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.

Sintético

sintético único

Eje mide aproximadamente 2.5 desde el arco

Malla superior

Atenúa choque durante el impacto fase y permite para una transición sin problemas a medio cuerpo.

A plantilla que se puede quitar para acomodar una ortesis médica.

Invertida las orejetas proporcionan cuesta arriba y tracción cuesta abajo en todos los tipos de terreno.

Acrónimo para ASICS High Abrasion Rubber. Colocado en crítico áreas de la suela por excepcional durabilidad.

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 Pixel para correr 14012 ASICS Gel-Venture Zapatillas 6 para mujer Negro/ Pixel Rosa a2849cbUpdated 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 Pixel para correr 14012 ASICS Gel-Venture Zapatillas 6 para mujer Negro/ Pixel Rosa a2849cb

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 Under Armour Speedform Slingshot para mujer - SS16 Negro / Rojo Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Volt / Blanco Zapatillas de running adidas Alphabounce Beyond W, para mujer Ecru Tint / Ash Pearl / Ash Pearl Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Dark Obsidian / Barely Volt , Zapatillas de running Newton Kismet II para mujer Negro / rosa , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow , Zapatillas de running NIKE Flyknit Racer NIKE para hombre Volt / Black-secoya Zapatillas de running New Balance Boracay V3 para mujer Plata Metálica / Trueno , Zapatillas de running Mizuno Wave para mujer Inspire 13 Plata / Azulejo Azul / Griffin Nike Hombres Air Zoom Pegasus 33 Wolf Gray / Blanco-cool gris , Zapatillas para correr Nike Flex Experience RN 5 Volt Black White 700 , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris , Brooks Womens Ravenna 9 Rosa / Plum / Champagne , Zapatillas de running Saucony para mujer Stretch N Go Breeze borgoña Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow Zapatillas de running Skechers Performance para mujer Go Run 400 Action Blanco negro , Zapatillas para correr Nike Women s Free Rn Distance 2 Hot Punch / Blanco polarizado azul Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco NIKE WMNS Zoom Fly Women 897821-411 Provence púrpura / negro , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , Zapatillas de running Vibram para mujer V Fiery Coral PUMA Men s Roma Basic Sneaker Blanco-gris violeta-cinta roja , Zapatillas de running Nike Tri Fusion Run para mujer Black White Pink Foil , Zapato de tenis ASICS Gel-Dedicate 5 para mujer Mónaco Azul / Blanco , NIKE Men s Flex 2017 RN Antracita / Clear Jade / Black / Blustery , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Blanco Zapatilla de deporte Nike Roshe One Hyp Br para mujer Gamma Blue / Gamma Blue-blue Lagoon , Zapatillas de running adidas Solar Boost para mujer Negro / Blanco / Rosa Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul ,

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 Uneek O2-w de mujer KEEN Aura Astral / Libertad , Zapatillas de running NIKE Juvenate para mujer Blanco negro Zapatillas de running NIKE Juvenate para mujer Armería Navy / Hot Punch / Blanco Wolky Comfort Jewel naranja , Wolky Comfort Jewel Black Croco Sandalia Omnium de mujer Teva Pizarra , Nike Men s Air Presto Essential Negro / Negro-blanco Zapato de running Mission XR para mujer Salomon Puntuación Azul / Muy Púrpura / Pop Verde , Zapatillas de running Topo Athletic Runventure Trail - Mujer Vino / gris , Deflector de gravedad Alivio del dolor comprobado Desafío G de las mujeres Mighty Walk - Mejor Zapatos para el talón Dolor, dolor en el pie y fascitis plantar Verde negro , Bota de tobillo Yama para mujer de FLY London Diesel Oil Suede , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul , Entrenador cruzado 611v1 de New Balance para mujeres Blanco negro Dr. Zen Jordan Comfort Profundidad extra terapéutica para mujer Cuero para calzado / malla con cordones Negro / Plata , New Balance Women s Wx715cw3 Carbón , Zapatillas de entrenamiento cruzado WOM s Tazon 6 Accent WN de PUMA para mujer Quarry-puma White-aruba Blue , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Blanco Blanco Vans Old Skool Unisex Adults Low-Top Trainers Negro (negro / blanco) , Zapatos de golf AsJei para mujer FootJoy Negro , Sandalia Bobbi Slide de Dolce Vita para mujer Ante amarillo , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Tinta azul marino Sandalia Dolce Vita Jene Slide para mujer Tela blanca a rayas Zapato de baile con esfera de gel y cohete para mujer ASICS 7 Onyx / Plata , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1682- Tan satinado , 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2824- Ivory Satin & White Glitter, tacón de aguja , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA7036 Tacón de 3 pulgadas + plegable Paquete de cepillo Patente negra , Bomba de vestido LifeStride Sevyn para mujer Denim oscuro Kenneth Cole New York Genevive Boot - Mujeres Tela Metálica / Multi , FRYE Botas de mujer Ray Grommet OTK Slouch Madera , Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Negro ,

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 Pixel para correr 14012 ASICS Gel-Venture Zapatillas 6 para mujer Negro/ Pixel Rosa a2849cb 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