1 Mathematical Tasks: Mathematical Tasks: educating awareness educating awareness John Mason & Anne Watson John Mason & Anne Watson Toulouse Toulouse June 2010 June 2010 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking
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1 Mathematical Tasks: educating awareness John Mason & Anne Watson Toulouse June 2010 The Open University Maths Dept University of Oxford Dept of Education.
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The Open UniversityMaths Dept University of Oxford
Dept of EducationPromoting Mathematical Thinking
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Generating further exploration
Getting a sense of composite functions
Using coordinates to read graphs
Combining FunctionsCombining Functions
Making mathematical sense of phenomena
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Generating FunctionsGenerating Functions
What functions can you make by What functions can you make by composing composing ff and and gg repeatedly? repeatedly?
f x( ) =2x+ 3 g x( ) =3x+ 2
What functions can you make by What functions can you make by composing composing ff and and gg repeatedly? repeatedly?
f x( ) =x+1 g x( ) =x2
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Drawing BackDrawing Back
What types of questions What types of questions seemed to be rich or fruitful?seemed to be rich or fruitful?
Learning from experienceLearning from experience– One thing we don’t seem to learn One thing we don’t seem to learn
from experience …from experience …– is that we don’t often learn from is that we don’t often learn from
experience alone!experience alone!
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PossibilitiesPossibilitiesSmooth functions
1 63 4 52
No. of zeros
No. of local extrema
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2
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Tangent PowerTangent Power
Imagine the graph of a smooth function fImagine the graph of a smooth function f The tangent power of the point P relative The tangent power of the point P relative
to f, to f, is the number of tangents to f through P is the number of tangents to f through P
What are the possible tangent powers, and where are they located?
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QuinticQuintic
Encounter: -inflection tangent
-sense of curve for large values of |
x|
-shift from single point to following tangent Invariance in the midst of change