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

Hecho en EE. UU. O importado

Eje mide aproximadamente Parte baja del arco

Parte trasera del pie GEL acolchado Sistema: Atenúa choque durante el impacto fase y permite para una transición sin problemas a medio cuerpo

Desmontable Plantilla: una plantilla que puede ser eliminado para acomodar una ortesis médica

AHAR Suela: acrónimo para ASICS High Abrasion Rubber. Colocado en crítico áreas de la suela por excepcional durabilidad

Para kilometraje moderado corredores, el GEL-Excite 4 zapatos ofrecen comodidad y durabilidad. Nuestra firma Rearfoot y gel para el pie tecnología de amortiguación suaviza el impacto enérgico, mientras nuestro ahar objetivos de suela zonas de alto impacto para una mayor resistencia úsese y tírese. En 1949, el Sr. Kihachiro Onitsuka comenzó su calzado deportivo compañía (Onitsuka) Co., Ltd.) por fabricación zapatos de baloncesto fuera de su vida habitación en Kobe, Japón. Él eligió el nombre ASICS para su compañía en 1977, basado en un famoso latín frase

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 deportivo ASICS deportivo Gel-Excite 15098 4 para mujer Zapato Aruba Azul/ Plata/ Acuario fbb96e3Updated 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 deportivo ASICS deportivo Gel-Excite 15098 4 para mujer Zapato Aruba Azul/ Plata/ Acuario fbb96e3

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 para correr Nike Women s Free Rn Flyknit 2017 Platino blanco / negro-sigiloso puro Calzado informal NIKE Darwin de mujer naranja , Zapato deportivo ASICS Gel-Cumulus 18 para mujer Azul Indigo / Plata / Rosa Brillo Zapato de running ASICS para mujer Torrance Gris piedra / helado Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running NIKE Air Max 2017 para mujer Concord / Blanco / Violeta Persa , Zapatillas de running TOPO FLI-Lyte 2 para mujer Verde / Coral , Zapatillas de running adidas Springblade Razor para mujer Power Teal / Frost Mint / Sol Gold , Supra Martine Run Skate Shoe para hombre Blanco negro , Zapatillas para caminar Skechers Go Run-Mojo para mujer Negro Zapatillas de running adidas Adistar Boost ESM para mujer Púrpura , Zapatillas de running Kinvara 8 Saucony para mujer Rojo blanco NIKE Women s Tanjun Partícula rosa / blanco Calzado deportivo NIKE CK Racer para mujer negro Zapatillas de running Pure Cloudfoam de adidas para mujer Carbono / carbono , Zapatillas de running Nike Lunarstelos negro / metalizado plata / azul luna / aluminio para mujer - Zapatillas de running adidas Alphabounce Em W para mujer Easy Coral / Sun Glow / Gray One Zapatillas de running New Balance 1340v3 para mujer Blanco / morado , Zapatillas de running adidas Originals Cf Superflex W para mujer Blanco / negro / negro Zapatillas Reebok Women s Runner 2.0 MT Track Negro / violeta violeta / peltre , Zapatillas para correr Saucony Koa Tr para mujer Vio / Aqua Zapatillas de running adidas Pureboost Xpose Clima para mujer Easy Coral / White / Light Maroon , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosado Zapatillas de running NIKE Mens Lunarglide 8 Racer azul / negro , Zapatillas de running adidas Women s Element Refresh 3 w Negro / Utility Black / Utility Black Calzado informal NIKE Darwin de mujer Algas Digitales Rosa 300 Zapatillas de running cross cross para mujer ASICS Cross Faire 2 para mujer Polvo azul / blanco / azul marino Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco-negro-corredor Azul , Camiseta adidas Swift Run W para mujer Aero Pink / Blanco / Core Black ,

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 de running NIKE para mujer Lunarepic Low Flyknit 2 Verde Palm / Negro / Vapor Verde Zapatillas de running adidas Cf Qt Racer Mid W para mujer Gray Two / Gray Three / Crystal White , Zapatillas para correr Saucony Freedom Iso Unisex Oro blanco , Converse Chuck Taylor All Star Seasonal Color Hola Gris frío Sandalia de susurro de mujer Keen Malaquita / Abedul plateado , CLARKS Sandalia de pescador de melón Tuvia para mujer Gris sintético , Calzado de running Inov-8 Roclite 305 para mujer Gris / Negro / Coral Salomon Women s X-Mission 3 W-w Acuario , Zapatillas de trail running Merrell para mujer Bare Access Arc 4 Elegante / Vibrante Coral , Zapato para caminar Metrolyte para mujer ASICS Negro / Regatta Blue / Regatta Blue Easy Spirit Twist2 Sneaker para mujer Mezclilla Skechers Principio Color de malva Sandalias de punta en punta con sandalias de mujer de YDN Sandalias de tacón bajo con cordones de zapatos de diapositivas Blanco negro , Zapatillas de running Mizuno Wave para mujer con sombra Fucsia Púrpura / Plateado / Azulejo Azul Sandalia Kinsley de Naturalizer para mujer Papaya roja / tostada Zapatilla Ascot de hombre UGG Dusty Olive / Sahara , Zapato de malla Nohea de OLUKAI - Hombre Dark Shadow / Dark Shadow Entrenador cruzado Hydrosphere Ryka para mujeres Bougainvilliea / Hyper Pink / Sconce Gray , Zapatillas de deporte Romy de Easy Spirit para mujer Gris blanco , Zapatos de golf AsJei para mujer FootJoy Bígaro , Sandalias Skechers para mujer Rumbler Wave-Ibiza Summer Slide Blanco Sandalias planas para piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural Salón de baile para mujeres Dance Shoes Party Práctica de la salsa Zapatos 1643EB Cómodo -Muy bien 2 \ de 5} Brown Nubuck & # 47 Mesh , Mizuno Women s 9-Spike Advanced Sweep 3 Metal Softball Cleats - Blanco y Blanco CLARKS Mujer McKella Morris Sneaker Nubuck gris sintético SOREL Women s Out N About Slippers Negro / Cantera Chanclas Deportivas Cooper Slide de Marc Jacobs para mujer Aqua / Red Grand Crosscourt Ii de Cole Haan para mujeres Azucar de arce Bota Khola de mujer de Penny Loves Kenny Gris / Multi , Arcopedico Mujeres Ls Titanio

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 deportivo ASICS deportivo Gel-Excite 15098 4 para mujer Zapato Aruba Azul/ Plata/ Acuario fbb96e3 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