During these final weeks of the summer term, we begin teaching our AS students some of the A2 material. For single Maths students, this comprises the algebraic fractions and functions from Core 3. For Further Maths students, we crack on with the rather daunting complex numbers chapter of FP2: de Moivre’s theorem, loci and transformations.
However, this week is our so-called “block week”: each day the students spend a whole morning with one of their subject teachers. The purpose is to enrich their studies rather than full steaming ahead with next year’s material. Today I had the pleasure of keeping my further mathematicians amused for about three and a half hours (plus breaks).
Having fun with A2 Further Maths?
Speaking diplomatically, our department doesn’t have the best systems in place for talking about practice and sharing our best ideas. The instruction/agreement for block morning was that we would do the “chi squared stuff”. Now, I’m only in my second year here and I wasn’t entirely sure what fell under the “chi squared stuff” umbrella. I was proffered the chapters of notes from CIMT (Goodness of Fit Tests and Chi-Squared) and a copy of a copy of an activity based on the colours of M&Ms.
Maybe it’s a result of my limited concentration span or maybe it’s just empathy for my students and a hope of giving them an interesting morning, but either way I couldn’t bring myself to spend three and a half hours on chi-squared.
Begin with Purpose
I settled on the following intentions for my morning:
- Activities that students would enjoy engaging with
- Tables blocked together so students are sat in tables of 4
- Activities that would provoke small group discussion and debate
- Some competitive elements
- Help the students develop their intuition of and an appreciation for significance level and hypothesis tests, without the formal details
- Have a game or two on hand that relate to probability or strategy
- Have additional topics on hand in case we needed a break from the statistics
As you can see, these are not lesson objectives in the formal sense but simply the elements that I hoped to build together.
Here is a brief summary of the activities that actually took place. Further details below.
- Least positive integer competition
- Discussing the likely score on a ‘true/false’ quiz with 5 questions
- Discussing a pass mark for true/false quizzes with more questions and looking at binomial tables
Lesson 2 (mix up student groups)
- Least positive integer competition
- Chi-squared test for distribution of colours in a bag of M&Ms
- Playing Yahtzee
- Calculating the probability of scoring ‘Yahtzee’
Lesson 3 (mix up student groups again)
- Least positive integer competition
- Plotting a polar graph
- 10 question ‘maths challenge’ speed round
Least positive integer competition – this is something I learned about from the Tweetup at #mathsconf4. It was the perfect ‘ice breaker’ to make the students feel at ease on their group tables, to get them used to the competitive elements of the day, and to get them thinking straight away. I had plenty of mini bags of Haribo on hand as prizes for the morning! It went so well at the start of the first lesson that we re-ran the competition as a settler at the start of the second and third lessons, too.
5 question true/false quiz – using the following slide as a prompt, this activity was intended to engage the students in discussion in their pairs and around their tables. I made sure I gave plenty of time and little input for this. They needed to realise I was not looking for a ‘quick answer’ but a well-thought out explanation.
After sufficient discussion time, I gave each table the opportunity to explain their thinking and reasoning. Fortunately one group in particular had calculated the probability of achieving 5 correct answers by luck, 0.55, which is approximately 3%. I then introduced the concept of a significance level and the basic principle of carrying out a hypothesis test.
I wanted to give them a 5 question true/false quiz after this discussion: to relax them, get them competitive again and just for a bit of fun! One student did get all the questions correct – and she won the obligatory Haribo. (A quick Google search uncovered a number of fun questions. I picked 5 from this book.)
Discussing a pass mark – back to small group discussions again. This time I wanted the students to discuss how they would set a ‘pass mark’ to discriminate between luck and knowledge. To keep them focussed, I suggested quizzes with n questions with n = 1, 5, 10, 20, 100. In brief, they discussed this thoughtfully and had interesting ideas about how to set the ‘pass mark’. Some opted for a fixed percentage, others worked on a sliding scale of decreasing percentage for the increase in the number of questions. I wrote their suggestions on the board. I then asked them to calculate some probabilities for a 10 question quiz: P(all 10 correct), P(9 correct), P(8 correct). As they hadn’t studied the binomial distribution, this was a very thought-provoking challenge for them.
Binomial tables – I then issued some binomial tables and asked them to highlight the n = 10, p = 0.5 column. I then asked them to work out what the tables actually meant. They did it! We then used a 5% significance level to identify ‘pass marks’ for n = 5, 10, 20. I left them wondering about n = 100 – it’s healthy for students to go away with unsolved problems.
Chi-squared with M&Ms – One of my colleagues passed me the details of how to run this activity. Essentially some data provided by Mars indicated the manufacturer’s intended proportions of colours – presented in my beautiful hand-drawn pie chart. I quickly explained the process of taking a sample and completing a table that structured the chi-squared calculation.
The students appreciated this opportunity to do some structured ‘number-crunching’ – that’s what a ‘normal’ maths lesson feels like to them. It puts them back in their comfort zones.
Once again, we discussed significance levels and made reference to tables of critical values to draw our conclusions. I then had the students pool their data to see the effect of analysing a larger sample.
Playing Yahtzee – To be honest, this was more an opportunity for them to enjoy playing a game. I think game-playing is important for all students as it develops so many skills: turn-taking, strategy, mental arithmetic, and just the simple enjoyment of playing a game involving chance. None of them knew this game before so I did my best to explain the rules with examples. I did have to visit each pair and ensure they had grasped the rules correctly. (Yahtzee score sheets are easily found online and I keep a bag of 100 dice in my desk for such occasions.) Typically, it was during this activity that one member of SMT decided to visit. Fortunately, I could use him to make up a pair and get involved. He left as I gave the students the (rather fiendish) challenge of determining the probability of scoring a ‘yahtzee’.
Plotting a polar graph – Even the third round of our least positive integer competition didn’t revitalise the students enough to get them back on the yahtzee probability challenge. As we all would in that situation, I abandoned that idea and moved on to something completely different. I introduced them to the idea of polar graph paper (printable from this amazingly useful site) and connected it with the polar form of complex numbers that we had already studied in our regular classes. I set them up with the challenge of plotting r = 5 sin(3θ) as a quiet task they could take some care over.
Amusingly a number of them tried to guess (incorrectly) after completing one petal. The first two to produce a correct graph were awarded yet more Haribo. I explained that in Further Maths next year they would learn techniques for finding the area and length of such graphs which I think captivated their (future) interest.
Maths challenge speed round – This was the perfect activity for an awkward 20 minute empty patch at the end of the morning. I’d printed the first ten questions from an American maths challenge competition. They are not too difficult, starting from very easy and building up to mildly taxing. To make it slightly more exciting, they were playing as a team and the time limit of around 15 minutes meant they would have to think about sharing the workload. The time flew by; I pointed out in the binomial tables the minimum score that would indicate they hadn’t just guessed and they quickly marked each other’s papers to determine the winners of the final packs of Haribo.
Apart from being exhausted at the end of the session, I was rather pleased with how well it had all gone. I’d kept the students’ interest throughout almost all of the morning – they only really flagged with the P(yahtzee) problem which we quickly moved on from. More importantly, however, I met my aims of getting them to solve novel problems and discuss and debate situations for which they had not previously been taught formal methods.
The work on hypothesis tests should give them something on which to base next year’s S2 and S3 learning. The work on the polar graph should have piqued their interest about some pure techniques next year. The maths challenge problems at the end saw them share responsibility for completing a task under pressure.
Time for me to have a well-earned break and my own little pack of gummi bears…