going to push them towards the top as much as I can. This means doing some SERIOUSLY interesting maths.

My first chance at this was when I saw ‘The Counting Principle’ in the SoW. It actually featured as a tiny mention at the bottom of the ‘Order of Operations’ lesson. To me this suggested a brief mention at the end of a lesson, telling them the multiplicative rule and leaving it there… but I felt that it deserved more than this, especially as many of my students in this class will be looking at Maths A Level next year. I would like them to be as prepared as possible for this.

Having looked at the specimen paper exam questions and a blog post on JustMaths I was assured that my thinking was correct. For some ‘part b’ questions, knowing the basic rule wasn’t enough. Students need a sound understanding of systematic listing, combinations and permutations at a basic level. I set to work trying to find the best way to set this lesson up – JustMaths was the start point, followed by a look though the New GCSE Support Page at Resourceaholic. I found Colleen Young’s blog particularly helpful in stepping through the topic from the basics, with the video embedded from Transum Maths to be one of the turning points in my planning. In the end I spent a double lesson on this content, covering everything from listing to those permutations ‘part b’ questions.

I started with the basics, an example of systematic listing – a staple of the maths GCSE since I took it myself, and probably for decades before that.

countin1

The class looked at me like I was insane at this point – ‘Oh God, she’s planning an hours lesson on how to list things!’ From here I started to make use of some of the Maths Challenge style questions, helpfully collated by the fabulous Dr Frost Maths in his lesson for this topic. They found these much harder and suddenly realised that this ‘systematic’ approach wasn’t as easy as they first thought. They also started to wonder if there was a quicker way… that was my cue!

Here I modified a question from the Just Maths exam question collection, and I started to make a case for needing to know a quicker method – my numbers suddenly got much bigger and harder to handle.

counting2

I let them have a go. There was a bit of panic as they all started to label Boy 1, 2, 3,…. & Girl 1, 2, 3… and pair them up. I put them out of their misery after about 30 seconds/1 minute. “We’re going to come back to this question later, let’s do some easier ones first.”

At this point, the classic ‘T Shirts & Trousers’ combinations came out. I made some jokes about my capsule wardrobe (which the girls laughed at and the boys looked bemused at…) and started mapping out my choices on ‘probability trees’ so students could see my amount of choices at each stage. “For every top, I have a choice of 2 pairs of trousers… that is 5 lots of 2.” I extended this to then start thinking about various accessories I could include in my looks as they became more confident. We then talked about the counting principle and its definition, then went back to our ‘nasty’ example of the roles in the play. Unsurprisingly they were all very relieved we now had a faster method of doing this.

We ended this section with the exam question concerning Pavel and his combination locks. Many students at this point were wanting to do calculations such as 4×10 as Pavel made a choice of 10 digits 4 times, rather than 10^4, prompting some great discission about the multiplication including the options for each choice, not the number of individual choices.

This led nicely into the next section of the lesson, where we discussed repetitions. First I asked the class to look for the differences in these two questions:

counting7

Once we had this straight, we ended the lesson with a group task to figure out the possible combinations of ‘words’ created using 4/5/6 letters of the alphabet taken from Dr Frost Maths once again.

I started the next lesson with an idea from the end of the JustMaths blog – deciding which set up would be best for some new numberplates which went well and gave them some real life context into where this maths could be used. I then gave the class a similar question to above, this time concerning permutations vs combinations. I should make this clear now – at no point did I use this language with my class – for them it was a simple case of ‘Does order matter here?’

counting8

We mapped these two possibilities out explicity as there were only 5 people to choose from. Students very quickly saw that they were ‘double counting’ if they didn’t choose people for specific roles and so their options were ‘halved’, but this was not the case for the goalkeeper and striker. I then used the ‘Ordered or not’ slides from Dr Frost Maths to see if they could spot when to divide by two in different scenarios. It had mixed success for the students but they were starting to discuss this issue really well in groups and pairs using examples of names around them – “If I chose you then Kate, or Kate then you, it wouldn’t matter because you’re both still picked – one of you doesn’t have a better job than the other”. I think this is the best way to deal with these questions and I was happy to support these conversations where I could.

As a result, I let them just have a go from this point using the Dr Frost Maths questions. (Which I am eternally grateful for because I cannot come up with scenarios for this type of thing for the LIFE of me!)

We ended with one of the specimen exam questions which concerned soloists in a choir, with a permuatations style part b.

Some of the students were still struggling with knowing when to divide by 2 (this is as hard as our permutations got) and so came to catch up later in the week for additional practice. However, this meant finding them more questions on a topic that is quite underresourced. I turned to @MrMattock ‘s blog  here for some amazing tips on converting questions that already exist into ones that are useful for the topic, and picked up an old S1 textbook for some inspiration. The result was my ‘Multiplicative Counting worksheet which can be found on TES – these are a jumble of combinations and permutations questions which really keep students on their toes, thinking about their methods. By the end of the hour in catch up, they were all getting the method right after a read-through of the question and fist-pumping the air when they got the answers correct.

If I had a chance to do it over again, I’d pick up the pace on the first hour if I could and spend a lot more time pulling questions apart to decide on a combinations/permutations approach, as I really think this is the bit they struggled with the most. But overall, I was happy with the progress made and the level at which my students were communicating mathematical ideas. Many of them also commented that they felt they’d learned an awful lot in one day and felt more challanged than having just recapped the knowledge from last year. In all of these ways, I felt very much like I’d succeeded in how I’d approached the lesson and in the things I had wanted them to achieve. I don’t think you can ask for more than that!

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