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.

El talón mide aproximadamente 3.9 pulgadas

Tacón alto medida: aprox. 10 cm

En una Acabado sintético, estos dan una sexy y confiado borde mientras que también proporciona apoyo y comodidad.

Estos sadnals con estilo Está compuesto de el dedo abierto, alto tacones de aguja y deslícese.

Exhibición un muy simple y manera elegante, para crear el mejor exquisito sandalias a tu próxima cena sofisticada, negocio formal invitar.

Todos Zapatos YDN están terminados por estrictamente artesanal. Servicio de personalización de tamaño, color, altura o más están disponibles.

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

YDN Zapatos Mujeres Open Toe High 15355 Heels Bombas Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Turquesa a8317f8Updated 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.

YDN Zapatos Mujeres Open Toe High 15355 Heels Bombas Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Turquesa a8317f8

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 Tiffany de BareTraps para mujer Rosa roja David Tate Wish Sandalia para mujer Negro Salvatore Ferragamo Mujer Varina Oxford Blue Sandalias Rosalyn de Steve Madden para mujeres Negro / Multi Sandalia Uneek o2-w de mujer KEEN Blanco negro DAWGS Womens Arch Support Z Sandals Lima suave CLARKS Sillian Stork Fisherman Sandal para mujer Negro Marc Fisher LTD Mujeres Glenna Gris claro gamuza Sandalia Celeste-Cvl Slide de Loeffler Randall para mujer Eclipse Sandalia de Bling con tachuelas Macayla UGG para mujer Negro Bomba de plataforma FLY London Yadi718fly para mujer Pearl Cool SAS Mujeres, Sanibel Thong Sandal Caramelo Salvatore Ferragamo Womens Belluno Armada Mujeres nacidas - Emmy Band Cuero Ginger Full Grain Bomba de plataforma Camper Women s Damas 21923 Blanco ALMA NATURAL Ballet femenino femenino Poliuretano liso azul marino , J.Renee Womens Mcwayfalls Plata Drew Womens Element Nubuck marrón , Kenneth Cole REACTION Sandalia de cuña con correa baja en el tobillo de mujer Maravilla Bomba Amie Naturalizer para mujeres Negro , Sandalia de tacón Riki de mujer Alegria Asher OLUKAI Pua Sandal - Mujer Deep Red / Bean , Sandalia plana Halle de Walking Cradles para mujer Cuero Agata Negro , Sandalia de mujer Teva W Terra-Float Livia Negro Negro , NAOT Mujeres Fiona Verde azulado nobuck / Nubuck azul aceitoso / Cuero lustre plateado David Tate Wish Sandalia para mujer champán , Salvatore Ferragamo Mujer Varina Oxford Blue , Sandalia de tacón Riki de mujer Alegria Asher Sandalia Nora Ryka para mujer Gris oscuro

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 Mizuno Wave para mujer Inspire 10 Blanco Zapatillas para correr Saucony Koa St para mujer Verde / Negro / Aqua La Sportiva Women s Wildcat 2.0 GTX Trail Running Shoe Gris / rojo , Salomon Women s Sonic Pro Azul Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Thistle / Artic Sky , solo mujer Hebilla de correa de tobillo Cutout Gem Clear Stiletto High Heels Gladiador transparente Sandalias de tiras con Rhinestones B Rose Gold , Sandalia de vestir de mujer Mephisto Francesca Pitón pardo oscuro Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Onyx / Carbono CLARKS Mujer Gracelin Mara Flat Berenjena Skechers Women s Appeal Flex New Image Sneaker Blanco negro J.Renee Desdemona Sandalia para mujer Rosa / Pastel Multi Easy Spirit Women s Traveltime Mule Gris claro gamuza / malla , Easy Spirit Women s Traveltime Mule Plata NIKE Men s in-Season 7 Cross Trainer Gris oscuro / Barely Volt - Aurora Verde Vans Herren Authentic Core Classic Sneakers True White Vans Unisex Old Skool Zapatillas clásicas de skate Madder Brown / Blanc De Blanc , Zapatillas de ciclismo híbridas ECM Saddle para mujer ECCO Negro , Sandalia para mujer Ivanka Trump Earin Slide Natural , Zapato Tap-Flex para mujer, de Bloch Dance Broncearse , 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5008- Cuero blanco UGG Mujer Alena mocasín castaña , LifeStride Women s See This Dress Pump Tierno Taupe Bomba de Angel II para mujeres estilo suave Elegancia negra , Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Patente negra U-lite para mujeres Zapatilla de lana de piel caliente Casual Slouch de invierno Piso plano y al aire libre Loafer, pantuflas de lana para mujeres Negro Tamaris Women s Vivi 28325 Oxford Flat Coñac , ED Ellen DeGeneres Womens Waide Oaktone / Dark Chive Ventura Pebbled Polo Ralph Lauren Kids Ranger Hi II Bota de moda Nubuck azul marino CLARKS Jocolin Vista Boat Shoe para mujer Champagne Metálico Sintético Ballet Tali de mujer Cole Haan Bow Bow Maple Sugar Leather ,

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 YDN Zapatos Mujeres Open Toe High 15355 Heels Bombas Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Turquesa a8317f8 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
1930726

2948710

6626405

9176710

12660743

13718457

17691546

18343158

19009461

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