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

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

I.G.S. (Orientación de impacto) Sistema) Tecnología: Filosofía de diseño ASICS que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.

ComforDry Plantilla: extraíble plantilla que proporciona amortiguación rendimiento y absorbe la humedad para un refrigerador, secador, un entorno más saludable.

SpEVA Material de la suela intermedia: Mejora el rebote características de la espalda y disminuye la entresuela Descompostura.

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 Zapato running/ ASICS Negro GT-1000 6 para mujer Negro/ Blanco/ Atlantis 49f3bf3Updated 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 Zapato running/ ASICS Negro GT-1000 6 para mujer Negro/ Blanco/ Atlantis 49f3bf3

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

Short NIKE Tempo para mujer Deporte Rojo / Blanco / Negro / Blanco Zapatillas de running Gel-Moya para mujer de ASICS Azul negro Nike Hombres Air Zoom Pegasus 33 Equipo Naranja / Blanco-negro-puro Platnum , Zapatillas de running NIKE Air Max Flair para hombre Blanco negro Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Gunmetal / Thunder NIKE Men s Roshe Two Running Shoe Industrial Blue White 402 , Zapatillas para correr NIKE Women s Air Zoom Structure 21 Gris oscuro / lobo blanco gris Saucony Triumph ISO 3 Running Sneaker para mujer Blanco / verde azulado / Citron , Zapatillas de running NIKE Women s Air Zoom Structure 20 Antracita / Lobo gris / Rosa fuego / Blanco , Under Armour Women s HOVR Sonic NC Gris Nublado (101) / Blanco Nike Men s Air Presto Essential Hueso claro / gris oscuro , Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / Begonia rosa Amortiguación 460v2 New Balance para mujeres Gris , Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Blanco Zapatillas para correr Hoka One One Stinson 3 ATR Fuchsia / Fusion Coral para mujer - Zapatillas de running adidas Alphabounce Em W para mujer Misterio Ruby / Bahia Magenta / Gray Five , Zapatillas de running Swift para mujer adidas Originals Hielo rosa / blanco / negro Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Blanco / Negro Calzado de running NIKE RN 2017 2017 para mujer Negro / Metálico Rojo Bronce , Zapatillas para correr W630V5 para mujer New Balance Rosado Zapatillas de running ST-2 para mujer Topo Athletic Azul blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Láser Naranja / Gamma Azul / Cool Grey / Black , Calzado de running para mujer New Balance Cypher Castlerock / White Zapatillas de running NIKE Air Vapormax Flyknit 2 para mujer Cactus negro / caliente Punch-blanco-polvoriento Zapatillas para correr Mizuno Women s Wave Creation 15 Blanco / Mar Rosa / Azul deslumbrante , Zapatillas NIKE Women s Free Run 2017 negro / azul de hidrógeno / gris oscuro-lobo gris Negro / Hidrógeno Azul-oscuro Gris-lobo Gris , New Balance Women s w990v4 Running Shoe Gris / rosa

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 NIKE Men s Benassi Just Do It Athletic Sandal Sepia Stone / Sepia Stone / Summit White , Sandalia deportiva Chaco Zvolv 2 para mujer Pino anidado Saucony Women s Ride 10 zapatillas para correr Blanco azul , Sandalia Dusty Sport mujer muy volátil Gris Salomon Xa Pro 3D GTX W Trail Runner Tahitian Tide / Eggshell Blue / Estanque Reflectante Zapato de running Mach de HOKA ONE ONE para mujer Blanco / Hibisco , Salomon Women s Speedcross 4 W Trail Runner Rojo tibetano / sangría / negro Zapatillas de running adidas Women s Cf Superflex W Mystery Ink / White / Energy Aqua Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco , Sandalia Vionic Unisex Wave Toe Post negro , Sandalia de puño de 2 piezas Cobb Hill Women s Hollywood Cuero negro , Sandalias Skechers Reggae Slim-Vacay para mujer Negro Sandalia estampada Benikei JDI de mujer Nike Piedra negra / adoquín Zapatilla Ascot de hombre UGG castaña Zapato de malla Nohea de OLUKAI - Hombre Negro Negro , Zapatilla para mujer Dr. Comfort Katy Verde y Turquesa Under Armour Threadborne Push para mujer Gris Nublado (102) / Rosa Tropic , Vans Unisex Old Skool Zapatillas clásicas de skate (Lienzo) Negro / Blanco verdadero Zapatillas de deporte Fila Philadelphia s Cage Delirium Dark Shadow, Pink Glo, Royal Blue PUMA Womens Mostro Glacier Grey / Glacier Grey / Puma White Sandalia Buxom Slide de Charles David para mujer Plata , Zapatos de danza del salón de baile de las señoras muy finas EK9691 con 1.3 \ Satén marrón claro Egos Copenhagen House Slip on Slippers Negro , Zapatilla para fumadores de la plataforma Mujer Soludos Broncearse Vionic Women s, Josie Low Heel Pumps Bronceado suave Muck Boot Tremont Wellie Tall Snow para mujeres Negro , Ellie Shoes Women s Juliet Platform Sandalia rojo UGG Mujer Alena mocasín Anochecer ED by Ellen Womens Langston - Botines de cuero con puntera cerrada Rosa Avalon Nappa TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Marine Blue Heritage Canvas ,

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 Zapato running/ ASICS Negro GT-1000 6 para mujer Negro/ Blanco/ Atlantis 49f3bf3 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