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.

Textil y sintético

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

Para el corredor necesita Estabilidad

Ventilador Wave Plate para máxima estabilidad y lo máximo amortiguación

Ajuste en movimiento superior

X10 suela para ultimate tracción

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 para correr Mizuno B06XJY9XDK Blue Women/ s Wave Paradox 4 Patriot Blue/ White d37b22eUpdated 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 para correr Mizuno B06XJY9XDK Blue Women/ s Wave Paradox 4 Patriot Blue/ White d37b22e

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

Brooks Womens Adrenaline GTS 18 Blanco / morado / gris Zapatillas de running Gobiv2 para mujer New Balance Poisonberry / Black , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco , Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello NIKE Women s Revolution 4 Running Shoe Blue Recall / Pure Platinum / Purple Slate , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Carbon / Black / Aruba Blue Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal Zapatillas para correr Nike Downshifter 6 para mujer Negro / antracita / blanco , MBT Shoes Mujeres GT 17 Athletic Shoe Cuero / Malla Con cordones Gris / morado , Calzado de running NIKE Flyknit Lunar 2 para mujer Rosado , Brooks Women s Dyad 9 Ombre Blue / Festival Fuchsia / Mood Indigo Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Solar Rojo / Negro / Mango Under Armour Women s Thrill 3 Lapis Blue / Mdn / Icelandic Rose , New Balance W880gb7 para mujeres Azul claro , Saucony Women s Guide 8 Running Shoe Blanco / Crepúsculo / Rosa Zapatillas de running NIKE Air Huarache para hombre Hyper Turquesa / Blanco Zapatillas de running Reebok Print 2.0 para mujer Negro / azul polar / peltre Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Gris oscuro / Azul Racer / Negro , Zapatillas de running Nike Kaishi para hombre Negro / Metálico Plateado-blanco , Zapatillas de trail running Topo Athletic MT2 - Hombre Azul / Mango , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Blanco Zapatillas de running NIKE RN 2017 para mujer gratis Platino blanco / negro puro , Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple , Calzado de running NIKE RN 2017 2017 para mujer Black / Hot Punch / Aurora Green , Zapatillas de running Mizuno Wave para mujer Wave Horizon Plata / rosa , Zapatillas de running Nike Air Huarache Run Ultra Br para mujer Crimson total / Crimson total Zapatillas de running adidas Ultraboost para mujer - SS18 Blanco , Zapatillas de running adidas Women s Edge Rc W Blanco / gris claro / 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 Zapatillas de running ASICS Gel-Cumulus 19 para mujer Blanco / negro / azul violeta Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave Earth Origins Sophie Women s Sandal Azul marino , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Gris Piedra / Carbono / Rosa Concha , Zapatillas de running adidas Springblade Razor para mujer Blanco , Salomon Sense Ride Running Shoe - Mujer Blue Bird / Deep Lagoon / Blazer azul marino , Reebok Women s Ridgerider 2.0 Trail Runner Lila Shadow / Coll. , Zapatillas de running Gel-Moya para mujer de ASICS Azul negro Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Gris oscuro / Metallic Platinum / Fire Pink , Zapatillas de running ASICS Gel-Kayano 23 para mujer Púrpura / Plata / Menta OTBT Nomadic Amarillo dorado Splendid Women s Brooklyn Sandal Plata , Sandalia Imogene Gladiador de Naturalizer Camelot , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Negro / ondulado / estiramiento , Zapatillas de tenis Wilson Rush Pro 2.5 para mujer Púrpura Toni Pons Womens Noa Multi Cordelia , Clásicos de base Slip-on (Tm) para hombres de Vans Carmesí / blanco , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Pergamino / blanco Sandalia Mila Slide de mujer Naturalizer Ante negro Sandalia de la diapositiva de China Girl My Girl Lollipop Red Suede , Sandalia Cato Dolce Vita para mujer Orchid Suede , Sandalia con tacón Florence Block de Via Spiga para mujer Patente de arena Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa 16612EB Cómodo-Muy Fino 2.5 \ de 5] Cuero negro y malla negra , 50 tonos de oro 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) S92309 Trenza dorada y plateada , Zapatos muy finos Dance Sneakers Series SN016 Rosado , Grishko 0509/2 2007 Pro-Flex , XYD mujeres Slingback medias gatito talón bombas punta estrecha hebillas zapatos de vestir con elegantes arcos Pasto verde Botín de ante TOMS Sevilla para mujer Dark Amber Suede / Volantes Zapatos OLUKAI Pehuea - Mujer Trench Blue Honu / Trench Blue Zapatos antideslizantes clásicos de lona de mujer de TOMS Caoba roja ,

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 para correr Mizuno B06XJY9XDK Blue Women/ s Wave Paradox 4 Patriot Blue/ White d37b22e 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