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

Suela: Caucho sin marca suela con un concha de mar inspirada diseño de la vaina de tracción.

Plantilla: Densidad única Plantilla de la PU con un inserto de gel y suave absorción de la humedad plantilla de microfibra cubrir con hula inspirado digital impresión. y lavable.

Extraíble Plantilla: OluKai el calzado está diseñado como el perfecto ambiente para tu pie. esto en mente, nosotros hemos diseñado nuestra plantillas para ser fácilmente extraíble y lavable.

De grano completo cuero

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

OLUKAI Alohi - Mujeres Trench Blue Trench/ Alohi Trench Mujeres Blue 4fda7e4Updated 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.

OLUKAI Alohi - Mujeres Trench Blue Trench/ Alohi Trench Mujeres Blue 4fda7e4

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

Propet Walker de la estabilidad de las mujeres Blanco deporte , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Carbón , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco Zapatillas para caminar Go Walk 4 Super Sock 4 de Skechers Performance para mujer Gris pardo , New Balance WW813 Hook and Loop Walking para mujer Gris Zapatillas para caminar Reebok Ultra V DMX Max para mujer Negro / gris plano , Drew Shoe Women s Paradise II Velco Slip-on Taupe ternero , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Blanco negro , Vivien Flat de Bella Vita para mujeres champán , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Plata / Metálico , SAS Womens Sunny Plata , Skechers Women s Go 4-14918 Calzado para caminar Negro , I-RUNNER Eliza Women s Therapeutic Athletic Extra Profundidad Zapatilla de cuero / de malla con cordones Gris azul CLARKS Womens Trigenic Flex Esmeralda , Skechers Women-14960 Sneaker Negro , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Armada , Calzado para caminar CUSH + WA365v1 New Balance para mujer Gris blanco Zapato para caminar WW813 para mujer New Balance Negro Hush Puppies Power Walker Sneaker para mujer Cuero de abedul , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Estaño Zapatillas Slip-on de mujer Skechers You Inspire gris , Zapatillas Erika Propet para mujer Hueso Calzado deportivo G-Defy Gamma-Ray para mujer Gravity Defyer Gris, rojo Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Negro OLUKAI Nohea Nubuck - Mujeres Basalto / Tapa Sandalias de mujer Ros Hommerson Mellow rojo Cobb Hill Rockport Mujeres Petra Mary Jane Flat Negro , Trotters Mujer Simmy Mary Jane Flat Sage Veg Tumbled Leather , I-RUNNER Maria Puntera terapéutica para mujer con profundidad extra Calzado de cuero / malla con cordones Gris / Salmón

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 Airstream Sequent 3 para mujer Borgoña-gris-blanco Adidas Adizero Tempo 9 W femenino Res. Alta Naranja / Alta resolución Azul / Legado , Sandalias Ahnu Women s W Tilden V Athletic Negro Brooks Womens mazama 2 Rosa / ciruela / azul marino Zapatillas de running adidas Cloudfoam Super Flex W para mujer Choque rosa / negro / blanco Zapatillas de running NIKE Zoom para mujer Gris , OluKai Ohana Tropical / Hielo Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth Zapatillas de running NIKE Lunar Skyelux para mujer Aluminio / Binario Azul / Azul Mediano , Zapatillas de trail running Ryka Women s Intent XT 2 SR Blanco / plata cromada Jambu Women s Spain Flat Mezclilla , Sandalia Tonya de mujer Trotters Equipaje Reebok Crossfit Nano 7.0 Track Shoe para mujer Neón / blanco / negro / amarillo solar , Diadora Unisex I.C. Jungle Green , Zapatillas de golf Skechers Women s Go Golf Birdie Black / Hot Pink Heathered Zapatillas de golf ECOM Biom Hybrid 3 Gore-Tex para mujer Blanco negro Zapato de golf NBGW1006 para mujer New Balance Gris / rosa , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro , Sandalia Victoria Slide de mujer SCHUTZ Tuerca tostada Sandalia Joden Women s Slide de Joden Blanco Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Satén marrón , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7010-gris , El arranque inicial MuckBoots MuckMaster original Negro , Cole Staan WP de Cole Haan Women Marca Brown Bomba Sadie Wedge 40MM de Cole Haan para mujer Patente negra , Sandalia Birkenstock de cuero Florida para mujer Sfb Black Birko-flor , Xelement 2505 Righteous Womens Black Zipper Botas para moto - 7 - , Botas de combate Belleville para mujeres con clima cálido y verde caliente - USAF, F600 Verde oliva Converse Chuck Taylor All Star Core Ox Blanco óptico , CLARKS Women s UN Lory Loafer Marron oscuro

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 OLUKAI Alohi - Mujeres Trench Blue Trench/ Alohi Trench Mujeres Blue 4fda7e4 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