On Exponential growth and decay

This sequence of learning activities, designed with the development of students’ mathematical maturity in mind, is intended for use by secondary teachers to strengthen and deepen students’ understanding of compound interest, move into exponential growth and decay, and gently via iteration onto the exponential function.

Piggybacking learning on an interrogation of the bank balance of a 20th century pizza guy who finds himself in 31st century New New York after a cryonics lab accident (Futurama clip below), via a paper-folding journey across the observable universe and a mathematical modelling of the potentially memetic spread of Random Acts of Kindness, we end up at Euler’s number, e.

Four one-page outlines of learning activities are provided below (and as png images at the end of the post), each centred on a particular theme or hook — i.e. Futurama, Paper folding, Pay it Forward, Euler’s number — and each containing key questions and links to video and other online provocations, prompts, etc.  Resources such as student practice tasks are also provided on each outline in the form of hyperlinks or QR codes to scan, to facilitate fuss-free downloads or projections onto a screen as desired.  Note that all the following downloads are provided in pdf format, unless shown otherwise.

• Download 'The Futurama Function'
• Download practice tasks and problems
• Download 'The Paper Folding' function
• Download practice tasks (past GCSE exam questions)
• Download 'The Pay it Forward' function
• Download 'Euler's Number, e'
• Download practice tasks and problems
• Download solution to the 'Primerica' question
• Download solution to the finding e problem (xls)

Each one-page outline does not necessarily correspond to one lesson — the sequence in full typically takes me 10-12 hours teaching time.  Neither are the outlines intended for teachers to follow prescriptively; should a teacher use the ideas and resources presented here, they should of course adapt them in terms of where their own students' learning is currently situated, and where they are aiming to get to.

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Work on the activities presented in this sequence focuses learning on the following aspects of the National Curriculum Mathematics Programmes of Study for key stage 4 in England:

• Ratio, proportion and rates of change: Solving and interpreting answers in growth and decay problems, including compound interest and work with general iterative processes.
• Algebra: Recognising, sketching and interpreting graphs of… exponential functions y = k^x for positive values of k.  Plotting and interpreting graphs (including… exponential graphs)… in real contexts