ISBN:
9789462098602
Language:
English
Pages:
Online-Ressource (VI, 342 p, online resource)
Series Statement:
Dutch Design in Mathematics Education
Series Statement:
Educational Research E-Books Online, Collection 2005-2017, ISBN: 9789004394001
Series Statement:
SpringerLink
Series Statement:
Bücher
Parallel Title:
Erscheint auch als Geometry with Applications and Proofs: Advanced Geometry for Senior High School, Student Text and Background Information
Keywords:
Geometry
;
Education
;
Education
Abstract:
Preliminary Material -- Geometry between application and proof, a general introduction /Aad Goddijn -- Geometry, classical topics and new applications /Martin Kindt -- Given: circle with butterfly or: how do you learn proving? /Aad Goddijn -- Distances, edges and domains -- Voronoi diagrams -- Reasoning with distances -- Computer practical Voronoi diagrams -- A special quadrilateral -- Exploring isodistance lines -- Shortest paths -- Example solutions -- Worksheets part I -- Thinking in circles and lines -- Using what you know -- The circle scrutinized -- Finding proofs -- Conjectures on screen -- Proving conjectures -- Clues for chapter 3 and 5 -- Conflict lines and reflections -- Preface -- Edge and conflict -- Parabola, ellipse and hyperbola -- Analytic geometry -- Conic sections.
Abstract:
This book shows how geometry can be learned by starting with real world problems which are solved by intuition, common sense reasoning and experiments. Gradually the more formal demands of mathematical proofs get their proper place and make it possible to explore new applications. This process helps students to feel the need for precise definitions and procedures, to contribute to the construction of an axiomatic system, and to experience the power of systematic reasoning. The course is designed for students in a Nature & Technology strand which prepares for studying the sciences or technology at university level. Its goal was basically to reintroduce ‘proof’ in a meaningful way in the late 1990s Dutch secondary education curriculum. Following the educational view of the Freudenthal Institute this is not done by stating Euclid’s axioms on page one, but rather a starting point is chosen in students’ intuitions and tentative solutions of problems that are experienced as real and relevant. The photograph on the cover shows students exploring one of the problems from the midpart of the course in the computerlab
Description / Table of Contents:
Contents; Geometry between application and proof, a general introduction; About this book; Geometry in Dutch education; Mathematical contents of the course; A short note on axioms and deduction; Dynamic geometry software; The aftermath of the Profi project; The authors; Geometry, classical topics and new applications; Modeling, abstracting, reasoning; The Dutch geometry curriculum; Geometry of Territories; How to prove 1 and 2?; The discrete parabola; Some conclusions; Literature; Given: circle with butterfly or: how do you learn proving?; What came before?; Form as tool; Heuristics
Description / Table of Contents:
RecognizingLearning to note; Find a link; ????????? Yesssssss!!!!!; Translating; Conjectures and Cabri (or Geogebra); Finally: teacher and student; Geometrical footnotes; Literature; Distances, edges and domains Advanced geometry, part I; Chapter 1: Voronoi diagrams; 1. In the desert; Summary of chapter 1; Preview; Extra exploration exercises: Recovering the centers; Chapter 2: Reasoning with distances; Introduction to this section; Starting-points: triangle inequality and Pythagoras; The perpendicular bisector; Perpendicular bisectors in the triangle; The circumscribed circle
Description / Table of Contents:
Chapter 3: Computer practical Voronoi diagramsSummary; Chapter 4: A special quadrilateral; Summary of chapter 4; overview of theorems in this chapter; Chapter 5: Exploring isodistance lines; Summary of chapter 5; Chapter 6: Shortest paths; shortest road length; meeting more lines on shortest routes; The principle of Fermat; billiards problems; Summary of chapter 6; Example solutions; Chapter 1: Voronoi diagrams; Chapter 2: Reasoning with distances and angles; Chapter 3: Computer practical Voronoi diagrams; Chapter 4: A special quadrilateral; Chapter 5: Exploring isodistance lines
Description / Table of Contents:
Chapter 6: shortest pathsWorksheets part I; worksheet A: Folding to Voronoi; worksheet B: Exact Voronoi diagram for the desert; worksheet C: map of the Netherlands; worksheet D: isodistance lines round a square); worksheet E: triangle, feet, sectors; worksheet F: an ant on a shoebox; worksheet G: triangles and mirrors; Thinking in circles and lines Advanced geometry, part II; Chapter 1: Using what you know; 1. In advance; 1. Finding arguments and writing down proofs; motivations, given; Chapter 2: The circle scrutinized; an application with angle bisector and perpendicular bisector
Description / Table of Contents:
Chapter 3: Finding proofslinks in short; 'splitting conditions' in short; finding proofs; preview; Chapter 4: Conjectures on screen; thinking up conjectures and investigating; movement with left traces; Chapter 5: Proving conjectures; Clues for chapter 3 and 5; Clues for chapter 3 Finding proofs; Clues for chapter 5 Proving conjectures; Conflict lines and reflections Advanced geometry, part III; Preface; Chapter 1: Edge and conflict; 1. Borders under water; Chapter 2: Parabola, ellipse and hyperbola; Chapter 3: Analytic geometry; from equation to figure
Description / Table of Contents:
the intercept form equation for a straight line
Note:
Description based upon print version of record
DOI:
10.1007/978-94-6209-860-2
URL:
Volltext
(lizenzpflichtig)
URL:
Volltext
(lizenzpflichtig)
Permalink
