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Exploring Inter-Rootal Distances and ZigZag Graphs in Mathematical Learning

This paper from John Mason at St. Patrick’s College, Dublin, discusses innovative approaches to understanding mathematical concepts through conjecturing and graphing. It highlights the importance of creating an engaging atmosphere for learning, where questions drive inquiry and support among learners. The exploration includes sketching quadratics with specified inter-rootal distances and examining complex function graphs such as y = ||x-1|-2|. It bridges theoretical insights with practical challenges, aiming to deepen understanding of mathematical variations and graphical characterizations.

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Exploring Inter-Rootal Distances and ZigZag Graphs in Mathematical Learning

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  1. Functioning Mathematically John Mason St Patrick’s CollegeDublin Feb 2010

  2. Conjecturing Atmosphere • Everything said is said in order to consider modifications that may be needed • Those who ‘know’ support those who are unsure by holding back or by asking revealing questions

  3. Inter-Rootal Distances • Sketch a quadratic for which the inter-rootal distance is 2. • and another • and another • How much freedom do you have? • What are the dimensions of possible variation and the ranges of permissible change? • What sets of numbers can be the set of inter-rootal distances for a polynomial?

  4. ZigZags

  5. ZigZags

  6. Notation y = |x2 – a| y = |x – a| • What might the graph of • y = ||x-1|-2| look like?

  7. ZigZags Can you make these? What can and cannot be made? Characterising

  8. ZigZags • What might the graph of y = ||x-2| – |x-5|| look like?

  9. Towards the Blanc Mange function

  10. Reading Graphs

  11. Powers • Specialising & Generalising • Conjecturing & Convincing • Imagining & Expressing • Characterising • Distinguishing & Connecting • Assenting & Asserting

  12. Didactic Transposition Expert awareness is transposed/transformed into instruction in behaviour(Yves Chevellard)

  13. More Ideas For Students (1998) Learning & Doing Mathematics (Second revised edition), QED Books, York. (1982/2010). Thinking Mathematically, Pearson, London For Lecturers (2002) Mathematics Teaching Practice: a guide for university and college lecturers, Horwood Publishing, Chichester. (2008). Counter Examples in Calculus. College Press, London. http://mcs.open.ac.uk/jhm3 j.h.mason@open.ac.uk

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