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.

Nylon

La parte superior está hecho de una mezcla de peso ligero y muy transpirable materiales.

El Suela de Phylon con un Waffle inspirado el diseño hace para excelente tracción.

Longitud completa Entresuela de Phylon con un Solarsoft suministros de plantilla confort premium.

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

Zapatillas de running NIKE Roshe One B017N2SZ5Y azul-antracita-cool Print para 19755 tobillo y tela para mujer Negro/ Brgd azul-antracita-cool gris ffd5f7dUpdated 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.

Zapatillas de running NIKE Roshe One B017N2SZ5Y azul-antracita-cool Print para 19755 tobillo y tela para mujer Negro/ Brgd azul-antracita-cool gris ffd5f7d

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 New Balance 1500v3 para mujer Porcelana / Alpha Pink , PUMA Men s Roma Basic Sneaker Blanco / Peacoat / avena , Zapatillas de running NIKE Air Max Flair para hombre Piedra Pómez / Legión Verde Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Brt Mng / Wlf Gray-pch Crm-plst P Zapatillas de running NIKE Lunar Skyelux para mujer Golpe caliente / Marrón nocturno , PUMA Men s Roma Basic Sneaker Blanco / gris claro NIKE Men s Roshe Two Running Shoe Midnight Navy / Black-sail-volt NIKE Mens N98 Tribute Track Jacket Blanco negro Zapatillas NIKE para mujer Air Zoom Mariah FK Racer PRM Running , Short NIKE Tempo para mujer Dinastía púrpura / Turf Naranja / Gris lobo , Zapatillas de running adidas Women s Edge Rc W Gris / Granito / Coral Fácil Zapatillas de running Saucony PowerGrid para mujeres Gris / Coral Zapatillas de running ASICS Gel-Contend 4 para mujer Gris Medio / Mar Glaciar / Plata Skechers Performance Women s Go Run 400 Obstruct Gris / azul , Zapatillas de golf adidas W CC Ballerina II para mujer Clear Aqua / Running blanco / plata mate Zapatillas para correr ASICS Gel-DS Trainer 22 para mujer Acuario / Aqua Splash / Flash Coral , Short NIKE Tempo para mujer Blk / Blk / Vivid Sky / Wolf Gr , Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird PUMA Men s Roma Basic Sneaker Negro / Blanco / Plata Under Armour Women s Drift 2 Sigilo gris (008) / rosa islandesa , Brooks Womens Hyperion Negro / Dewberry / Aruba Blue Zapatillas de running adidas Women s Cf Racer Tr W, mujer Gray Two / Gray Two / Black , Brooks Womens Ghost 9 Silver / Blue Atoll / Lime Punch , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris Zapatillas de running TOPO FLI-Lyte 2 para mujer Amarillo negro Zapatillas de running NIKE Men s Free 5.0 Tr Fit 5 Fabric Brght Crmsn / Prm Rd-atmc Pnk-wh , Brooks Women s Adrenaline GTS 17 Carbón metálico / negro / oro rosa Short NIKE Tempo para mujer Armada , Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell

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 New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado , Mujer adidas Originals Ultraboost St Negro / Plata Metálico / Carbono Calzado deportivo Inov-8 Roclite 315 para mujer TEAL / NEGRO , Zapatillas de running adidas Alphabounce Cr W, mujer Legacy / Legacy / Ash Pearl Zapatillas de deporte Onitsuka Tiger Mexico 66 Karamari / Sumi , Zapatillas para correr Saucony Women s Cohesion 10 Gray Coral , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro , Calzado de running NIKE Zoom All Out Low para hombre NIKE Crimson total / Crimson total , Zapatillas de running NIKE Zoom All Out Low para hombre Negro / gris oscuro / antracita Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Navy , Propet Walker de la estabilidad de las mujeres Blanco deporte Prop & eacute; t Zapatillas para caminar TravelFit para mujer Verde , Calzado de senderismo Go Joy 15603 de Skechers para mujer Negro / gris , Sandalia Maya Miz Mooz para mujer Negro , Saucony Triumph ISO 3 Running Sneaker para mujer Púrpura / azul / Citron Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Black Suede / White Mesh , SAS Womens Sunny Azul profundo Mercado de Tierra Santa Adultos unisex / niños piel genuina Sandalias Bíblicas / Flip fracasos (Jesús - Yashua) Jesús - Estilo Yashua II Camel Trademark Zapatillas de deporte Xelero Matrix Comfort Therapeutic Extra Deep Sneaker Shoe Leather con cordones Negro Zapato de entrenamiento New Balance Unisex-Adult UX200V1 Tóxico Stuart Weitzman Sandalia Rosita para mujer Tpu blanco Sandalia Cato Dolce Vita para mujer Cuero metalizado / azul Sandalias planas Nori de Easy Street para mujer Pewter Metallic BareTraps Women s Cherilyn Slide Sandal Blanco , ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ S9206 Vegan Copper , Así que Danca CH99 3 Heel con T-Strap Character Shoe Caramelo , Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Poliuretano negro , Zapatilla de zapatilla de zapatilla 100% auténtica de lujo de las señoras de Snugrugs con suela suave ciruela , Wedge del desierto de las mujeres de TOMS Black Suede W / Shearling B.O.C - Mujeres - Peggy Chocolate engrasado ,

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 Zapatillas de running NIKE Roshe One B017N2SZ5Y azul-antracita-cool Print para 19755 tobillo y tela para mujer Negro/ Brgd azul-antracita-cool gris ffd5f7d 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