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

Goma suela

Medidas de talón aproximadamente 1 1/2

Plataforma mide aproximadamente 3/4

Medicare / HCPCS código A5500 diabético zapato

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

Zapatillas de viaje 7927 Protic W3910 B07DHMZYTB Vista Walker de para mujeres Blanco suave 6f7b8a5Updated 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.

Zapatillas de viaje 7927 Protic W3910 B07DHMZYTB Vista Walker de para mujeres Blanco suave 6f7b8a5

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

Sirena de mujer Merrell Flip Q2 Athletic Sandal Negro , OluKai Kulapa Kai Papaya / Negro Chaco Mujer ZX3 Classic Sport Sandalia Melocotón de jardín Sandalia para mujer Terrell Ari Wrap Sport de Merrell Baja Azul , Sandalia Uneek O2-w de mujer KEEN Opal azul / azul provincial Columbia Women s Barraca & shy; Sunlight Athletic Sandal Fósil, Natural , Zapatilla Slip-On Hydro-Life W de Teva para mujer Gris oscuro , Sandalia atlética de Rka Women s Glance Negro / rosa , Sandalia Kea Vent mujer Columbia Fósil, Cervatillo , Sandalia de susurro de mujer Keen Shiitake / Malaquita , Wolky Comfort Jewel 70100 canales blancos Sandalia deportiva W Sirra de mujer Teva para mujer Azulejo azul Sandalia de susurro de mujer Keen Shiitake / Malaquita Sandalia Omnium de mujer Teva Pizarra , Chaco Women s Zx3 Classic Athletic Sandal Aqua Mint , Wolky Women s Rio Sandalia Azul Sandalias KEEN Venice II-W para mujer Bleacher / Raven , Chaco Women s Z1 Classic Athletic Sandal Gaulteria , Sandalia KEEN Clearwater CNX para mujer Anémona / acacia , Sandalia original Teva Women s Original Negro Sandalia de mujer KEEN Rose Imán / Gárgola , Wolky Comfort Jewel Jeans Blue Multi Color Fantasy Oasis Women s Zoey Fisherman Shoes marrón Sirena de mujer Merrell Flip Q2 Athletic Sandal Negro MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 melocotón Sandalia Yucatan ECCO Mujer Polvo blanco , Revs New Classic Masaje de reflexología Sandalias para hombres & Mujer. Mira. Diseño. Terapia. Cojín Comfort Con plantilla de masaje & Absorción de impactos Único. Negro , Chaco Women s Wrapsody X Athletic Sandal Key Teal , Wolky Comfort Jewel Beige ,

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 Sequoia para mujer Mark Nason Angeles Lila , NIKE Men s Flex 2017 RN Negro / Industrial Azul / Azul Leyenda / Blanco Encaje de quimera de Columbia para mujer Arrecife / Nivel del mar , Sandalia NIKE Benassi JDI para mujer White Wolf Gray Metallic Silvr , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Acero / Trueno Skechers Reggae Fest Fest-Squirt-Pescador Slingback Casual Zapato de agua Gris oscuro , Rocket Dog Women s Simm Shimmy Pu Sandalia Rainbow Multi Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul , Zapatillas para caminar Skechers Performance para mujer Go Step Lite Ultrasock gris , Sandalia de cuña Karmen de mujer Alegria Unidad Negro / Blanco Alpargata de tela \ Negro , Zapatillas para caminar Skechers Gostyle 15606 para mujer Azul marino / turquesa , Easy Street Adagio Wedge Slide para mujeres Azul marino Patnet / Stretch , Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Gris / Coral Skechers Women s Go Step Lite-Indulge Loafer Flat Carbón Apex Women s Ambulator Biomechanical Negro adidas Originals Mujer Superstar Blanco / Onix / Metálico / Dorado , Vans Auténtico Rosa neón Vans Unisex Old Skool Zapatillas clásicas de skate Lux Cuero Porcini Chocolate Afeitado , Drew Shoe Women s Tulip Comfort Estiramiento negro / negro , Supra Vaider LC Sneaker Negro / Oro / Oro / Blanco , Sandalia Dolce Vita Jene Slide para mujer Denim azul claro , CLARKS Sandalia Leisa Lacole Slide para mujer Cuero de arena , Zapatillas de samobal Swar 5 Fastpitch Cleat de Mizuno para mujer Azul marino / blanco , Zapatillas Laurin Moccasin de Tempur-Pedic para mujer gris Reebok RCF Lite TR TXT Cross Training Porcelana / plateado verde para mujer , Botín de tobillo Hadley de Bella Vita para mujer Gamuza almendra , Drew Shoe Blazer Mujer Oxford Taupe Nubuck , TTY de Oxford, aprobado por correo, de Rocky Women Duty Negro Kmeioo Slip On Mocasines, mujeres Bordado mula Zapatos con felpa Terciopelo de piel de cordero Sin respaldo Toe Mule Slides Tacón negro sólido

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 Zapatillas de viaje 7927 Protic W3910 B07DHMZYTB Vista Walker de para mujeres Blanco suave 6f7b8a5 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