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.

Malla

Importada

Caucho suela

Malla superior para una óptima transpirabilidad

Suave entresuela de espuma ofrece liviano, sensible amortiguación

Goma ofertas de suela excelente tracción

Underlays en el vampiro y punta de punta oferta de apoyo y estructura

Horizontal fuelles son cortados en la espuma para respuesta agregada

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 NIKE B06XPQQP8Q/ Women blanco s Revolution 3 para correr Gris oscuro/ blanco/ negro 13039b4Updated 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 NIKE B06XPQQP8Q/ Women blanco s Revolution 3 para correr Gris oscuro/ blanco/ negro 13039b4

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 deportivas 574S Sport New Balance para hombre Driftwood / White Zapatillas de running Adidas Originals Adizero Prime Parley Carbono, Blue Spirit S, Ftwr Blanco Zapatillas de running NIKE Roshe Two para mujer Blanco / Blanco-Puro Platino , Zapatillas de running ASICS Gel-Scram 4 para mujer Negro / gris oscuro Under Armour Women s RailFit 1 Negro (001) / Blanco , Adidas Women s Edge Lux 2 W Tinte Orquídea / Plata Metálico / Gris Sólido Oscuro Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Nobink / Core Black , Sandalias OLKAKAI Ohana - Hombre Negro NIKE Men s Lunar Fingertrap Cross Trainer Antracita negra Topo ST-2 para mujer, zapatos para correr y ejercicio, banda para la cabeza, paquete Frambuesa negro , Zapatillas Drew Shoe para mujer Paige Cuero blanco Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco negro Zapatillas NIKE Women s Revolution 3 para correr Gris oscuro / blanco / negro , Under Armour Women s Charged Rebel Black (004) / Purple Rave NIKE Men s Lunar Fingertrap Cross Trainer Wolf Gray / Midnight Navy , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco / Blanco / Verde , Zapatillas para correr Under Armour UA Flow Criss Cross para mujer Acero , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Mono negro , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Hot Punch / Volt-total Orange , Zapato de running ASICS GEL-Venture 5 para mujer Gris plateado / turquesa / sacador de lima NIKE Mujeres Air Zoom Pegasus 33 Negro / gris frío / gris lobo / blanco , Zapatillas de running New Balance Coast V3 para mujer Negro / Imán Roxy Women s Set Session Zapatillas atléticas para caminar Oro rosa , Zapatillas de deporte casual Gazelle de adidas para hombre Negro / Negro / Oro Metálico Zapatillas deportivas 574S Sport New Balance para hombre Azul marino / Vintage Indigo Zapatillas de deporte casual Raleigh Mid W de adidas Neo para mujer Blanco / Blanco / Blanco , Zapatillas de running Kinvara 7 Saucony para mujer Verde , Zapato de running ASICS GEL-Kayano 22 para mujer Onyx / Plata / Carbón de leña Zapatillas de running Questar Tnd W, para mujer, de adidas Gris crudo / blanco / azul Aero

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 deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Negro / Laguna Abismo de pulso verde Zapatillas de deporte Onitsuka Tiger Colorado Eighty Five Blanco Blanco , Chaco Mujer ZX1 Classic Sport Sandalia Dart Peach , OluKai Kulapa Kai Dark Java / Dark Java , NIKE Kids gratis Rn (grande) Negro / Blanco / Gris Oscuro / Antracita Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Cool Grey / White / Volt / Blue Glow Zapatillas de running NIKE Zoom All Out Low para hombre Azul Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Plata / rosa En la ejecución de calzado de mujer en la nube Mulberry / Velvet , SAS Mujeres, Sanibel Thong Sandal Serpiente negra , Drew Womens Ariana Rosa / Oro Floral CLARKS Sandalia plana Brizo Sammie para mujer Microfibra perforada negra , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Heather gris / azul , Drew Shoe Athena para mujeres Crema / Combo Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Negro Negro , Sandalia de cuña Logan Walking Cradles para mujer Cuero Antanado Negro Suave , Zapatillas Superga 2750 Cotu para mujer Caoba rosa , Sam Edelman sandalia de mujer Sera Slide Cuero metálico yute , Sandalia de cuero Nirveli de Nine West para mujer Cuero blanco Sandalia de girasol Aerosoles para mujer Mezclilla , 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) 7017- Serpiente dorada VANELi Womens Hattie Tela vidriosa negra , El deslizador caliente Smokin de mujer FLEXX Gunmetal / Black Bruga / Cashmere , OluKai Ohana Pewter / Black Made For You Women & rsquo; s Zapatillas de lana con Amapola hecha a mano Flor, Cómodo con goma antideslizante Soporte de Sole y Arch Plantilla Beige Bota de señora Pleaser para mujeres Adore-2023 / B / PU Poliuretano de franja negra / negro mate SOREL Women s After Hours Lace Up Sandalia de cuero y gamuza Negro Propet Travelactive Slip-On Oxford para mujeres Heather negro / blanco Trotters Liz Loafer para mujer Rosado

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 NIKE B06XPQQP8Q/ Women blanco s Revolution 3 para correr Gris oscuro/ blanco/ negro 13039b4 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