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This event focused on the educational aspect of mathematical tasks, imparting awareness and promoting mathematical thinking. John Mason and Anne Watson presented strategies to combine functions, read graphs using coordinates, and explore composite functions to deepen understanding. Attendees engaged in generating functions through compositions of f and g functions, uncovering rich questions, and learning from practical experiences. The session also delved into the concept of smooth functions, zero points, and local extrema, along with tangent power in relation to graphs. The Quintic encounter explored inflection points, curve senses, and invariance amid changes.
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The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking Mathematical Tasks:educating awareness John Mason & Anne Watson Toulouse June 2010
Combining Functions Making mathematical sense of phenomena Using coordinates to read graphs Getting a sense of composite functions Generating further exploration
Generating Functions • What functions can you make by composing f and g repeatedly? • What functions can you make by composing f and g repeatedly?
Drawing Back • What types of questions seemed to be rich or fruitful? • Learning from experience • One thing we don’t seem to learn from experience … • is that we don’t often learn from experience alone!
Possibilities Smooth functions No. of zeros No. of local extrema 1 6 2 3 4 5 1 2 3 4
Tangent Power • Imagine the graph of a smooth function f • The tangent power of the point P relative to f, is the number of tangents to f through P What are the possible tangent powers, and where are they located?
Quintic Encounter: -inflection tangent -sense of curve for large values of |x| -shift from single point to following tangent Invariance in the midst of change
