Polygons triangles and capes Aad Goddijn FIsme TSG
- Slides: 32
Polygons, triangles and capes Aad Goddijn (FIsme) TSG 34 (FIsme) Freudenthal Institute for Science and Mathematics Education Utrecht University, The Netherlands, A. Goddijn@fi. uu. nl 1
A design for a one day team task • Educational setting • Less known Mathematical content • Design process 2
Protruding vertex In - Polygons, triangles and capes Simple polygon: No selfintersections! ! ! diagonal 3
Situation: Math – B day • • Senior Highschool A-lympiad, Math B-day teams of 4 students, no teacher interference One day; 9. 00 – 16. 00 hrs, 27/11/2007. – from initial problem exploration to final essay • Competition + Regular school task • Numbers: (why ? ? , PA) – 180 schools, each 6 -10 teams of 4, about 5000 students (out of 10000) – 118 teams go for competition – 10 winning teams, get a price * *sponsored by TEXAS INSTRUMENTS* 4
Design ‘constraints’ and ideals for the Math B-day assignment • No try-out possible – competition …. . • Teams of students – Debate, argumentation – division of labour • Safe start and adventurious open ending – Regular task + competition; full day – do not send them home in tears • Mathematics as a ‘research’ activity, rounded of by written essay – Guided Reinvention 5
The assignment for november 23, 2007 • 15 pages A 4 • How was it created? 6
Design process timeline • Design team: – 4 Teachers, 1 research mathematician, 2 professional edu-math designers • One-day (24 hours) session of the design team – Wild idea, selecting, exploring, and more • Realisation by 2 team members • Finishing touches by everbody 7
Wild idea • Polyeders of ’s only – Count number of p’s with 10 triangular faces! – Switching edges: Do you get them all by switching? 8
From wild to focus • Debate : – – Exciting! Topology, we never do at school Half-way problems difficult to find Too far away from regular curriculum • Rejected, but: – keep polygons + ’s on board – Related sourcematerial was present in meeting 9
Design team explores mathematics of polygons • Web-source Computational geometry (Ian Garton) – triangulations of polygons stay in center – (Art gallery theorem as a possible application) • Unknown problems for most members! • Design team explores new mathematics like students should do 10
Inspiring van Gogh-like headings at Ian Garton‘s website Ear = Cape 11
Application field: Computational geometry Proofs are not existen ce proofs but Explicit me thods (algo rithms) to fi ‘something nd ’ within a ce rtain time Math B-day ’s choice: - no algorith mic order e stimates 12
End of the day: • Basic exploration of main problems of the subject ready • Rough outline of content • Results : – – possible and impossible problems and solutions, obstacles located tried-out survival hints additional (new) problems • 2 designers promise to write a first version 13
Overview of the assignment • A: Introduction material – definitions, sum of inner angles of simple polygon • C: Basic part: the triangulation theorem – each simple polygon with n sides can be decomposed in n-2 triangles (… a triangulation) • D: Optional part : convex polygons – What is the number of different possible triangulations of a convex n-gon? • (F: Difficult part: The Art Gallery Theorem) – Each simple polygon with n vertices can be guarded by p cameras that are mounted in the vertices, where p is te smallest whole number for which p > n/3 - 1 14
A: Introduction to the field total angular sum of n-gon. Students work: 15
Can’t they do shorter? • Probably! • But: To be handed in: Make a continuous report of your work that is easy to read and can be understood by your fellow students even those who have never heard of simple polygons. 16
A(2): Introduction to several concepts Construct all kinds of (counter)examples of m u im n i m a s. e s l i g e n Ther ruding a t o r p 3 y! h w Show 17
C: Reinventing the triangulation theorem and a proof Raising doubts about (n-2) 180 A B C Triangulation theorem (to be proven) for each simple polygon with n vertices there is a triangulation with n - 2 triangles 18
Divide and conquer; a hint! di l na o ag Triangulatable q-sided Triangulatable p-sided 19
• Part of the design team: proof is ok. • Prediction: Most students will not ask: Is there always a diagonal? • But probably they see the point of it and pick up a suggestion. 20
Diagonals exist always. Proof: Find a protruding vertex. A Cape or Not a cape (Students work) diagonals Next question: is there always a protruding vertex ? 21
Guided(? ) reinvention, remarks about • Strong hint about main idea of proof – Main point first: divide and conquer – Students keep overview on the whole story • Proof unrolls in reverse order: – Div+conquer ? ? Diagonal ? ? Protruding vertex ? ? Yes! • In contrast with (axiomatic) deduction: – Polygons -> protruding vertex -> diagonal exist -> divide and conquer -> induction -> theorem -> QED. • Local deduction (Freudenthal) • Explicit proofs rather late in Dutch curriculum, this can be first time – Students show many different levels of retelling the proof-story • Design team explored possible students behavior 22
D: Counting triangulations in the CONVEX case (optional, but recommended) • Problem formulated during design-day: How many different triangulations are possible in a convex 5, 6, 7 -gon? • Initial design motivation: There should be accessible problems for everyone • [Name of this number: Tn ] 23
Student work in progress; T 6 = 14 ? Yes! 24
Design team at mathwork again • T 5 = 5, T 6 = 14 • Find T 7 from T 6. • General: – find Tn from earlier Tm’s. • (there is a convolution - recursion formula) • (numbers are the so-called Catalan-numbers) 25
Divide and conquer, first attempt in design team How many triangulations are there with the red LINE included? 5 * 14 14 55 (9 -sided) 14 5 A dead end: no clear way to go on. 26
This Halfway-question is a better hint! How many triangulations are there with the gray triangle included? (10 -sided) 27
Final splitting up comes in many different representations (1) ? 28
And on many levels ( 2) 8 -gon 9 -gon 29
Final formula found in many styles (3) Difficult spots in process 30
Final formula found in many styles (4) 31
Final conclusion • the hard core of this design process was the mathematical activity of the design team • which is a kind of reinventing of problems and solutions • The resulting guided reinvention in the task – – Prevents some possible disasters helped students exploring a route in the problem field On many different levels … And did not steer them totally from 0 to 100. 32
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