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: Haga’s theorem
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)

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 – Haga’s 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: Haga’s 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. Student’s 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 …… S’

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