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.

Cuero

Importado

Talón mide aproximadamente 1 pulgada

Esto el estilo es un Medicare y Diabetic aprobado zapato.

incomodidad es una cosa de pasado con esto estilo acogedor, por lo prepárate para verdaderamente disfruta tu Gratis ¡Tiempo!

Suave y transpirable parte superior de cuero con perforaciones y detalles tejidos.

Tradicional cierre con cordones para un seguro y ajuste ajustable.

Ligeramente lengua acolchada y collar.Tongue está asegurado a revestimientos con gore elástico. Suave y transpirable revestimientos textiles.A extraíble SAS CoolStep & trade; Características de la plantilla una antifricción forro que proporciona una superficie fresca bajo los pies un absorbente de humedad capa inferior diseñada para mantener tus pies fresco y seco. resistente a los olores SAS Tripad & reg; Cojines diseñados para consolar a todos tres presiones puntos: el interior pelota, fuera de la pelota y talón para todo el día soporte.Durable y ligero no sli

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 San Zapatilla Antonio Mujer SAS, Zapatilla de B071ZZMDLR 19949 deporte Freetime con cordones Carbón 966b699Updated 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 San Zapatilla Antonio Mujer SAS, Zapatilla de B071ZZMDLR 19949 deporte Freetime con cordones Carbón 966b699

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

Zapato de caminata Skechers Performance para mujer Go Walk 4 Pursuit Azul marino / gris , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Verde Drew Shoe Women s Tulip Comfort Brown Print , Zapatillas de senderismo Salomon Odyssey Pro - Hombre BELUGA / LIME GREEN Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Grey Suede / White Mesh Zapatillas deportivas para caminar Spira CloudWalker para mujer con resortes Blanco negro , Skechers Women s Walk Walk Joy-Invite Sneaker gris , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Negro , Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Blanco negro , Dr. Comfort Marla Mujer diabético terapéutico Extra Profundo Calzado Lycra Velcro Negro SAS Womens Sunny Plata Walking Cradles Womens Henna Nubuck negro estampado , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Baya Skechers Sport Relajación de mujer Breathe Easy Moneybags Sneaker Negro Zapatillas para caminar New Balance 85v1 para mujer Tan / joya , Zapatillas para caminar Skechers Performance para mujer Go Fit TR Gris / rosa fuerte Zapatillas para caminar Ryka Women s Devo Plus 2 Negro / púrpura , Vionic Women s Flex Sierra con cordones Blanco negro , Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Navy Marina Zapatillas de paseo Reebok para mujer Reeshift DMX Ride Wide D Negro / Remache Gris / Sol , Drew Shoe La alegría de las mujeres Azul cielo ante , OLUKAI Nohea Nubuck - Mujeres Dark Java / Tan Orthofeet 877 Women s Comfort Diabetic Therapeutic Extra Profundidad Calzado Lycra Velcro Easy Spirit Twist2 Sneaker para mujer Porcelana blanca , Zapatillas de deporte Arizona Trotters para mujer Desnudo , Zapatillas clásicas Vionic para mujer Walker Negro Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Negro Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Negro Negro , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco

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 Corkys Rock Mujer Slip On Azul Multi , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Negro / Negro / Gris Oscuro , Zapatillas de running AMPLICA para mujer ASICS Gris Medio / Plata / Aruba Azul Flip-flop Teva Women s Mush II La Magenta Green , NIKE AIR Max 97 QS (GS) Silver Bullet 2017 US Release - 918890-001 , Sandalia de tacón de gamuza Salomé de mujer Anne Klein Dark Natural , Zapatillas de running mujer HOKA ONE ONE Clifton 4 Castle Rock / Asphalt Zapato de running FUZETORA para mujer ASICS Negro / Negro / Blanco Imagine Sandalia de tacón IM-Dailey de Vince Camuto para mujer Negro , Zapatillas para caminar Go Step Lite-Solace de Skechers Performance para mujer Gris pardo Bomba de plataforma Camper Women s Damas 21923 Marrón medio , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Give Teal Glitter , Zapatillas para caminar Brooks Women s Addiction Walker Blanco , Sandalia Amara 2 de Carlos Santana para mujer de Carlos Ópalo Sandalia Tonya de mujer Trotters Equipaje , Converse Chuck Taylor All Star Core Ox Blanco negro Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Piso de mujeres de Coral Pies extra anchos Diabético ortopédico Zapatos atléticos Sneakers para caminar Negro Skechers Performance Women s Define Wide Sneaker Blanco negro Sandalia Mila Slide de mujer Naturalizer Gamuza rosa Sandalia de la diapositiva de junio de las mujeres de Propet Plata , Sandalia de diapositivas de ocio para mujer de Seychelles Azul , Sam Edelman Women s Bay Slide Sandal Marfil negro / negro Sandalias Hush Puppies para mujer Dachshund Slide Cuero negro Sandalia Maria Slide de Nanette Lepore para mujer Blanquecino UGG Womens Dakota Sunshine Perf castaña , Blundstone Unisex Original 500 Series Stout Brown / Navy Tartan , Footprints By Birkenstock Southport Zapato de cuero Braun / Brown Hush Puppies Chardon Oxford para mujer Gris oscuro / metálico Propet Women s Tour Walker Strap Sneaker Negro , Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Negro / gris suave ,

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 San Zapatilla Antonio Mujer SAS, Zapatilla de B071ZZMDLR 19949 deporte Freetime con cordones Carbón 966b699 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