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.

Gamuza

Sintético único

Lanzamiento el día en este intrigante y deportivo casual

Suave / gamuza cuero, tela / patente cuero o tela / liso parte superior de cuero

Corbata cierre

Absorbe la humedad Forro DRI-LEX®

Choque Plantilla de espuma

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

Easy B01MSWBI2Y Spirit ON CUE Oxford Easy para Negro-gris mujeres Negro-gris oscuro ea2a4ceUpdated 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.

Easy B01MSWBI2Y Spirit ON CUE Oxford Easy para Negro-gris mujeres Negro-gris oscuro ea2a4ce

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

Alexis Oxford de Eastland mujeres Negro Sanuk Women s Vee K Shawn Chukka Boot Carbón de leña lavado , Thorogood 534-6333 Código de la Mujer 3 Oxford Shoe Black Negro Sesto Meucci Mujeres Nonnie Ternero manchado negro , Zapatillas SAS Women s Tour con cordones Zapato de confort activo Blanco , Mocasín Oxford de Miz Mooz Harmony para mujer Desnudo Zapatillas de deporte Aika de Cloud Footwear para mujer Burbujas , Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Raya de peltre Cole Loan de Conductor de borla de rodeo de mujer de Cole Haan Cornwall Blue Nubuck / Ivory , Tiempo libre para mujeres de SAS Hueso La oficina de Angela Scott Sr. Smith Red & Pink Striped Oxford Granate , Walking Cradles Dara Sneaker-Inspired Oxford para mujeres Niño cepillado azul marino Tiempo libre para mujeres de SAS Blanco , Prop & eacute; t Nessie Oxford para mujeres Blanco Sandalia Sneaky Sport mujer muy volátil Negro , ara Mujer Kay Oxford Rose Gold Caruso Ros Hommerson Annie Zapato informal para mujer con cordones de cuero Negro / Combo TOMS Kids 10009918 Alpargata-K Birch Heather Jersey / Faux Shearling , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Café rosa , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Azul , Zapato con cordones Brogue negro para hombre de Dr. Martens Negro Zapatos con cordones Wolky Comfort Gobly 40150 Gamuza Taupe , Tiempo libre para mujeres de SAS Blanco , Dr. Martens unisex-adulto 1461 3 ojo zapato Black Smooth Skechers Women s Lite Step-Reactive Oxford Taupe oscuro , Tiempo libre para mujeres de SAS Blanco , Ros Hommerson Camp Women s Oxford Negro , Zapatillas para mujer Del Rey TOMS Black.chambray Bota para mujer Hana de Zerogrand de Cole Haan Espejo de patente-plata impermeable blanco óptico ,

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 de deslizamiento ultra Comfort NIKE para mujer Mango brillante / negro , Converse Chuck Taylor All Star Seasonal Color Hola Rojizo-Marron , Sandalia original premier universal Teva W de mujer Aceituna Zapatillas de correr Mesh Lone Peak 3.5 de Altra para mujer Púrpura / Naranja , Mephisto SANO de Women s Escape Sport Cordones Acero Sandalia plana Halle de Walking Cradles para mujer Tabaco Agata Cuero Prop & eacute; t Olivia Oxford para mujeres Hueso Sandalia Natalia Ryka para mujer Azul / menta , Calzado informal NIKE Darwin de mujer Negro Negro Blanco Zapatillas de running W520v2 New Balance para mujer Armada Drew Shoe Women s Tulip Comfort marrón Sandalia Uneek o2-w de mujer KEEN Malaquita / Everglade Chaco Mujer Zcloud 2 Sport Sandalia Angora de llama Zapatillas para caminar Skechers Performance para mujer Go Fit TR Negro / gris Zapato de mujer PUMA Fashin Alt Twill para caminar - Calzado para caminar de mujer Mephisto Cross Cuero gris Bucksoft , Zapatos de caminata SAS Low Me de cuero para mujer Blanco , Zapatillas de deporte especializadas Gola para mujer Blanco apagado / blanco Zapatillas de tenis 1296v2 New Balance para mujer Amarillo / gris , Zapatillas deportivas Superga Womens 2750 Cotu Canvas Blanco negro Zapatillas de entrenamiento Nike Air Zoom Strong 2 Metallic para mujer US) Negro / Multi-color-puro Platino , Vans Herren Authentic Core Classic Sneakers Negro Negro , Supra Mujer / Hombre Noiz Sneaker gris , Zapato de golf Biom híbrido ECCO para mujer Blanco / caramelo , Sandalia Maria Slide de Nanette Lepore para mujer Blanquecino J Slides Sandalia de mujer Naomi Slide Estaño Sandalias planas Cammie de Jessica Simpson para mujer Desnudo Blush Ariat Womens Heritage Contour Field Zip Tall Riding Tierra de siena , Botín de tobillo de Dallas para mujeres de Naturalizer Grano de café , CLARKS Mujer Kinzie Light Loafer Flat Negro

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 Easy B01MSWBI2Y Spirit ON CUE Oxford Easy para Negro-gris mujeres Negro-gris oscuro ea2a4ce 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