Mathematics, Origami and GeoGebra

1
Mathematics, Origami and GeoGebra
Shi-Pui Kwan Lecturer The Hong Kong Institute of
Education
2
A model of teaching
TPCK
3
TPCK

• Content (Mathematical) Knowledge
• Pedagogical Knowledge
• Technological Knowledge
• hands-on manipulative
• virtual manipulative

4
GeoGebra
Geometry Algebra
5
GeoGebraSpreadsheet
6
GeoGebra3D
7
Problem 1 Hagas theorem
The lower left hand corner C is folded upward to
touch a point F on DE. If DF1/n of DE, what
fraction Is EN of DE? (n 2, 3, , 9)
8
Problem 2 mouhefanggai
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?
9
my teaching notes Hagas theorem

• origami simulation by GeoGebra
• some construction techniques
• error free measurements
• guided discovery through measuring, tabulating,
  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

10
my teaching notes mouhefanggai

• a solid formed from the circumscribing squares of
  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

11
math proof outline Hagas theorem
12
math proof outline mouhefanggai (mhfg)
13
ScreenCasts
14
Learning in different styles

• 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.

15
Students drawings
16
GeoGebra

• 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)

17
GeoGebra

• 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

18
Thank You!
Thanks to the teachers students involved in
the try-outs! Thanks to the GeoGebra
developers! Thank You!