These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.

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Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015

Vince mujer Camuto Sandalias Vince Ellanna Ellanna Slide para mujer Rojo cereza 5338950Updated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.

The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.

This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.

Includes GSP-3 files

Includes GSP-4 files

Includes GSP-5 files
Contents:

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.

Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.

Vince mujer Camuto Sandalias Vince Ellanna Ellanna Slide para mujer Rojo cereza 5338950

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

J Slides Sandalia de mujer Naomi Slide Oro rosa , Sandalias planas para piscina Walking Cradles para mujer Envoltura Metálica Multi / Cork , kensie Sandalia Dominic Slide para mujer Negro CLARKS Sandalia Leisa Lacole Slide para mujer Peltre Metálico Cuero , Sandalias Fifi19 Slide de Donald J Pliner para mujer Plata Sam Edelman Women s Slide Sandalia Red Kid Suede Leather Sandalia Betsey Johnson Misty Slide para mujer Rojo / Multi Zapatilla Ascot de hombre UGG castaña , Prop & eacute; t Womens Aurora Slide Rubí Sandalias de plataforma deslizante con plataforma Forella de Gentle Souls para mujer Plata , Sandalias Sandalias Bashan moldeadas Sandalias Sandalias Jerusalem Broncearse Sandalia Mason para mujer Naturalizer Slide Cuero metalizado dorado claro , Sandalia Dolce Vita Delana Slide para mujer Cuero rojo , Sandalia de cuero cosida SAS para mujer Negro , Athena Alexander Womens Soraya Negro , Sam Edelman Women s Bay Slide Sandal Peltre Metálico Cuero , Sandalia de la diapositiva de Florene de las mujeres de Lucky Brand Denim ligero Sandalias Hush Puppies para mujer Dachshund Slide Cuero marrón , Kensie Sandalia para mujer Digby Slide Blanco , UGG Mujer W Joni Niebla de Lavanda , Stuart Weitzman Landslid femenino Jeans ante , Sandalia de tobogán plano de piscina Iona de Gentle Souls Laguna , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro , Sam Edelman sandalia de mujer Sera Slide Cuero metálico yute Sandalia de diapositivas de ocio para mujer de Seychelles Azul Sandalia de mujer Hana Slide de Via Spiga Cuero negro , UGG Mujer W Joni Ocre suave Sandalia Kai Dolce Vita para mujer Cuero Caramel ,

trigonometry, and probability. Fair warning: The discussion revolves about
a figure that appears only when the sketch is animated. Without Geometer’s
Sketchpad, this one will make little sense at all.
Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.

The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.

Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.

Inversion Geometry Sandalia para mujer Terrell Ari Wrap Sport de Merrell Negro Sandalia Yucatan ECCO Mujer Biscaya , Skechers Women s Go MEB Razor 2 Turquesa , Teva Winsted Solid - 1017425RASP Rosado , Zapatillas de running Saucony para mujer Peregrine 8 Vino / Playa , Zapato para caminar con cordones Go Step de Skechers Performance para mujer Rosado Sandalia de mujer UGG Holly Cerlse Zapatillas de running New Balance 990V3 para mujer Negro Skechers Women s Walk Walk Lite-Daisy Loafer Gris pardo , Sandalias de tacón de soñador Easy Street para mujer Broncearse , Zapatillas running para mujer Saucony Liteform para mujer borgoña Earth Origins Hadley Azul cobalto , Sandalia universal W Ysidro de mujer Teva Gris pardo , Inov-8 Mujer F-Lite 195 V2 (W) Cross Trainer Gris oscuro Zapato para caminar WW577 para mujer New Balance Blanco New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Silver Mink / Dusted Peach Zapatillas de entrenamiento CUSH + WX577V4 New Balance para mujer Negro / Heather , Zapatillas de tenis Clay Court 996 V3 para mujer New Balance Deep Ozone Blue / Ozone Blue , Clásicos de base Slip-on (Tm) para hombres de Vans Negro / Marshmallow , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / de peltre Sandalia Ryana Marilyn Slide para mujer Blanco negro Sandalia de tacón Alto Posey de Tahari para mujer Gris pardo Sandalia Heather Slide de mujer Vía Spiga Ladrillo de gamuza Bota de montar de Donnily de Calvin Klein para mujer Ante negro 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 7013- Satén gris plateado , CLARKS Bomba de cuña Flores tulipán para mujer Ante azul marino , Arcopedico 1361 Shocks 5 Botas para mujer Negro 055398210 Timberland PRO - Botas de trabajo TiTAN para mujer - Café marrón , Zapatilla de lona lavada Sayel Clew de Sperry Top-Sider para mujer Blanco Propet Women s Tour Walker Strap Sneaker Blanco / Berry

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.

Linkages

Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.

On the Square

The challenge: Given four arbitrary, unique points, *A*, *B*,
*C*, and *D*, construct one line through each point such that the four lines form a square.

Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.

The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.

Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.

Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.

Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.

The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.

Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.

Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.

Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured Vince mujer Camuto Sandalias Vince Ellanna Ellanna Slide para mujer Rojo cereza 5338950 it out, and I got a chance to try out this new graphics software.

Resources

The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge
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and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

For email to reach me, the word