In this article we explore a couple of great modelling activities designed and road tested by our Core Maths pioneers. You can find out more about what Core Maths is here.
The question to ask is “Why wouldn’t you teach it?”, says Colin Prestwich, Yorkshire Ridings Maths Hub Lead.
As schools and colleges around the country are picking up the baton of Core Maths (known by a different name in each exam board but identifiable as ‘Level 3 Core Maths’) we take a look at a couple of activities that demonstrate what Core Maths can offer that traditional maths courses do not.
In our research for this article, it has become clear that Core Maths has generated some fierce advocates amongst its teachers. When we tentatively put out a call for contributions, we were surprised by the number of busy teachers keen to share their experiences and resources. Here, Colin Prestwich (Harrogate Grammar School) and Dominic Nice (West Sussex College, tweeting as @NiceMaths) each share one of their most successful series of lessons. First, Dominic’s task asks hypothetically whether Santa could exist. This task contains the sort of frivolity and light-heartedness that both students and teachers appreciate at this time of year, while addressing the ‘Modelling Cycle’ aspects of the curriculum. Second, by asking students to reflect on what they spend money on, Colin’s task has them calculating what gross salary they will need to earn to finance their lifestyles, addressing the ‘Personal Finance’ aspect of the curriculum.
Dominic Nice: Is Santa Real?
At West Suffolk College, we are in our second year of running Core Maths, with a Year 2 cohort of 18 students and a Year 1 cohort of 22. Over these first two years we've been building up resources unique to Core Maths, which engage students in a different way than we are used to from GCSE.
One such activity which students enjoyed last year was our task, 'Is Santa Real?'. Students were given the very broad task of determining mathematically whether the Father Christmas we are all familiar with could feasibly exist (there's also an accompanying sheet of teacher notes). They might choose to discover whether he could possibly travel fast enough to visit every home in the world, or might calculate how many reindeer it would take to pull a sleigh containing thousands of tons of presents.
How many reindeer will be needed to pull a sleigh?
click image to see students' reasoning
They can make any assumptions they like (provided they're stated) and can use computers to research anything they need to. At the end of the project, they present their findings to the group. This task should preferably be left as open as possible, to allow students to find their own inspiration and to encourage unique projects, though a prompt sheet of possible ideas is provided for teachers needing to offer students more guidance.
Some of our students blew me away with what they produced: I had students researching the origins of Saint Nicholas and the population density of the Roman Empire in the fourth century; some were looking into employing a team of elves; while others were finding the number of calories in a billion mince pies, and what amount of exercise could possibly burn that number of calories before the next Christmas (see below - also available to download in PowerPoint format). They each took the question in a different direction, and while more support was needed for some groups, they all engaged with the activity and produced something really impressive.
If students need more guidance, there are countless examples of similar calculations online, which students can look up themselves, and of course they can always ask for a bit of help from their teacher to point them in the right direction. However, I feel this really does work best with as little teacher intervention as possible, at least until students have decided what they want to discover. I was initially wary of giving them so much freedom, thinking they wouldn't know where to start, but from my experience with the task, they engaged with being able to direct their own projects, and discover and calculate whatever they thought might be interesting.
In terms of tying this activity to Core Maths objectives, it certainly had students thinking about how to set out a mathematical argument, research independently, critically analyse, and to an extent, Fermi estimation came in to play too. Not all Core Maths topics lend themselves to this type of task, and not all of our lessons are run this way. However, our ethos is to plan tasks to be this kind of open-ended investigation wherever possible, where students can each reach a different answer or conclusion, and can each be right for a different reason. It's a breath of fresh air to teach, which I genuinely enjoy planning and teaching every week.
Colin Prestwich: How Much Do I Need to Earn?
How can students leave school without being able to calculate net salary from gross salary? Or indeed manage their debts properly?...
Students readily recognise the importance of understanding personal finance issues, the ability to analyse large amounts of information and the confidence to tackle problems that are new to them.
Here is a series of three lessons based on 'How much do I need to earn?' that, using a problem solving thread, links a number of key big ideas:
- web interrogation
- use of assumptions
- wisdom of the crowd
- summary calculations
- fluency with net income calculations
- calculating and using percentages in context
- use of spreadsheets
- appreciation of household expenditure.
(designed to meet AQA 1350 Mathematical Studies specification paper 1 refs 3.1 Analysis of Data, 3.2 Maths for Personal Finance and 3.3 Estimation).
You are 25, single, and looking for a sales and marketing job in Leeds.
What advertised salary should you be targeting?
Text in bold takes us through the steps of the project, indented text describes how my students responded to the task.
The first step is to agree what a 25-year old will spent money on. It is a good idea to ask a student to chair this discussion.
The students found this fun and engaged readily. The discussion was very enlightening, and I was surprised at the student responses - they had not thought about this before! They came up with: rent, food, entertainment, clothes, energy, transport, phone. There was plenty of disagreement before they realised that all their views could be represented.
Next, students are asked to come up with estimates for all the spending categories using real life searches, pure guesses, or other methods that they can justify. The ‘wisdom of the crowd’ principle is useful for dealing with pure guesses and Marcus du Sautoy's YouTube clip is an excellent addition here. This is a rich task and will involve many discussions.
I let students come up with their own estimates and supported individuals as necessary.
“What are rates for?” “Why do you need to pay council tax?” “I thought that water was free!” were just a few of the questions my class asked as they went along.
So how are the results to be summarised?
The students found this hard to answer. Eventually putting the results in a spreadsheet was suggested.
“Sir, what is a spreadsheet?”
I was shocked at how few of them were able to do this efficiently. Indeed, many simply used it as a table in which to enter values - even the totals and means were calculated by hand! By allowing them to do this, inefficiently at first, they were then able to appreciate the power of a spreadsheet when shown some of the basic functions. At this stage I gave them some example questions to help them understand the data and bring in different types of percentage work.
By lesson two, students should have agreed a net monthly income required.
My students settled on £1140 by taking averages of their various sums. The students thought that this was a king's ransom! They started to panic.
Then students are set on job searches on the internet for marketing jobs. Local newspapers, corporation websites etc can all be used – the students first need to find them.
Working backwards from net income to gross income is very challenging and it is valuable to discuss possible approaches.
Initially, my students decided to select a gross income from the adverts and then see if it gave a net salary that would be sufficient. We divided up the calculations and pooled the results. After encouragement the students realised that a spreadsheet would enable these calculations to be done much more quickly.
In lesson three, students set about producing a spreadsheet to do the appropriate calculations using up-to-date tax and National Insurance bands.
My students found this very hard and we needed a number of attempts, but I did not show them an example until all had tried hard to come up with a working spreadsheet with a number of gross and net incomes so that they could sensibly address the original question.
With a well-constructed spreadsheet, it is easy to scroll down to exceptional salaries and provoke meaningful discussion about tax contributions of high and low earners.
For my class, this was a big wow factor. I took the spreadsheet and scrolled down to an income of £100 000, £250 000, £500 000, £1 million: at this point they were shouting for more: a five-minute discussion about footballer, film star and maths teacher salaries took place.
The final step is in asking the students to reflect on what they have achieved.
Many of my students quickly came to the conclusion that most people would just try to get a job and then look at their pay packet at the end of the month and just hope it was enough, rather than working backwards as we had done. The project allowed them to see how they could use mathematics to support their everyday life and enable them to do the calculations required to plan ahead – to be proactive rather than reactive with their finances.