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 y Textil

Importado

Sintético suela

Plataforma mide aproximadamente 0.5 pulgadas pulgadas pulgadas

Cuero Y parte superior tejida

EVA Plantilla para cojín

Merrell Air Cushion en el talón absorbe choque y agrega estabilidad

Neopreno para comodidad acolchada

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

Sandalia para Helecho mujer Wrap Terrell Ari Wrap de Merrell para Helecho c281b4aUpdated 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.

Sandalia para Helecho mujer Wrap Terrell Ari Wrap de Merrell para Helecho c281b4a

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

Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul , Propet Women s TravelActiv Ss Sandal Oro / negro , Fabulicious Women s Caress 401Dm sandalias de diapositivas Claro Rhinestone negro / peltre Sandalia Drew Angela para mujer Dusty Multi Metallic Sandalia KEEN Women s Venice H2 Gruta Celeste / Azul , Earth Origins Sophie Azul marino OluKai Kulapa Kai Blanco negro Sandalia Shaye Sport mujer Donald J Pliner Platino , Wolky Comfort Jewel Estampado de flores blancas Mordenmiss - Sandalias Huarache de cuero, para mujer, de verano, zapatos planos de tacón Estilo 1-verde Sandalias Newport H2 de KEEN para mujer Negro pizarra / Cantón , Sandalia para mujer Terrell Ari Wrap de Merrell Resquicio de esperanza Sandalia Terrell II de Merrell para mujer Gris pardo , Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Negro / Negro / Blanco Tiza Sandalia Yucatan ECCO Mujer Chili Red / Concrete / Black , Sandalia NIKE Kawa Slide para mujer Cloro Azul / Azul Glaciar / Foto Azul OluKai Ohana Hibisco / Negro Chaco Mujer Diana Sport Sandalia Avo florentino KEEN sandalia de Venecia para mujer Mineral azul / ámbar , BZees Sandalia con cuña de beso para mujer Blue Multi Print , Sandalias adidas de mujer Adilette Cf + Armad Athletic Slide Maroon / Ice Purple Colegiate Navy Sandalia Teva Terra-Float Nova para mujeres Gris Wolky Comfort Jewel Acero azul liso , Sandalia deportiva Zvolv X de Chaco para mujer Manta Blues , Sandalia Merrell para mujer Terran Ari Post Sport Helecho , Sandalia Belmar Athletic de Ryka para mujer Gris oscuro , Sandalia Kea Vent mujer Columbia Candy Apple, Earl Grey Sandalia Drew Angela para mujer Negro Sandalia Archie Fisherman del Dr. Scholl Monumento Nubuck

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 Reebok Women s Club C 85 Blanco / gris claro Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Cool Grey / White / Volt / Blue Glow , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Vino de Oporto / Deadly Pink / Tea Berry / Black , Brooks Women s PureFlow 6 Antracita / Plata / Flor de Cactus Púrpura Sandalia Drew Angela para mujer Beige-multi , Sandalias de tirantes Sunstone para mujer Merrell Sodalita , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Blanco / blanco / amapola , Zapato de running ASICS para mujer Torrance Mysterioso / Negro / Aluminio Zapatillas de running adidas NMD_CS2 PK Originals para mujer Lino Zapatillas Merrell Women s Vapor Glove 2 Barefoot Trail Running Mar Azul / Coral , Sandalia Drew Angela para mujer Negro Zapatillas de running Skechers Performance para mujer Go Run 400 Action Carbón de leña / azul Tecnología Vionic Men s Orthaheel White / Navy Walker Gris oscuro Amortiguación 460v2 New Balance para mujeres Gris , New Balance W880gb7 para mujeres Azul claro , Zapato de running GEL-Kinsei 5 para mujer ASICS Golpe caliente / blanco / real FLY bomba de cuña Yett de mujer de Londres Plata / Blanco Borgogna / Damani NAUTE Mujer Beverly Sandalia Nubuck esmeralda aceitosa / Nubuck azul aceitosa / Silver Mirror Lthr Sandalias planas con cordones para mujer SAS s Clover Open Toe Armada , Sandalia de enseñanza Grecian de cuero para adultos, T8900 Broncearse Zapatillas de entrenamiento Puma Mujeres Feroz Nubuck Naturals Rosado , Calzado de tenis de bádminton Yonex Aerus 2 LX para mujer Gris , Zapatillas altas con cordones All Star de cuero Converse Chuck Taylor Chocolate , Jig Foo Latin Salsa Rumba Chacha Practice Ballroom Zapatos de baile para mujeres Negro , Zapatillas de ciclismo Pearl Izumi para mujer W Select Road v5 Studio Negro Negro Steve Madden Womens Georgie Patente negra , Twisted X Boots Mens Lace up Impermeable Old Brown Hiker marrón , Zapatilla deportiva perforada con láser Mephisto para mujer Hielo blanco liso / plateado Betunia Loafer de mujer Aerosoles Oro metálico , Sam Edelman Loraine Loaine para mujer Amarillo / Multi Gingham

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 Sandalia para Helecho mujer Wrap Terrell Ari Wrap de Merrell para Helecho c281b4a 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