## Math 420: Investigations & Proof in Geometry

High School Geometry Revision & Self-Testing

a) Download free Grades 10-12 Mathematics PDF Textbooks for the South African curriculum or consult them online with embedded videos, simulations, powerpoint presentations, etc. at Everything Maths. Alternatively, access the following online texts specific to geometry:
Euclidean geometry
Introduction
Triangles
The mid-point theorem
Proofs and conjectures
Chapter summary
End of chapter exercises

Measurements
Area of a polygon
Right prisms and cylinders
Right pyramids, right cones and spheres
The effect of multiplying a dimension by a factor of k
Chapter summary
End of chapter exercises

b) Download the following chapters from a USA High School Geometry Textbook for revision:
Chap 1: Lines and Angles
Chap 2: Reasoning and Proof
Chap 3: Parallel and Perpendicular Lines
Chap 4: Triangles
Chap 5: Relationships in Triangles
Chap 6: Proportions and Similarity
Chap 7: Right Triangles and Trigonometry
Chap 9: Transformations
Chap 10: Circles
Chap 11: Area and Volume
Chap 12: Surface Area
Chap 13: Volume

d) For revision and self-testing of your school level geometry knowledge use the Self-Check Geometry Quizzes from Glencoe Mathematics. Tests are sorted by chapter & section, standardized or mixed tests, with test results that can be e-mailed to any person.

e) View or download a range of Free Geometry Video Lectures on high school geometry.

f) Download an older Geometry Chapter for Grade 12 with some circle geometry from the Free High School Science Text series (2008) - for the South African curriculum at the time.

Sketchpad 4 Demo and Workshop Guide

Rethinking Proof Sketchpad 4 Sketches and Further Investigations
Custom Tools
Rethink Chap 1
Rethink Chap 2
Rethink Chap 3
Rethink Chap 4
Rethink Chap 5
Balancing Weights on Vertices of Quadrilateral (No need to unzip)
Area proof for median concurrency, discovery and proof of Ceva's theorem(No need to unzip)

Visit the Sketchpad User Group site, Sketchpad SketchExchange, to download free Sketchpad sketches for various topics at school ranging from primary to high school and college. You can also join the site if you want to contribute, discuss or ask questions from other teachers and users in the Discussion Forum.

To help you better understand how to mathematically find the "best position" for the water reservoir for four villages if the four villages are not concyclic in Question 3 under the Further Exploration of the Water Supply activity in Rethinking Proof on p. 30:
1) look at the similar "least squares" technique for finding the straight line of "best fit" through a set of data points with this nice applet at Linear Regression
2) download this PDF paper on Straight Line Fitting which shows how the formulae for m and c can be derived
3) and have a look at this Linear Regression Exploration.

For some more challenging, geometry explorations on your own, download the free 2006 book (2.5 MB) Exploring Advanced Euclidean Geometry with Sketchpad by Gerard Venema, Dept. of Math & Stats, Calvin College, Michigan USA.

Download and print these 3D Worksheets, to build a model of a square pyramid which can be used to show how 3 of them can be assembled into a cube; hence explaining why the volume formula of a pyramid is 1/3 area base x height. (A template is on the first page). You can also download a folder with Cabri 3D sketches from Cabri 3D sketches 1 and Cabri 3D sketches 2.

Read the NCTM Standards for Geometry (2000) at Geometry Standards.

Read John Tabak's "History of Geometry: The Language of Space and Form" (2004) at History of Geometry(1.4 MB): specifically read the chapter on Non-Euclidean Geometry. Also read the Non-Euclidean Geometry chapter (582 KB) from the classic book "The Mathematical Experience" by David and Hersh (1981).

Read "Studies in Geometry Education" - UNESCO (1986) - 11.9 MB.

Read mathematician and Mathematical Gazette Editor, Gerry Leversha's talk at the Mathematical Association Meeting (2009) "The Importance of Teaching Geometry" 930 kb.

Read Scott Steketee's invited talk at the Asian Technology Conference in Mathematics (2010) 'Geometric Functions', which is about developing the function concept using dynamic geometry. 672 kb.

The NCTM Illuminations website contains a large list of useful Weblinks for online activities on Geometry at List of online geometry activities.

My book: Some Adventures in Euclidean Geometry 219 pp. @ USD 4.00 to download - printed copy also available @ USD 19.99- payment via PayPal or Credit Card. (This book seeks to actively involve the reader in the heuristic processes of conjecturing, discovering, formulating, classifying, defining, refuting, proving, etc. within the context of Euclidean geometry. Extensive attention is also given to the classification of the quadrilaterals from the symmetry of a side-angle duality. Many examples lend themselves excellently for exploration on computer with dynamic geometry programs such as Sketchpad.)

Visit my site Explorations for Students which has many interactive Java applets ranging in level of difficulty, from interesting investigations, demonstrations and simulations, to some which are likely to challenge your inventive faculties. In particular, look at the following investigations and activiities:
Classifying, Exploring, Constructing & Defining Quadrilaterals
Experimentally Finding the Median and Centroid of a Triangle
Generalizing the concepts of perpendicular bisectors, angle bisectors, medians and altitudes of a triangle to 3D
Some hexagon explorations
Miscellaneous Dynamic Transformations (of Geometric Figures & Graphs)

* The role and function of proof in mathematics (1990)
* The role and function of a hierarchical classification of quadrilaterals (1994)
* To teach definitions or teach to define? (1989)
* Some reflections on the Van Hiele theory for learning geometry (2010/12)
* The Role of Proof in Investigative, Computer-based Geometry: Some Personal Reflections (1997)
* Defining in Geometry - Chapter in 2009 NCTM Yearbook on Geometry
* Flashback to the Past: a 1949 Matric Geometry Question (In press)
Some general papers on modelling in mathematics:
* The role of technology in mathematical modelling (1994)
* Place Kicking Locus in Rugby (1999)
* Mathematical Applications, Modeling & Technology (2007)
Some recent papers on crossed quadrilaterals:
* Slaying a Geometrical Monster: Finding the Area of a Crossed Quadrilateral (2014)
* Poem: I have a dream ... (2014)
Many more papers on geometry, mathematics and mathematics education are available at my Homepage at: Downloadable Articles, Papers & Materials.

A 2006 research study on young children's experience of the dragging tool and their conceptualization of quadrilaterals is available at: Young students experience the dragging tool.

Read (or subscribe to) the latest: Math e-Newsletter with information about new math books, websites, conferences, quotes, humour, and challenging explorations for students.

Download Sketchpad sketches, Mathematica notebooks and graphing applet for Math 320: Differential Equations from: Differential Equations sketches and notebooks.

The July 2014 issue of the Learning & Teaching Mathematics journal of AMESA (Association for Mathematics Education for South Africa) is available at:
(click on pic below)
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