I’ve been thinking about some of the new content in the GCSE and wanted to delve a bit deeper into some of the topics – not only to make sure that my understanding of the “learning objectives” are correct but if I’m honest I also think I have a hang up about subject knowledge, because of my experience trying to get onto my PGCE course when it was strongly suggested that maybe I would be better suited to teaching Business Studies (I have an Engineering & Business Studies degree from Warwick). Additionally, I’m very conscious that I sometimes just teach a “process” for several reasons: sometimes the students just aren’t interested, sometimes it doesn’t add anything to their understanding and sometimes it’s down to time constraints. So, my intention is to start writing a weekly blog post about various topics within the new 9-1 GCSE … just for fun! I therefore apologise in advance if this is teaching some of you to suck eggs but my intentions are about making me understand the maths behind the prescriptive programme of study from the DFE, in order I suppose to improve my teaching!
In the aforementioned document under “Number: Structure and calculation” is the below assessment objective:
- apply systematic listing strategies including use of the product rule for counting
The inclusion of the product rule for counting was new to the 9-1 GCSE and just to remind you that “Only the more highly attaining students will be assessed on the content identified by bold type. The highest attaining students will develop confidence and competence with the bold content” which means that it’s a Higher tier concept only (that’s not to say I would limit teaching it to those doing the higher tier – I may not choose to call it the “product rule” with some groups or even when we’re using initially using it but that’s my pedagogical choice)
In the context of “just” an assessment objective, with systematic listing, you’re looking at “here’s 3 digits how many different numbers could be made” or “here are some fruit, Jo is going to choose 2 pieces” blah, blah, blah type of questions. However, in my opinion, this is a useful skill to get embedded from a very early stage and the use of the word “systematic” is something I use and get students to understand from the very start of year 7 (or at least I like to think I try to do – I’ve written about this before ) The type of topics that lend themselves well to demonstrating this are the obvious ones like listing factors of a number in “pairs” instead of randomly trying to list them all, HCF/LCM (including multiples in context i.e. buses to Acton and Barton 😉 ) using sample space diagrams to link with the idea that “listing” isn’t just a straight “up and down” list it can be pictorial too and a link to the product rule through the use of more “problem solving/context” questions can be used too.
It’s quite common to use the example of the letters ABCD being arranged into different orders and listing them systematically – I always start with just ABC as a class discussion and then get students to work on the ABCD problem so that they can see every combination available to them and look to see if we can draw out that the number of combinations in the first is 3x2x1= 6 and the second one is 4x3x2x1=24. I also like using this example of how, sometimes, simplifying a problem (i.e. using easier numbers or a smaller amount of “things”) can sometimes provide a strategy that can be scaled back up to solve the original bigger problem.
I may then go onto look at using say, shirts and ties as an example where Mr Seager has 3 shirts and 4 ties and how many different “looks” he can have. For each shirt there are 4 different ties he can wear and so there are 3 x 4 = 12 possible combinations. This can be extended to introduce 5 pairs of trousers he could wear: 3 x 4 x 5 = 60 possible combinations.
When looking at situations involving counting it is often not practical to count all the things and the above example illustrates the “product rule for counting” also known as the “fundamental rule for counting” – I always stress that methods such as this have been developed to help us to do “maths” more efficiently:
If there are m ways of doing one thing and for each of these, n ways of doing another thing, then the total number of ways the two things can be done is m x n ways.
This can be extended (but note that this isn’t mentioned in the programme of study) but I’d certainly extend the topic to include three tasks, each of which can be done in m, n and p ways respectively, then the number of ways all three tasks can be completed is m x n x p.
Having delved into the origins of this area of maths, I now know that this is an area of “combinatorics” which is an area that I’ve been curious about since I looked at Maths Dobble and flirted with combinatorics a little at the time. There are many sub area but I’d define it as the “art of counting”. I don’t mean that light-heartedly or in any way do I mean to dismiss other sub-topics but I am in awe of the complexity and beauty of maths that I continually come across and combinatorics, is so much more than the teeniest bits I’ve homed in on below. Just needed to say that!
So, combinatorics makes use of permutations and combinations and are linked strongly to the study of probability. Think about it: probability looks at how likely an outcome is – it involves counting/calculating how many way this outcome can happen. For the many circumstances where we need to count the number of outcomes there are two diﬀerent counting situations – permutations and combinations:
- Where the order matters it’s a permutation – basically an ordering of a group of things e.g. 213 is a permutation of the numbers 123, as is 321 etc.
- Where the order doesn’t matter, it’ll be a combination e.g. 10 students chosen out of a group of 250 students – regardless of the order in which the students were chosen they will represent a combination of 20 of the 250 students in the year group.
Got it? If not, think about how many ways there are of getting two letters from: a, b, c. There are ab, ac, ba, bc, ca and cb.
- When we ask the question “how many permutations?” … there are 6 because ab and ba are considered to be different because of the “order” of them.
- When we ask the question “how many combinations?” … there are now only 3 because ab and ba are considered to be the same as we no longer care about the “order” of them. It makes sense to think about the permutations and eliminate the “duplicates”.
Even if, like me, you’ve never (or rarely) referred to the product rule per se, you’ll probably find that it is the context of the question that students will struggle with and whether they need to adjust the multipliers they use as you would for some situations and the one way to overcome this is through deliberate practice of these type of questions.
I liked the idea of introducing this to my year 9 group through a real-life context (I admit it’s more pseudo context!) rather than using an abstract context so I put together a little activity shown in the pic below, looking at an alternative way of producing number plates and introducing the task through the increased number of new cars being registered in 2013 and 2014. I’ve also pulled the questions from Edexcels exam wizard together (here -> Product Rule Exam Questions) and put some solutions together that you may find useful (here-> Product Rule Exam Questions – SOLUTIONS)
It’s an area of maths that has its roots going back 1000’s of years to about 1550 BC with the earliest known reference in problem 79 of the Rhind Mathematical Papyrus which is considered to be one of the best-known examples of Egyptian mathematics. Fast forward through the next three and half thousand years will see this area of maths being discovered/rediscovered in multiple places across the globe with contributing work from so many names I must admit to not knowing enough about: Cardano, Mersenne, Descartes,,Leibniz, Fibonacci, Pascal and by the 18th century graph theory is born in the wake of Euler’s work on the Königsberg bridge problem and his formulation of the polyhedron formula.
There is so much more to learn than just a simple line in a programme of study!