Mathematics, Origami and GeoGebra
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Shi-Pui Kwan
Lecturer
The Hong Kong Institute of Education
A model of teaching
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TPCK
TPCK
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 Content (Mathematical) Knowledge
 Pedagogical Knowledge
 Technological Knowledge
 hands-on
manipulative
 virtual manipulative
GeoGebra
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Geometry + Algebra
GeoGebraSpreadsheet
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GeoGebra3D
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Problem 1: Haga’s theorem
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The lower left hand corner C
is folded upward to touch
a point F on DE.
If DF=1/n of DE, what fraction
Is EN of DE?
(n = 2, 3, ……, 9)
Problem 2: mouhefanggai
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A mouhefanggai is formed by
cross sections which are
circumbscribing squares of the
circular cut sessions of a sphere.
How does it look like?
Without knowing the volume
formula for the sphere, how to
determine the volume of the
mouhefanggai?
my teaching notes – Haga’s theorem
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 origami simulation by GeoGebra
 some construction techniques
 ‘error free’ measurements
 guided discovery through measuring, tabulating,
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conjecturing and proving
Record to Spreadsheet  select ‘value’ in algebra
view  move slider ‘ denominator ’ to obtain data
observation: EF increases with DF
computation of fractional values
problem extension
my teaching notes – mouhefanggai
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 a solid formed from the circumscribing squares of
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the circular cut sections of a sphere
visualization of the mouhefanggai
questions for discussion
the ancient Chinese way in determining its volume
1/8 mouhefanggai and its complementary part inside
the circumscribing cube(r3)
introduction and visualization of Yangma (a square
pyramid)
problem extension
math proof outline: Haga’s theorem
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math proof outline: mouhefanggai (mhfg)
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ScreenCasts
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Learning in different styles
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 The cursor jumping from one spot to another during
the process somehow distracts me so that I cannot
concentrate on the picture
 and cannot easily exert my own imagination.
 I prefer to stare at a static picture and think about it,
 better yet, draw my own picture if that is possible.
- Prof. Siu M.K.
Student’s drawings
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GeoGebra
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 is a freeware
 is ‘error free’ (in compare with measurement)
 can generate lots of data (time saving)
 can help students in visualization (for spatial sense
development)
 is a useful tool in identifying the variants and the
invariants (particularly in learning geometry)
 is effective for problem extension
 provides insights in making conjectures (inductive
thinking)
GeoGebra
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 files are more long lasting than origami models
 construction is itself a learning process (more
suitable for secondary students)
 is a new learning/teaching tool which provides
students/teachers with various modes of lesson
delivery
 demonstration/illustration
 interaction/exploration
 project learning
 ……
Thank You!
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Thanks to the teachers & students involved
in the try-outs!
Thanks to the GeoGebra developers!
Thank You!