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

Ligero suela de poliuretano absorbe el choque y se flexiona con su pie.

Poliuretano plantilla de espuma de memoria distribuye la presión sobre la plantar superficie para más suave pasos.

Suave cuero o cuero / patente superior, con un respirable forro de punto de nylon, ayuda a la irritación del pie.

Cuero o cuero / patente superior, parte superior de nylon revestimiento, poliuretano plantilla / entresuela / suela exterior...

Poliuretano plantilla de espuma, metal caña. 1 pulgada.

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

Trotters para mujer Dea Slingback Oro Pump mujer Acolchado Oro Acolchado cebe58fUpdated 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.

Trotters para mujer Dea Slingback Oro Pump mujer Acolchado Oro Acolchado cebe58f

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

Bomba Stillson para mujer de Sam Edelman Black / Multi Venezia Metallic Jacquard , Zapato clásico Haflinger para mujer Natural , TOMS Clásicos de mujer Lienzo rojo , Cobb Hill Rockport - Aubrey Dress Pump para mujer Armada , Zapatilla bordada mula de Ibiza de las mujeres de Soludos Sand Multi , Trotters para mujer Dea Slingback Pump Blanco / negro Soft Kid / patente Naturalizer Michelle s Dress Dress Pump Armada , Trotters Lucy Ballet para mujer Hueso , Bomba de vestido Pointe para mujer Easy Street Negro Negro , Bomba Francesca de mujer Adrianna Papell Blush Satin , TOMS - Slip-On de lienzo clásico para hombre Toffee Microfiber Monocromo , Zapatillas Iris Easy Spirit para mujer Azul Zapatilla Little Bootah de mujer Sanuk Arco iris vintage Marc Jacobs Sienna Espadrille Ballet Mujer plana Black Multi Pantuflas de Katy Perry para mujer Negro , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro , Zapatilla Peluchey Flexus de Spring Step para mujer gris Minnetonka Womens Chrissy Slipper Bootie Canela J. Rene233; Gris pardo , Zapatilla Teresa Stud de mujer Teresa Sana Broncearse mejor sentido Unisex Lana clásica Zapatilla para Adultos - Toda la lana natural - Ultra cómodo - Muchos tamaños y Colores Medoc rojo Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra Azul rojo , Pantuflas de seda de mujer color verde de Patricia Green Rosa 1 , Bansi Platform Pump Jessica Simpson para mujer Barra de labios Pat , Dualyz Fit Unisex 100% lana Zapato con suela extraíble Gris oscuro / amarillo SOREL Women s Out N About Slippers Cantera , Zapatilla de piel de oveja desgastada 441169 de Old Friend Women Marron oscuro , Haflinger Women s en Tahoe Slip on Slipper Gris , Imagínese la zapatilla Casele de Vince Camuto para mujer Negro ,

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 Reebok Mujer Print Her 2.0 Camo Track Camo - Negro / Blanco Converse Chuck Taylor All Star Seasonal Color Hola Monocromo negro , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial azul marino / azul ceniza , Supra Martine Run Skate Shoe para hombre Vintage Khaki-blanco Converse Chuck Taylor All Star Seasonal Color Hola Lava blanca y negra Calzado de running adidas Pureboost X Element para mujer Gray Five / Core Black / Real Coral , Calzado deportivo de mujer Fila de Statique Dark Shade / Black / Purple Cactus Flower Clorts Women & Men s Water Shoe Sandalia para caminar de secado rápido WT05 Rosado Sandalia de cuña Karmen de mujer Alegria Dazzler de oro , Zapatillas de running ASICS GT-2000 6 para mujer Brillante rosa / negro / blanco Zapatillas de running adidas Supernova Sequence 9 W para mujer Utilidad Black / Tech Rust / Easy Green Zapato de malla Nohea de OLUKAI - Hombre rojo Skechers Women s Go Step Lite-Indulge Loafer Flat Carbón , Zapatillas Cross-Trainer WN de PUMA Women s Fierce Lace WN Puma Black-puma White New Balance 608 Blanco / rosa , Atletismo femenino de Apex Negro Sam Edelman Women s Bay Slide Sandal Denim Blue / New Blue Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ El arranque inicial MuckBoots MuckMaster original Negro Bota de montar de pantorrilla ancha Christine de Franco Sarto para mujer Negro Mizuno Women s 9 Spike ADV Finch Elite 2 Juego de sordina de softbol moldeado rápido Real / Blanco , Bomba de punta abierta Nina Carolyn para mujer Negro , Bota de tobillo Juniper Bree de Sperry Top-Sider para mujer Negro , Klogs USA Moro Sneakers Perdiz Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 28 , Dr. Martens Women s 1461 Mono Smooth Oxford Guijarro negro , Quería mujeres Downey Oxford Patente negra , Duckfeet Sj & aelig; lland Desert Boot Granate , CLARKS Women s CloudSteppers Sillian Jetay Flat Black Perf Textile , CLARKS Mujeres Sillian Bella Mary Jane Flat Negro sintético

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 Trotters para mujer Dea Slingback Oro Pump mujer Acolchado Oro Acolchado cebe58f 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