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

Nube onda ofrece un suave, animado y liviano sentirse

Diseñado Malla superior para un calcetín como todavía ajuste transpirable

u4iCx la amortiguación mejora sensación de nube

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 19313 correr Mizuno Women High s Wave Rider B0722XFYKN 21 High Rise/ Gray Stone d69889aUpdated 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 19313 correr Mizuno Women High s Wave Rider B0722XFYKN 21 High Rise/ Gray Stone d69889a

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

Camiseta adidas Swift Run W para mujer Traza púrpura / Traza púrpura / Traza púrpura Zapatillas para correr Zante V3 para mujer New Balance Poisonberry / Dark Mulberry , WMNS Zoom Fly Apenas gris / Sequoia , Zapatillas para correr Pearl NZUMi Women s W EM Tri N 1 Blanco / Paradise Pink , Zapatillas para correr Mizuno Women s Wave Creation 16 Mulberry Purple / Silver , Zapatillas de running Under Armour para mujer Toccoa Soldadura (300) / Gris opaco , Zapatillas de running ASICS Gel-Contend 4 para mujer Gris Medio / Mar Glaciar / Plata Skechers Sport Skech Air Run High Fashion Sneaker Carbón de leña / verde , Calzado informal NIKE Darwin de mujer Pure Platinum Wolf Gray 001 , Zapatillas para correr Mizuno Women s Wave Rider 17 Celosia / Purple Passion / Green Flash Calzado de running NIKE Dualtone Racer para mujer Blanco / puro platino , Nike Hombres Air Zoom Pegasus 33 Negro / blanco / antracita / gris fresco , Zapatillas de running Gel-Moya para mujer de ASICS Azul blanco , Nike Hombres Air Zoom Pegasus 33 Negro / blanco / antracita / gris fresco Amortiguación 635v2 New Balance para mujer Gris Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Gris oscuro / Blanco / Negro / Sigilo , Calzado de running NIKE RN 2017 2017 para mujer Negro / Hematita Metálica / Antracita / Gris Oscuro New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple NIKE Men s Flex 2017 RN Pure Platinum / Black / Wolf Gray / Cool Gray ASICS Womens Gel-Fortitude 8 Gris medio / blanco / azul de porcelana Zapato deportivo para acolchado 635v2 New Balance para mujer Silver Mink Zapatillas de running NIKE Air Max 2017 para mujer Té Berry Puerto Vino Lobo Gris 601 Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Blanco / Fresco Grey-metallic Silver , Mochila para correr Newton Women s Motion 7 Fuschia / Teal 9 - , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Phantom / imán , Under Armour Women s HOVR Sonic NC Negro (003) / Marfil , Zapatillas de running Under Armour Speedform Intake 2 para mujer Gris Zapatillas de running Saucony Women s Breakthru 3 Azul / menta

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 Roshe Two Running Shoe Gimnasio rojo / vela negra-voltio Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco / Verde Brillo Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Core Green / Chalk White / Easy Green , Sandalia KEEN Women s Venice H2 Cascade / Misty Jade Zapatillas de running NIKE Zoom All Out Low para hombre Paramount Blue / Black-black , Rookite 325 Trail Runner de mujeres Inov-8 Gris oscuro / rojo oscuro , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Azul Drew Womens Element Naranja quemada , Nike Hombres Air Zoom Pegasus 33 Blanco , Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado Zapatillas de entrenamiento Primus Lite para entrenamiento de mujer Vivobarefoot Verde / Heather YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Azul , Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro , Saco de gladiador Gentle Souls Break My Heart 3 para mujer Grafito SESTO MEUCCI Womens Eirlys Sandal Mustard Soft Nabuk / Coral Soft Nabuk / Elástico Negro , Zapatilla Ascot de hombre UGG Negro 2 Chaco Zcloud 2 Sport Sandal para hombre Varsity Sun Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Carbón New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Silver Mink / Pigment Entrenador campeón Gusto Cross Trainer Mono azul claro , ASICS para mujer Gel-Fit Yui 2 SE Negro / Limelight / Plata , Vans Auténtico Horizonte abstracto NIKE WMNS Hyperdunk 2017 Tb Womens 897813-402 Azul marino Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Cuero de acción azul marino Zapatillas de deporte Reebok Women s Studio Basics Orquídea ahumada / Ciruela lavada / Blanco , Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Desnudo , Bomba de vestido LifeStride Sevyn para mujer Black Micro Trotters para mujer Dea Slingback Pump Negro Negro , Bota de mujer Shellys London Erin Hiker Espejo de peltre Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Blanco

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 19313 correr Mizuno Women High s Wave Rider B0722XFYKN 21 High Rise/ Gray Stone d69889a 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