Sketchpad 4 Sketches

(Updated 17 November 2005;

Go to Section C - Articles further down)

Michael de Villiers

http://mzone.mweb.co.za/residents/profmd/homepage.html

**IMPORTANT**:

1. To **view** and **dynamically manipulate** these *Sketchpad 4* sketches, requires a copy of *Sketchpad 4* or the demonstration copy of *Sketchpad 4*, which can be downloaded for free from:

http://www.keypress.com/sketchpad/sketchdemo.html

2. The sketches have all been zipped in BinHex 4.0 and requires WinZip or similar programme to unzip. (A free unzipping programme Stuffit Expander can be downloaded from www.aladdinsys.com/expander or use WinZip from http://www.winzip.com/ - also available for free at many other sites). After unzipping, and *Sketchpad* *4* or *Sketchpad 4 Demo* has been installed, the sketches can be opened from the File menu in the programme or by double clicking.

For ordering a full version of *Sketchpad 4* in Southern Africa, consult:

http://mzone.mweb.co.za/residents/profmd/homepage3.html

Some of the sketches below can be used as investigations while others are intended as mere demonstrations. Note that the sketches have been prepared on a Macintosh Powerbook, hence the screen-size and outlay may differ a little for IBM compatibles and other Macintoshes. If you experience any downloading problems, please let me know.

**A. SOME SAMPLE SKETCHES FROM MY BOOK "Rethinking Proof with Sketchpad 4", Key Curriculum Press, USA. **www.keypress.com

There are 9 Sketchpad sketches included in this sample (zipped to 20 K). They have been slightly adapted to be somewhat independent from the worksheets in the book. Go to page 2 of my Homepage for more information on my book.

http://mzone.mweb.co.za/residents/profmd/rethink2.zip

**B. SOME SAMPLE SKETCHES FROM "Let's Talk Mathematics!", a Grade 9 Textbook by E. Makae, M. Nhlapo, M. de Villiers, H. Glover, M. Shembe & M. Masimela. (2001). **(Available from Juta-Gariep Publishers, Tel: 021-7633500/3600; gariep@mweb.co.za or Juta-Gariep, PO Box 35890, Menlo Park 0102. Tel: (012) 362-5799 Fax: (012) 362-5744; or Juta, Woodlands House, 55 Wierda Rd. East, Wierda Valley, Sandton 2196; Tel: 011-2177200. Gjohannes@juta.co.za or Bndukumbini@juta.co.za. Website: http://www.juta.co.za).

This book follows an OBE problem-centered approach, in the sense, that interesting and challenging problems from the real world or mathematics are used to motivate and develop mathematical content from. The approach is also integrated in the sense that different mathematical content and processes like modelling, conjecturing, proof, etc. are involved in producing solutions to the sets of problems.

There are 7 *Sketchpad* sketches related to *Module 9: Water Supply*, which introduces the concepts perpendicular bisector, circumcircle, proof as a means of explanation, and maximisation and minimisation. These can be downloaded from:

http://mzone.mweb.co.za/residents/profmd/module9.zip

There are 9 *Sketchpad* sketches related to *Module 10: Best Positions & Shortest Routes*, which introduces the concepts angle bisector, incircle, proof as a means of explanation, conjecturing and generalisation. These can be downloaded from:

(Thanks to Kate Mackrell from Queen's University, Canada who kindly made the above conversions from GSP3 to GSP4!)

C. SKETCHES RELATED TO ARTICLES OR WORKSHOPS

The *Sketchpad* *4* sketches below refer to articles of mine, which have either been published or submitted/accepted for publication.

1. NEW!!! Feynman's Triangle: Some Feedback and More, an extension of some feedback in *The Mathematical Gazette*, 89 (514), March 2005, p. 107.

http://mysite.mweb.co.za/residents/profmd/feynman.zip

2. A dual to a BMO problem. *Mathematical Gazette*, March 2002, pp. 73-74. (2 KB).

http://mysite.mweb.co.za/residents/profmd/bmo.zip

3. A Sketchpad investigation of "Arithmetic Sequence" Parabola, AMESA 2003 Workshop.

http://mzone.mweb.co.za/residents/profmd/work.zip

4. All cubic polynomials are point symmetric. (Published in *ImstusNews*, 1989 & reworked for the AMESA *Teaching & Learning Mathematics* journal, May 2004).

http://mzone.mweb.co.za/residents/profmd/cubic.zip

5. The affine equivalence of cubic polynomials. *AMESA KZN Math Journal*, November 2003, pp. 5-10. (Click on appropriate tabs).

http://mzone.mweb.co.za/residents/profmd/cubic.zip

6. A generalisation of the nine-point circle and Euler line. (Article under preparation for *Pythagoras*).

7. Generalizations involving maltitudes. (Published in *Int. J. Math. Ed. Sci. Technol***.**, 1999, 30(4), 541-548.)

http://mzone.mweb.co.za/residents/profmd/maltitudes.zip

8. Clough's Conjecture: A Sketchpad Investigation. Workshop presented at AMESA 2004 in Potchefstroom and NCTM 2004, Philadelphia. (8 KB).

http://mysite.mweb.co.za/residents/profmd/clough.zip

9. An interesting property of a quadrilateral circumscribed around a circle. An excerpt from my *Some Adventures in Euclidean Geometry* book.

http://mysite.mweb.co.za/residents/profmd/circumquad.zip

D. READER INVESTIGATIONS from AMESA KZN Math Journal

1. Feb 2003, http://mzone.mweb.co.za/residents/profmd/reader03a.zip

(The problems can be downloaded from:

http://mzone.mweb.co.za/residents/profmd/reader03a.pdf)

E. PROBLEMS & CONJECTURES

1. Some (as yet) unpublished problems & conjectures from talk "Examples of experimental investigation & conjecturing with Sketchpad", MA Conference, Norwich, UK, April 2003.

http://mzone.mweb.co.za/residents/profmd/problem.zip

2. Prove that the circumcircle of the triangle formed by 3 tangents to a parabola always passes through the focus of the parabola.

http://mzone.mweb.co.za/residents/profmd/circumpara.zip

F. MISCELLANEOUS

1. NEW!!! A folding investigation producing an interesting result.

http://mysite.mweb.co.za/residents/profmd/fold.zip

2. An example of using the iteration facility of Sketchpad to create arithmetic sequences and series. (4 KB).

http://mysite.mweb.co.za/residents/profmd/arithmetic.zip

3. Dynamic Platonic solids by Kate Mackrell from Queen's University, Canada, who was inspired by the Cabri figures of Genevieve Tulloue.

http://mzone.mweb.co.za/residents/profmd/polyhedra.zip

4. Three introductory sketches to the affine transformation Shearing.

http://mzone.mweb.co.za/residents/profmd/shear.zip

5. Inspired by Escher's MetaMorphosis and other gradual deformations of tessellations to produce some of his art, I've recently started experimenting a little with Sketchpad with some ways of gradually and in a systematic way deforming the sides of the tile or tiles forming the tessellation. Here is one example I've called "*whirly-gig*", with the parameters controlling the deformation able to be animated to create a dynamic, "*rippling*" tessellation. The colours of the rainbow were used to show the gradual deformation of the one set of tiles into the next.

http://mzone.mweb.co.za/residents/profmd/whirly.zip

6. Conic Intersections. For those who have an interest in investigating various properties of conics, you may be interested in downloading a really useful (zipped) set of Sketchpad tools that you can use to construct intersections between straight objects and various conic sections.

http://www.keypress.com/sketchpad/downloads/sketches/Conic_Intersections.zip

7. Real Life Modelling. One can paste pictures of any object into *Sketchpad* and then for example use *Sketchpad*'s geometric tools to superimpose a geometric curve or shape over the picture or to show how a design was created. Here are two examples of respectively modelling the shapes of the Ridge Rd. Bridge, Durban and the Benjamin Franklin Bridge, Philadelphia, PA with parabolas and exponential functions (1.1 MB).

http://mysite.mweb.co.za/residents/profmd/bridges.zip

8. Conic Sections. These beautiful sketches by Gerrit Stols from the Math. Dept., UNISA show how the conics are obtained by slicing through a single or double cone, as well as the constant sum property of an ellipse and an application of Pascal's theorem showing that 5 points are sufficient to determine a conic. (4 KB).

http://mysite.mweb.co.za/residents/profmd/conics.zip

9. Advanced Constructions. This is a valuable 7 page pdf file by Michael Fox (mdfox@foxleam.freeserve.co.uk) from Leamington Spa, Warwickshire, UK, giving a description of advanced constructions with *Sketchpad*. It is useful for constructing and exploring properties of conics, involutions, harmonic polars, double elements of projectivities, etc. (100 KB).

http://mysite.mweb.co.za/residents/profmd/constructions.pdf