teaching h2 further maths - math.nie.edu.sgmath.nie.edu.sg/pctoh/conics/conics_session1.pdf ·...

TEACHING H2 FURTHER MATHS:CONIC SECTIONS

IME20833, 10, 17, 24 Oct 20172.30 – 5.30 @ AST

Resources• http://math.nie.edu.sg/pctoh/conics.htm• Feel free to send suggestions to

peechoon.toh@nie.edu.sg

H2 F-MathsSyllabus, Extracted Sept 2017

How Should We Teach Conics?

http://spikedmath.com/517.html

Discovering Mathematics“Learning mathematics is more than just learning conceptsand skills. Equally important are the cognitive andmetacognitive process skills. These processes are learnedthrough carefully constructed learning experiences. … Forexample, to encourage students to be inquisitive, thelearning experiences include opportunities where studentsdiscover mathematical results on their own.”

- Extracted from 2016 Pre-U H2 Further Mathematics Syllabus

Polya’s Three Principles of Learning1. For efficient learning, the learner should discover by

himself as large a fraction of the material to be learnt asfeasible under the given circumstances.

2. For efficient learning, the learner should be interested inthe material to be learnt and find pleasure in the activityof learning.

3. Learning begins with action and perception, proceedsfrom thence to words and concepts, and should end indesirable mental habits.

Polya (1963), On Learning, Teaching and Learning Teaching, The American Mathematical Monthly, v70,605-619

Conics• Menaechmus (380 - 320 BC)• Euclid (325 – 265 BC)• Apollonius of Perga (262 – 190 BC)

• Treatise on Conic Sections• named the various curves

• Pappus of Alexandria (290 – 350 AD)• Focus-Directrix property

Apollonius

Frontipiece of Treatise on Conic Sections

Socratic philosopher Aristippus, when shipwrecked and cast ashore on the island of Rhodes, said “Let us be hopeful for I see the traces of Man.”

Generating the conic sections

Conics (named by Appollonius)• Parabole (Latin) - comparison, to

set alongside, literally to throw alongside

• Hyperbole (Latin) - excess, literally to throw beyond

• Ellipsis (Latin) meaning “omission” or “falling short”

• “…” • The angle between the plane and

the base, when compared to the side of the cone

Photo Credit: Wikipedia

Parabola

• Double meaning of “to throw”• In Chinese

Photo taken @ Vivocity

Galileo (1564-1642)• Showed that projectiles followed parabolic paths

Drawing a parabola

Source: http://mathdemos.org/mathdemos/conic_via_locus/

Real life demonstration:https://www.youtube.com/watch?v=BafkJt9KA8I

Parabolic Pulpits

Sidney Bandshell (British Columbia)

Creative Commons: Taken by Paul Hamilton

http://www.flash-lights.com/maglite/maglite-flashlights/Focusability.htmlhttp://www.flashlightuniversity.com/wp-content/uploads/2013/08/spot_vs_flood.jpg

Straits Times 26 September 2016

Do not believe Star Wars!