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.

Importado

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

Amortiguación 635v2 New para Balance B0784712C4 635v2 para mujer Negro 382ca47Updated 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.

Amortiguación 635v2 New para Balance B0784712C4 635v2 para mujer Negro 382ca47

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

Zapatillas de running NIKE Air Max Flair para hombre Obsidiana / negro-obsidiana Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Gamma Blue / White / Rio Teal , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow Zapatillas de running Reebok Zprint Her MTM Wvn para mujer Blanco negro Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Blanco Zapatillas de running adidas Alphabounce Rc W para mujer Gris claro Heather / Light Solid Gray / Medium Solid Gray , Zapatillas para correr Brooks Cascadia 12 para mujer Baja Azul / Paradise Pink / Clearwater Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow Zapatillas de running ASICS Gel-Kahana 8 para mujer Negro / Verde hielo / Naranja caliente Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Pale Gray / Pale Gray-dark Gray , Zapatillas de running PUMA Women s Speed 500 Ignite Tobillo-Alta Blanco / Rosa Go / Plata , Zapatillas de running AMPLICA para mujer ASICS Negro / Negro / Blanco Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Supercell / Sunrise Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Primer gris / verde azulado victoria / plata , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / antracita , Saucony Women s Ride 10 Running Shoe Gunmetal / rosa Zapato de running ASICS Gel-Quantum 360 cm para mujer Gecko negro / blanco / verde Pantalón largo clásico NIKE Golf para mujer Tech Essentials Marina de medianoche , Zapatillas de running ASICS Gel-Contend 4 para mujer Diva Azul / Plata / Orquídea , Zapatillas de running ASICS Gel-Kayano 23 para mujer Poseidón / Plata / Cacatúa , Amortiguación 635v2 New Balance para mujer Plata Zapatos de running Flyknit de movimiento libre para mujer de Nike, negro / blanco - 9 B (M) US - , Zapatillas de running Saucony Women s Triumph ISO 2 Coral / Plata NIKE Womens Air Zoom Wildhorse 4 Running Shoe 880566-007 Negro / Volt Green Abyss / Menta-obsidian-vapor Green , Calzado de running ft. Perpetua ft de memoria Fila Women s Castlerock / Sugarplum / White Zapatillas para correr Go Run Forza Boston 2016 de Skechers Performance para mujer Blanco , Adicción a las mujeres de Brooks 13 Ébano / Plata / Pagoda Azul , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de running Kineta Relay para mujer Saucony Cianotipo ,

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 Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Verde Palm / Negro / Vapor Verde , Sandalia Yucatan ECCO Mujer Negro , Vibram FiveFingers Mujeres Vi-S Zapatos descalzos y paquete de Toesock Negro , Zapatillas de running Reebok Print 2.0 para mujer Azul polar / Gable gris / Niebla / Verde menta / Blanco / Plata , Zapatillas de running PUMA para mujer Ignite Omphalodes / Coral caliente / Ultramar Sandalia KEEN Ana Cortez W para mujer Brisk / Espresso , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Mica Blue / White / Racer Pink / Sport Fuchsia , Camiseta adidas Swift Run W para mujer Core Black / Core Black / White , Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima , Zapatillas de senderismo acuáticas para mujer Astral TR1 Trek Deep Water Navy Zapatillas para caminar Propell Mujer Travellite MJ Plata , Zapatillas de running NIKE Air Max Thea para mujer Rosa / Fucsia Zapatillas para caminar Go Walk 4 Super Sock 4 de Skechers Performance para mujer Gris pardo , Skechers Women s Go 4-14918 Calzado para caminar Azul marino / gris Vionic Rest Farra - Sandalias de apoyo para mujeres Patente negra Brooks Women s Adrenaline GTS 17 Negro / antracita Vionic Women s de Orthaheel, Shore Slide Sandal Bronce Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja Drew Womens Ariana Dusty Gold , adidas Athletics 24/7 Tr M Cross Trainer Acero sin Procesar / Acero Crudo / Núcleo Negro , Zapatillas Slip-on de mujer Skechers You Inspire Armada , Botas de mujer PUMA x Rihanna Sneaker Puma Blanco / Puma Negro / Puma Blanco , Zapato de tenis 7 de ASICS Gel-Resolution femenino Aqua Splash / Blanco / Diva Blue , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro sin blanco , Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Tela blanca a rayas , Zapato de voleibol Boost W Volley Response 2 de adidas Performance para mujer Negro / Blanco / Plata Metálica , 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 , UGG Mujer Ansley mocasín Crema L.B. castaña El dólar de las mujeres de Eastland Taupe Suede

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 Amortiguación 635v2 New para Balance B0784712C4 635v2 para mujer Negro 382ca47 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