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01-07-2011 | Computer Math Snapshots - Column Editor: Uri Wilensky*

Mean-Invariant Polynomial Intersections: A Case Study in Generalisation

Author: John Mason

Published in: Technology, Knowledge and Learning | Issue 2/2011

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Anyone knowledgeable about cubics knows that they are symmetrical by rotation through 180° about their inflection point. Slightly less well known is that the tangent to a cubic at the midpoint of two of the roots, passes through the third root¹ (see Horwitz undated). Indeed, Aude (1940) used this property in reverse to locate the real midpoint of a pair of complex roots of a cubic. Arne Amdal (private communication) reported the observation as arising spontaneously from a student in his high-school in Norway using dynamic geometry to explore cubics. Kaye Stacey (private communication April 2002) working with colleagues also found some of the results to be recorded here. …

previous article Development Process of a Praxeology for Supporting the Teaching of Proofs in a CAS Environment Based on Teachers’ Experience in a Professional Development Course

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For example, taken from the internet: two people tackling the same specific problem, probably course work from the IB: https://nrich.maths.org/discus/messages/67613/68454.html and http://nrich.maths.org/discus/messages/67613/69786.html.

http://mcs.open.ac.uk/jhm3/Applets%20&%20Animations/Applets%20&%20Animations.html and look for Lagrange & Cubics.

Aude, H. (1940). A note to the theory of equations. National Mathematics Magazine, 14(6), 308–310.CrossRef

Cuoco, A. (2005). Mathematical connections: A companion for teachers and others. Classroom resource materials. Washington: Mathematical Association of America.

Gattegno, C. (1970). What we owe children: The subordination of teaching to learning. London: Routledge & Kegan Paul.

Horwitz, A. (undated). Using mathematica to prove and animate a property of cubic polynomials. http://archives.math.utk.edu/ICTCM/VOL08/C035/paper.pdf. Accessed Nov 2008.

Lascoux, A. (2003). Symmetric functions and combinatorial operators on polynomials. CBMS regional conference series in mathematics, vol. 99. Washington: American Mathematical Society.

Macdonald, I. (1975/1999). Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. Oxford: Oxford University Press.

Title: Mean-Invariant Polynomial Intersections: A Case Study in Generalisation
Author: John Mason
Publication date: 01-07-2011
Publisher: Springer Netherlands
Published in: Technology, Knowledge and Learning / Issue 2/2011
Print ISSN: 2211-1662
Electronic ISSN: 2211-1670
DOI: https://doi.org/10.1007/s10758-011-9185-y

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Other articles of this Issue 2/2011

Computational Diversions: Slamming Door Comedy

Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space

Development Process of a Praxeology for Supporting the Teaching of Proofs in a CAS Environment Based on Teachers’ Experience in a Professional Development Course

The Inscription, Translation and Re-Inscription of Technology for Mathematical Learning

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