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Mathematics, Origami and GeoGebra

Mathematics, Origami and GeoGebra. Shi-Pui Kwan Lecturer The Hong Kong Institute of Education. A model of teaching. T PCK. T PCK. Content (Mathematical) Knowledge Pedagogical Knowledge Technological Knowledge hands-on manipulative virtual manipulative. GeoGebra. Geometry + Algebra.

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Mathematics, Origami and GeoGebra

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  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 insidethe 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 • ……

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

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