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 y malla

Goma único

Eje mide aproximadamente 3 desde arco

Flexible GEL de amortiguación absorbe el choque por suave, cómodo pasos.

Trusstic estabilizador del mediopié ofrece maximo apoyo como tú paso.

DuoMax sistema de apoyo entrega mejorado soporte y estabilidad.

sintético / malla elástica superior, sintético forro superior, ComforDry plantilla, Ortholite plantilla...

Espacio Caña Trusstic, AHAR (talón) / soplado caucho (antepié) suela.

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

Zapato de running GEL-Evolution Tormenta 6 Rayo para mujer ASICS 6 Grafito/ Rayo/ Tormenta 9d7e582Updated 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.

Zapato de running GEL-Evolution Tormenta 6 Rayo para mujer ASICS 6 Grafito/ Rayo/ Tormenta 9d7e582

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 Bondi LTR HOKA ONE ONE para mujer Vientos alisios , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , Brooks Womens Levitate Azul / Menta / Plata Zapatillas de running NIKE Air Zoom Vomero 13 para mujer Racer Blue / Blue Tint Zapatillas de running NIKE Zooms Winflo 4 para mujer Ruido Aqua / Blue Force / Glacier , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal , Zapatillas para correr NIKE Air Huarache Run Ultra SE para mujer Lt Orewood Brn / Lt Orewood Brn-blur-black Grau / Beige / Gold Zapatillas de running NIKE Lunar Skyelux para mujer Explosión gris oscuro / blanco-antracita-rosa , NIKE Men s Flex 2017 RN Negro / Metálico Hematite / Tough Red Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul Zapato deportivo ASICS Gel-Cumulus 18 para mujer Diva Rosa / Plata / Coral , Calzado de running Pearl iZUMi para mujer Em Road H3 Gris / rosa de frambuesa Zapatillas de running Saucony Women s Breakthru 3 Azul / menta New Balance Women s w990v4 Running Shoe Gray / Castlerock Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Negro / Flash Coral / Negro , Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Wolf Grey / Black-cool Grey-pure Platinum NIKE Women s Free 5.0 Running Sneaker naranja Zapatillas de running Laguna Zoot para mujer Aguamarina / maracuyá / gris , Zapatillas de running NIKE Lunarglide 9 para mujer Gunsmoke / Solar Red / Hot Punch / Vast Grey Amortiguación 635v2 New Balance para mujer Negro Saucony Triumph ISO 3 Running Sneaker para mujer Gris morado , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Trueno / Blanco , Zapatillas de running W520v2 New Balance para mujer Púrpura / gris Zapatillas deportivas para hombre NIKE Lunar Apparent Carbono / Obsidiana ligero NIKE Men s Roshe Two Running Shoe Binario Azul / Lobo Blanco Gris Zapatillas de running adidas Ultraboost W para mujer Ice Blue / Ice Blue / Blue Night , Zapatillas de running adidas Cloudfoam Ultimate para mujer Orquídea Tinte / Orquídea Tinte / Aero Pink Amortiguación 460v2 New Balance para mujeres Gris

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 de mujer KEEN Rose Aluminio / gris neutro Sandalia Savannah de Ryka para mujer Beet Red / Hyper Pink , Sanda Verra Teva para mujer Mar de Bering , Brooks Womens Revel Blanco / antracita / negro , Sanda Verra Teva para mujer Surf Taupe Multi Zapatillas deportivas 574S Sport New Balance para hombre Coco tostado / blanco Under Armour Women s RailFit 1 Azulejo azul (300) / grafito Amortiguación 635v2 New Balance para mujer Negro Zapatillas para correr Nike Flex Experience RN 5 Antracita / Carmesí total / Negro / Blanco Zapato de running ASICS Gt-2000 5 para mujer Diva Blue / White / Aqua Splash Zapatillas de running NIKE Lunarglide 9 para mujer Negro / Blanco / Antracita , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Gris blanco , Sandalia Barrial Blowfish Mujer marrón , Zapatillas de running Kineta Relay para mujer Saucony Té de menta Zapatillas de running adidas Solar Boost para mujer Rosa / Plata / Blanco , CLARKS Sandalia plana Brizo Cady para mujer Coral Zapatillas de running adidas Alphabounce Em W para mujer Easy Coral / Sun Glow / Gray One , Bomba de plataforma FLY London Yadi718fly para mujer Gris (Concreto 014) Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Wolf Grey / Black-lava Glow , Zapatillas de running NIKE Flex Run 2018 para mujer Cool Gray / White , Trotters Mujer Simmy Mary Jane Flat Moho , BareTraps Kaylyn Sandalia para mujer Armada , Zapato de lona KEEN Sienna MJ para mujer Malaquita , Zapatillas deportivas Under Armour Xc Brigade Spike Azul negro adidas Crazytrain Pro W Cross Trainer para mujer Noble Ink / Cyber Gold / Super Purple , Salón de baile de mujeres Zapatillas Dance Salsa Práctica latina Zapatos 1670cEB Cómodo-Muy Fino 1.5 \ de 5] Cuero negro , Zapato ecuestre Federica Suede de Marvin K. de Aquatalia by Marvin K. Negro Jeffrey Campbell Womens Rylance tachonado arranque Black Box Silver Zapatillas Glamour de mujer verde de Patricia borgoña , Mephisto Michaela Bootie para mujer 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 Zapato de running GEL-Evolution Tormenta 6 Rayo para mujer ASICS 6 Grafito/ Rayo/ Tormenta 9d7e582 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