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

Zapato de para campo Kilkenny traviesa Kilkenny XC5 Blanco para mujer Saucony Rosa/ Blanco e33df7eUpdated 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.

Zapato de para campo Kilkenny traviesa Kilkenny XC5 Blanco para mujer Saucony Rosa/ Blanco e33df7e

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

Under Armour Fusible para mujer FST Negro (001) / Blanco Zapatillas de deporte Reebok Women s Studio Basics Blanco negro Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Vizi Orange / Citron Saucony Women s Showdown 4 Zapatillas de atletismo Azul rosado , Zapatillas Reebok Solestead para mujer Blanco negro Zapatillas de deporte Reebok Women s Studio Basics Orquídea ahumada / Ciruela lavada / Blanco , Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Berry | Zapatillas Adidas Unisex Performance Adizero Lj Track Blanco / Solar Rojo / Plata Metálica , Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Berry | Zapatillas de atletismo Spitfire 4 Saucony para mujer Púrpura / rosa , Zapato de atletismo PG-GT 4 Lite-Show PG para mujer ASICS Flash Coral / Plata / Negro , Zapatillas de running adidas Unisex Performance Adizero HJ con puntas Blanco / Infrarrojo / Plata Metálica Zapatillas para mujer Saucony Endorphin Racer 2 para mujer Púrpura / verde azulado Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Vizi Orange / Citron , Under Armour Fusible para mujer FST Acero (102) / Blanco Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Teal | Zapatillas Reebok Women s Plus Lite ti Track Ti - Whisper Gray / White / Guava Punch / Sour Melon / Lilac , Under Armour Fusible para mujer FST Negro (002) / Glacier Grey Zapato de campo traviesa Kilkenny XC5 para mujer Saucony Lavanda | , Zapatillas de correr planos Saucony Mujer Carrera XC Plata / vizi rosa Zapatillas de pista WSD400V1 de New Balance para mujer Púrpura Zapato de pista Md500v5 New Balance para mujer Negro / oro rosa Under Armour Fusible para mujer FST Flint (101) / Glacier Grey , Zapatillas de running Adidas Performance Adizero Prime SP con púas Blanco / Infrarrojo / Plata Metálica Under Armour Women s Drift 2 X MNSWR Raisin Red (500) / Beige Canvas Zapatillas Reebok Women s Plus Lite ti Track Lila Shadow / Lilac Glow / Melocotón Twist / Cobalto , Mujer adidas Sprintstar W Calzado blanco / gris ceniza / real púrpura Zapatillas adidas Performance Adizero Finesse Track de Adidas Blanco / Solar Rojo / Tech Silver Metallic Under Armour Fusible para mujer FST Negro (001) / Blanco

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 KEEN Clearwater CNX para mujer Negro / Resplandor Oasis Women s Zoey Fisherman Shoes Broncearse Zapatillas para correr Saucony Women s Triumph 11 Gris / morado Sandalia Drew Angela para mujer Beige-multi Sandalias EVA Gizeh de mujer Birkenstock Rosa Eva Zapato de agua NRS Crush - para mujeres Azul celeste , WHITE MOUNTAIN Moon Sandalia para mujer Cuero negro OLUKAI Womens Eheu Thong Sandal Estaño / carbón Zapatillas de running W1500v3 New Balance para mujer Lime Glow / Vivid Jade / Dark Denim Calzado de natación paso-naval Go Skechers Performance para mujer gris , Prop & eacute; t Olivia Oxford para mujeres Hueso La sandalia de gladiador FLEXX Women s Band Together Nubuck azul marino , Skechers Women s Define Sneaker Negro , Sandalia mondo volátil para mujer Natural New Balance WW813 Hook and Loop Walking para mujer Blanco , ASICS Womens Weldon X Gris medio / Carbono / Blanco , Vans Old Skool Unisex Adults Low-Top Trainers Blanco negro , Zapatillas de trail running Salomon Speedcross 4 Nocturne GTX - Mujer Tarde azul / plata reflexiva / jugo de uva , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Cumbre Blanca / Obsidiana , Sandalia de mujer Sayuri Slide de Musse & Cloud para mujer Coral Sandalia plana Dionne de Easy Street para mujer Pewter Metal , Zapatillas de salón Yvette para mujeres de Bloch Natural , Dakota femenina de UGG castaña Tangas Birkenstock Azul , Bota original Chelsea Wellington Boot para mujer (WFS1017RMA) rojo , Adtec - Bota de trabajo marrón de 9 \ marrón Zapatillas de deporte sin mangas para mujer de bajo perfil de Mónaco Jiffy Black Ct / Shell White Tenis Grandpro de Cole Haan para mujeres Nubuck de arena brasileño / Flores de cuero de arena brasileña / Blanco óptico Botín Naomi para mujer de Steve Madden Ante negro Ballet Tali de mujer Cole Haan Bow Bow Brillo de oro / plata

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 Zapato de para campo Kilkenny traviesa Kilkenny XC5 Blanco para mujer Saucony Rosa/ Blanco e33df7e 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