Thursday 14 March 2013

Official Outstanding Lesson: Permutations and Combinations

In this post I am going to describe to you the lesson that I taught this week which was graded as Outstanding by an OfSTED inspector.

Obviously I can't guarantee that if you do this lesson you will get a 1, because a lot of it is in the delivery, and in how your students respond. There was a sort of magic spark in my lesson, which I think was there because of the good relationship I have with the group.

My lesson was structured as a series of problems relating to permutations and combinations. I didn't tell my students what perms and combs were until right near the end of the lesson. Not one of my students could have told you the learning objectives, until the very end. But who cares about learning objectives when you're busy doing PROPER MATHS? FYI, the only resources you need for this lesson are a deck of cards and, optionally, some mini whiteboards and pens.

1) The Dramatic Intro

Here is a rough transcript of what I said at the start of the lesson:
"All of you in here are extremely lucky. I am going to show you something no one has ever seen before. *dramatic pause*. This is a once in a lifetime experience, literally. In the whole history of the human race, no one has ever seen this before, and there's a very good chance that no one ever will. I'm not just doing this because we have a special visitor today (nervous glance at inspector) but because I love you all very much [I tend to babble when I get nervous]. Are you ready? *Another dramatic pause*"

Then I took out a deck of cards and shuffled them. Then I fanned them out and showed them all of the cards. Then I just looked at my students, paused, then said, "Amazing, yeah!" In my most excited voice.

There was kind of a confused awkward silence, followed by a chorus of "no"s. I feigned disbelief. Then I explained to them that that particular order of cards has never been seen before by anybody, living or dead, and probably won't be seen again for an extremely long time.

I then explained the maths. I said, let's assume that playing cards have been around for about 500 years, because that's roughly what Wikipedia suggests. Let's assume the population of the world has been a constant 6 billion for the past 500 years. And let's assume that everybody shuffles one deck of 52 cards every second. How many orders of cards would that make? I let them work it out. I gave them mini whiteboards to do their working out on, because pupils are much more likely to start attacking a problem if they're not worried about making mistakes. With MWBs they can erase all evidence of stupidity later. They worked out the answer (I leave that to you as an exercise!) and then I got them to calculate the number of possible orderings (permutations) of 52 cards. Both tables of pupils worked out very quickly that it would involve factorials. Again I will leave the answer to you as an exercise.

[*edited to add* I've just remembered a terrible joke I made. This is pretty cringey, but I'll include it anyway. A student asked the sensible question of what to do about leap years. I said to ignore leap years, and pretend that everyone gets the day off from shuffling cards once every four years. Then I said, "oh no, but that contradicts the song!" So they're all like, "What song?" and I said, "You know, Everyday I'm Shuffling!". Pretty good considering I thought of it on the spot! ]

We had a discussion about the hugeness of the result. The students were pretty amazed, I think. The inspector was too!

2) The iPhone Problem

I moved swiftly on to my next problem. Here is roughly how I introduced the problem:
"You're sitting with your friend, and their iPhone is on the table. They leave it there and get up to go to the toilet. You want access to their phone, because you want to frape them, or whatever [is frape an appropriate  word to use during an observed lesson?] but they have a passcode on it. It's four digits, and you happen to know each digit is different, so there are no repeated numbers. The question is, do you have time to try every possible passcode before your friend comes back from the toilet?

The thing I like about this question is that students have to make certain assumptions in order to answer it. Like how long the friend will spend going to the toilet (actually that ends up being kind of irrelevant once you've worked out the answer), and how long it takes you to type in the four numbers. My two tables ended up with different lengths of time that it would take to try all the possibilities, because they made different assumptions.

You can have a little discussion then about phone security and whether a four-digit passcode is enough. That will take care of your PMSSC. I didn't really go into it that much though.

3) The Baby Chris Problem

"Chris' mum gave Chris five wooden alphabet blocks to play with. She gave him the letters A, H, M, S and T. She left the room to make a cup of tea, and when she came back, baby Chris had spelt out the word MATHS. So Chris' mum is going around saying he's a genius. Do you think her claim is valid?"

The students calculated the number of possibilities and decide whether the probability is low enough to claim he's a genius. The class decided it was just a coincidence.

So then I gave them part two:
"This time Chris' mum gives him all 26 alphabet blocks to play with. She leaves the room, and when she comes back, he has taken those same five blocks and spelt out MATHS. Now do you think her claim is valid?"

This is trickier to calculate, and the first thing many of my students wrote down was 26!. I had to question them a bit to steer them in the right direction. The probability is about 1 in 8 million, so we concluded that he probably is a genius.

At that point my inspector left, but I'll tell you the rest of the lesson anyway.

4) An exam style question

Each of the digits 3, 4, 5, 6 and 7 are written on a separate piece of card. Four of the cards are picked at random and then laid down to make a four digit number.
a)  How many even numbers are possible?
b) How many odd numbers smaller than 5000 are possible?

I had to help them quite a bit with this question, but it's a nice one. FYI, the answers are 48 and 30 respectively. 

I then formalised the idea of permutations slightly by showing them the notation. I didn't give them the formula though, because I find the formula is quite unhelpful. I think it's easier to just think about the question and work it out. Someone said, "Oh! That's what that's for! I always wondered that!" Meaning the key on the calculator.

I only spent a couple of minutes on that, and moved swiftly onto the next problem.

5) The lottery

My students have all memorised by now that the odds of winning the lottery are 1 in 14 million. I suppose I must go on about it a lot! But none of them has ever thought to work out where that number comes from. I think they were quite excited to actually be able to work it out. The key thing I wanted them to notice is that it is a different type of question because in the lottery, the order of the numbers doesn't matter, so it's not the same as finding permutations.

They worked out that they needed to divide by the number of arrangements of 6 numbers. Then it took them a little to work out what that was. The most common guess was 36.

I took their workings out and demonstrated that what they'd done was divide 49P6 by 6!. I said that finding the number of combinations will always be like that. Then I said, "If only there were a key on your calculator which could work it out for you!" At which point they all realised that what they'd just done was work out nCr. They'd used nCr in binomial expansions in core maths but hadn't realised (or forgotten) the connection with combinatorics. They were very relieved that there was a shortcut! But I think it is so important to show them where the formula comes from. I proved the formula using the nPr formula that we'd already derived.

6) The Cricket Team Problem

This was just a question I took from the textbook. A cricket team consists of 6 batsmen, 4 bowlers, and 1 wicket-keeper.
These need to be chosen from a group of 18 cricketers comprising 9 batsmen, 7 bowlers and 2 wicket-keepers.
How many different ways are there of choosing the team?

This question is actually really easy using nCr, but it was useful because they could see how to use nCr rather than working it out fully themselves. This was a good question to end on, because they found it quick and easy, which felt like sort of a reward for having derived the formulae earlier. 

Then I set the homework. I didn't really feel like I needed a plenary, because we'd been plenarying all the way through. 

So there you go, my outstanding lesson on perms and combs. I really enjoyed teaching it, and I think my students enjoyed it too.

If you decide to use this lesson, please let me know in the comments below how it went for you.

Outstanding Emma x x x


  1. I love it! Will totally nick for next year (such a shame I've already done perms and combs this year). It would suit my teaching style very well, perhaps you could make lesson plans a more regular feature on your blog!?!

    PS congrats. Maybe now would be a good time to look back on your "inadequate" post and laugh?

  2. It's a bit annoying that my "inadequate" post has seven times as many views as my "outstanding" one!

    I could make my lesson plans more of a regular feature, but I suppose I rarely feel they're good enough to mention. Although to be fair, most of my A Level lessons are like the one above.

  3. So totally did nick it in parts for my final perms and combs lesson. I wanted to do one final summary of a mix of questions without context of what to do.

    How it went:
    The card thing at the start was cool. I presented it by chucking a giant set of cards over a 6 person set of tables by the door. All distracted and intrigued by it as they walked in. Students highly disliked having to make assumptions and tried to instead watch me do the work on the board rather than think it up themselves.One group did the approximations in their head to get an answer of "no". Interesting discussion where some weren't convinced at the end of the maths: The problem was the language we were using as some kids were saying the order HADN'T been seen before when in actuality it was simply HIGHLY UNLIKELY it hadn't been seen before. Many rightly pointed out that the maths we'd done didn't PROVE it hadn't been seen before which made me happy as I strongly emphasise what it means to mathematically prove something.

    I had taken only the most fun questions and made it into a worksheet entitled "Perms and combs: Some mildly fun questions". This was followed up by some other more exam style questions.They pointed out almost instantly that iPhones lock you out for a certain amount of time if you go wrong 10 times therefore putting an upper bound on attempts that is way more limiting that the time spent in the toilet. I will change the question to a made up brand of phone and state that it lacks that handy security feature. I may also invent a part b which tells you they're a girl and the'd gone to the toilets because the bloke they fancy had just come in. Hence she'll be gone for a good half an hour or so. I also had problems with the lazy boys who deliberately misinterpreted the questions as requiring a yes or no answer with no further reasoning. They got a telling off. Will change the wording. It also scared some girls who didn't like making assumptions. I'll therefore change it to say "state the assumptions you make" which is good practice for a mathematician anyway.

    In summary: while adequate as an end of topic summary, especially when the teacher arrived back from the french exchange the night before, is tired grumpy and ill, and hadn't planned a lesson, I bet this lesson kicks ass as an introduction to the topic and I look forward to doing a proper job of it next year. Thanks for sharing it!

  4. Funny how a different class of students can respond so differently to the same activities!

    The phone problem - I told them to assume there was no feature that locks you out after three incorrect guesses. Then afterwards we talked about why they decided to add that feature in!

    I'm glad you used some of my ideas. And hey, if you ever want to do a guest post where you share one of your lessons, please let me know!

  5. Angelica Meyers12 March 2018 at 12:26

    You're very talented and an inspirational person! I only started to learn English and I use translations services to read your posts. Nevertheless, your blog is amazing!