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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.
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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
