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Mathematical Tasks:Mathematical Tasks:educating awarenesseducating awareness

John Mason & Anne WatsonJohn Mason & Anne Watson

ToulouseToulouse

June 2010June 2010

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