1. 1 Mathematical Tasks: educating awareness John Mason & Anne Watson Toulouse June 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 Generating further exploration Getting a sense of composite functions Using coordinates to read graphs Combining Functions Making mathematical sense of phenomena

3. 3 Generating Functions  What functions can you make by composing f and g repeatedly?

4. 4 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!

5. 5 Possibilities Smooth functions 16 3 4 5 2 No. of zeros No. of local extrema 1 3 4 2

6. 6 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?

7. 7 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