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

Sintético suela

Altura del talón 1.79 pulgadas

Ortholite plantilla

Cojín Tecnología suave

Full Cuero de grano

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

CLARKS Wendy Wendy Alto Fisherman mujer Sandal para mujer Cuero 9677 negro cd55531Updated 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.

CLARKS Wendy Wendy Alto Fisherman mujer Sandal para mujer Cuero 9677 negro cd55531

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

Sandalia Natalia Ryka para mujer Blanco negro Sandalia Dominica de Ryka Azul marino / menta Sandalia con tachuelas Tali Mini Bow de Cole Haan para mujer Marea tahitiana , Chaco Mujer Zcloud X Sandalia Deportiva Echo Paloma , Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Sliver-b (estilete) Sandalia de vestir Desi Easy Street para mujer Patente roja / roja Taipán de mujer de David Tate Verona Cordero negro Sandalia de cuero de travesía New Balance para mujer Negro Chancletas bíblicas unisex de cuero genuino (Jesús - Yashua) Nazareth Style - , Sandalias mujer La Plume Jen Rhinestone de marfil Vionic Womens Paden Patente negra Sandalia Barrial Blowfish Mujer Lienzo de Birch Rancher Sandalias EVA Gizeh de mujer Birkenstock Gris Sandalia Fernanda de Calvin Klein para mujer Blanco suave , Naot Footwear para mujer Kayla Cuero de pizarra vintage , CLARKS Sonar Pilot Sandalia para mujer Peltre Metálico Cuero Vionic Rest Farra - Sandalias de apoyo para mujeres Negro Negro Drew Womens Ariana Rosa / Oro Floral Sandalia Dionay Calvin Klein para mujer Rojo carmesí CLARKS Womens Un Reisel Myra Slide Sandal Cuero Sage Sandalia de Santa Barbara de las mujeres de NAOT Cuero Buffalo , Sandalias con tachuelas de cuero para mujer de Royou Yiuoer Sandalias con tira en T Desnudo , Sandalia Tonya de mujer Trotters Negro , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Azul / Denim , Sandalia P-Vinton de mujer amarilla Negro Sandalia de Gladiador Conchlusion de mujer Aerosoles Safari Print , Sandalias de peonía Easy Spirit para mujer Rosado , Chancleta Yoga Sling 2 Sanuk para mujer Azul / Ikat Chancleta Yoga Sling 2 Sanuk para mujer Glacier Ditsy Floral ,

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 ECCO Terra 2S para mujer Titanio Zapatillas NIKE Femenil Flex 2016 Rn Running Pink Blast / Blanco / Negro / Eléctrico Verde , Zapatillas de running New Balance 1500v3 para mujer Porcelana / Alpha Pink , Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea , Sandalia Uneek O2-w de mujer KEEN Negro Negro Zapatillas de running NIKE Lunar Skyelux para mujer Negro / Blanco / Antracita , Sirena de mujer Merrell Flip Q2 Athletic Sandal Aluminio , Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Plateado / Flash Coral , Adicción a Brooks 12 Antracita / Bluefish / Silver , Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Blanco , Zapatillas de running New Balance Urgev2 para mujer Alpha Pink / Vintage Indigo , Zapatillas de running adidas Alphabounce Rc W para mujer Indigo crudo / Gris Uno / Noble Indigo , Botas de mujer Naturalizer Wanya Cuero Saddle Tan Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Azul marino / blanco , Zapato para caminar WW1865v1 New Balance para mujer Estratos , Zapato Cross-Trainer X Conviction Women de ASICS Aqua Splash / Diva Blue / White CLARKS Leisa Cacti Slide Sandalia para mujer Cuero rojo Skechers para mujeres Reggae Fest-Marlin-Fisherman Open Back Mule Ajuste relajado y espuma de memoria a / C Zapato de agua Negro Birkenstock Arizona Sandalia de ante con suela suave Taupe Suede , Bomba Parigi de LifeStride para mujer Black Micron Zapatillas de lana hervida de Haflinger para mujer - Zapatillas de casa europeas Babsy Cat para interiores rojo , Zapatillas de deporte con cordones y tacón tobillo de mujer de Calajet para mujeres con tacón en punta 7CM rojo , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2713- Ultra Silver , Botas Ellie Shoes para mujer 421-Groove Engineer Blanco , Mephisto Michaela Bootie para mujer Negro / grafito / azul marino Velcalf Premium , Zapatillas de entrenamiento Nike Women`s Studio Wrap 4 Negro Trotters para mujer Dea Slingback Pump Black Micro / Satin , Indigo Rd. marrón ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Pisos tachonado Zapatillas Zapatos Patente Verde , Zapatos de tierra serval Gris pardo ,

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 CLARKS Wendy Wendy Alto Fisherman mujer Sandal para mujer Cuero 9677 negro cd55531 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
906906

4776421

7594651

8760393

11046426

11564161

14132031

17307899

17805056

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