Mathematics, Origami and GeoGebra

Shi-Pui Kwan

LecturerThe Hong Kong Institute of Education

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A model of teaching2

TPCK

TPCK

Content (Mathematical) KnowledgePedagogical KnowledgeTechnological Knowledge

hands-on manipulative virtual manipulative

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GeoGebra

Geometry + Algebra4

GeoGebraSpreadsheet5

GeoGebra3D6

Problem 1: Haga’s theorem

The lower left hand corner C is folded upward to toucha point F on DE.

If DF=1/n of DE, what fractionIs EN of DE?(n = 2, 3, ……, 9)

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Problem 2: mouhefanggai

A mouhefanggai is formed bycross sections which arecircumbscribing squares of thecircular cut sessions of a sphere.How does it look like?

Without knowing the volumeformula for the sphere, how todetermine the volume of themouhefanggai?

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origami simulation by GeoGebrasome construction techniques‘error free’ measurementsguided discovery through measuring, tabulating,

conjecturing and provingRecord to Spreadsheet select ‘value’ in

algebra view move slider ‘ denominator ’ to obtain data

observation: EF increases with DFcomputation of fractional valuesproblem extension

my teaching notes – Haga’s theorem9

my teaching notes – mouhefanggai

a solid formed from the circumscribing squares of the circular cut sections of a sphere

visualization of the mouhefanggaiquestions for discussionthe ancient Chinese way in determining its

volume1/8 mouhefanggai and its complementary part

inside the circumscribing cube(r3)introduction and visualization of Yangma (a

square pyramid)problem extension

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math proof outline: Haga’s theorem11

math proof outline: mouhefanggai (mhfg)

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ScreenCasts13

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.

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Student’s drawings 15

GeoGebra

is a freewareis ‘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 extensionprovides insights in making conjectures

(inductive thinking)

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GeoGebra

files are more long lasting than origami modelsconstruction 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

……

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Thank You!18

Thanks to the teachers & students involved in the try-outs!

Thanks to the GeoGebra developers!

Thank You!