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 / sintético

Importado

Caucho suela

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.

Dinámico Soporte DuoMax Sistema: esta evolución del sistema DuoMax mejora la estabilidad y apoyo, con peso reducido y plataforma aumentada apoyo.

Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.

FluidRide Entresuela: FluidRide proporciona lo último combinación de recuperarse y propiedades de amortiguación con peso reducido y excepcional durabilidad.

Orientación Sistema Trusstic Tecnología: Esto Sistema Trusstic integra orientación Construcción de línea para una marcha mejorada eficiencia mientras proporcionar el mediopié integridad estructural.

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 ASICS B00CPGXCWU Aqua Gt-2000 5 para mujer Diva 5 Blue/ White/ Aqua Splash da643dbUpdated 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 ASICS B00CPGXCWU Aqua Gt-2000 5 para mujer Diva 5 Blue/ White/ Aqua Splash da643db

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

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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 deportiva Nolita para mujer Anne Klein Tela negra , Sandalias de encaje Ros Hommerson para mujer Black Snake Print , Zapatillas de running 430v1 New Balance para mujer Gunmetal / Seafoam , Zapatos de mujer Teva Hurricane XLT Regrind Sport Sandal Ciprés Sandalia deportiva Nolita para mujer Anne Klein Tela Coral , Zapatillas New Balance para mujer Fresh Foam Gobi Trail Running Blanco / oro rosa , Salomon Women s Speedcross 4 W Trail Runner Bluebird / Acid Lime / Black , Salomon Women s X-Mission 3 W-w Laguna profunda Sandalia deportiva Zvolv X de Chaco para mujer Kaleido Eclipse CLARKS Womens Wave Grip Sage Nubuck , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Blanco negro! Bomba de cuña Dovehouse para mujer J.Renee Estaño / Plata Saucony Women s Guide 10 Zapatillas para correr Azul marino Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Astilla-c (talón cónico) , Drew Smiles Slip para mujer Negro / Microdot Sandalia para mujer Marc Fisher Luz natural , Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Plata blanca Zapatillas para caminar Propell Mujer Travellite MJ Plata , Drew Shoe Women s Tulip Comfort Cobre Metálico Zapatilla de running Titan II para hombre Diadora Blanco / Granate Entrenador cruzado Minimus 20v7 de New Balance para mujeres Fuschia , Zapato de entrenamiento cruzado TR-4 Trainer Gel-Craze TRY de ASICS para mujer Gris Medio / Carbono / Rosa Diva , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro plata Sandalia de tobogán plano de piscina Iona de Gentle Souls Rosa Bota ecuestre de mujer de Franco Sarto Brindley Whisky , Bota de lluvia Pamp Sport Cuff Wpn para hombre de Palladium Chocolate / hierro forjado FLY London Yoss de mujeres Grafito , Bota de mujer FRYE Natalie Short Suede Engineer marrón Allrounder de Mephisto Women s Dakona Oxford Malla de aire blanca La emperatriz de las mujeres de Skechers se ve bien Rosa topo ,

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 ASICS B00CPGXCWU Aqua Gt-2000 5 para mujer Diva 5 Blue/ White/ Aqua Splash da643db 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