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.

Cuero y Textil

Importado

Sintético suela

Plataforma mide aproximadamente 0.5 pulgadas pulgadas pulgadas

Cuero Y parte superior tejida

EVA Plantilla para cojín

Merrell Air Cushion en el talón absorbe choque y agrega estabilidad

Neopreno para comodidad acolchada

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

Sandalia para mujer Wrap Terrell de Ari 11622 Wrap de Merrell Negro 6c89c7cUpdated 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.

Sandalia para mujer Wrap Terrell de Ari 11622 Wrap de Merrell Negro 6c89c7c

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

Chaco Women s Zx2 Classic Athletic Sandal Apex Gray Encaje de quimera de Columbia para mujer Zuraqua , Sandalia Uneek O2-w de mujer KEEN Negro Negro , OluKai Kulapa Kai Papaya / Negro , Sandalia KEEN Women s Uneek Slide-w Negro Negro CLARKS Womens Wave Grip Sage Nubuck Sandalia de cuña Rheta Sam Edelman para mujer Ante negro Sandalia Kea Vent mujer Columbia Iceberg, Fawn , Sandalia deportiva Chaco Zvolv 2 para mujer Eclipse estándar , OluKai Ohana Piña / Negro Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink , Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo , Chaco Women s ZX / 2 Yampa Sandal Pixel Weave Sandalia deportiva Tirra para mujer Teva Mar de Bering Earth Origins Sophie Platino Sandalia Archie Fisherman del Dr. Scholl Nubuck negro Sandalia Drew Angela para mujer Dusty Multi Metallic Wolky Comfort Jewel Estampado de flores blancas , Sandalia KEEN Clearwater CNX para mujer Celestial / Vapor , Sandalia Yucatan ECCO Mujer Orquídea , Sandalias Swiftwater Mesh de Crocs para mujer Negro Sanda Verra Teva para mujer Rayas modernas negro / blanco Sandalias de cuero KEEN Clearwater CNX para mujer Mineral azul / amarillo , Sandalia Uneek O2-w de mujer KEEN Acero Gris / Vapor , Sandalias de mujer con tirantes Peep Toe de Bordello by Pleaser para mujer Negro Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave Sandalia de mujer KEEN Rose Aluminio / gris neutro Sandalias Newport H2 de KEEN para mujer Raya Black ,

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 Air Zoom Pegasus 34 para mujer Sport Fuchsia / Lava Glow / Racer rosa / blanco , OluKai Ohana Pewter / Black Sandalia deportiva Kankitta de Vince Camuto para mujer Summer Cognac Combo Sandalia Yucatan ECCO Mujer Polvo negro , Zapatillas de running Lagetiva Wildcat Trail para mujer Gris / mantequilla , Salomon Women s X-Mission 3 W-w Teal Blue / Granny Green / Passion Purple , Zapatillas de trail running Saucony para mujer Xodus 6.0 Verde / rosa , Zapatillas de running New Balance 990V3 para mujer Gris , Zapatillas de running ASICS Gel-Kayano 23 para mujer Diva Blue / Silver / Aqua Splash , NIKE W Air Vapormax Flyknit C para mujer Ah8448-001 Atmósfera Gris / Blanco-blanco-caliente Punch , Zapatillas para caminar Terrex Voyager CW CP W de adidas Outdoor para mujer Gray Four / Black / Chalk White Crocs Flip Crocband unisex Armada Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco / negro / gris oscuro / verde azulado , Sandalias con tachuelas de cuero para mujer de Royou Yiuoer Sandalias con tira en T Beige Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Velcro Monterey Zapatillas de deporte Bay para hombre Caminando atlético Zapatos Negro , Sandalia Gigi de Naturalizer para mujer Negro Calzado de espuma fresca para mujer New Balance 1100V1 Blanco negro , NIKE Men s rsquo; s Zapatos casuales Darwin Zapatillas deportivas ligeras Confort Running Blanco adidas Baseline K Sneaker Blanco / Blanco / Coral Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Carbón Supra Vaider LC Sneaker Océano / blanco Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Negro Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Cuero negro de acción Sandalia Dolce Vita Hazle Slide para mujer Fuera de cuero blanco Zapatilla Ascot de hombre UGG Café exprés Stuart Weitzman Sandalia Rosita para mujer Dusty Pink Tpu Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Phantom / Gunsmoke-cumbre Blanco , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Brown 2 Duckfeet Sj & aelig; lland Desert Boot marrón , J. Adams Koko Muslo alto - Zapato de tacón alto - Precioso cordones sobre la bota de la rodilla Blue Denim

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 Sandalia para mujer Wrap Terrell de Ari 11622 Wrap de Merrell Negro 6c89c7c 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