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.

Textil y sintético

Importado

Caucho único

listo para el agua sintético y poliéster la parte superior de las correas es extremadamente durable y se seca rápidamente después de mojarse

Nylon caña estabiliza y apoya su pie en desigual terreno

-Contoured Soportes de la plantilla de EVA tu arco para mantener tu pie feliz y saludable

-Easy cierre de gancho y lazo entra y sale rápidamente y consigue el ajuste perfecto

-Spider Caucho original la suela es extremadamente durable y suministros gran tracción

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 deportiva Tirra para para mujer Teva Tirra Negro/ B00FS0CHS0 Blanco Multi 2c9870fUpdated 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 deportiva Tirra para para mujer Teva Tirra Negro/ B00FS0CHS0 Blanco Multi 2c9870f

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

Wolky Comfort Jewel Beige 1 , Northside Womens Savannah Sport Closed Toe Sandal Gris / Aqua Drew Womens Cascade Denim Blue Nubuck Propet Women s TravelActiv Ss Sandal Negro rojo OluKai Ohana Tropic Blue / Dark Java , Flip-flop Teva Women s Mush II Companera Blue , Sandalia de mujer Drew Alana Pewter-bronce , Chaco Z / Canyon 2 Aguacate disperso , Sandalia original Teva Women s Original Deep Sea Coral , Revs Seva Sandals, Sandalias de reflexología Para hombres y mujeres. Absorción de impactos, Cojín Comfort & Soporte del arco Negro , Sandalia Teva Terra Stella Lux de mujer Teva Negro , Earth Origins Sophie Women s Sandal Platino Sandalia Merrell Tetrex Crest Wrap Sport para mujer Negro Skechers Performance Women s The Go 400 Radiance Flip Flop Azul marino / blanco , CLARKS Womens Wave Grip Smokey Brown Sandalia Drew Angela para mujer Beige-multi , Sandalias de tirantes Sunstone para mujer Merrell Vertiver Sandalia Drew Angela para mujer Bone Smooth Leather Chaco Mujer Zcloud 2 Athletic Sandal Candy Grey , KEEN Sandalias Newport Eco-W para mujer Victoria / Dress Blues , Sandalia Drew Angela para mujer Beige-multi , Sandalias KEEN Mujer Ana Cortez Flip-W Negro , Sandalia Deportiva Jordan Spires de Via Spiga Paja de gamuza OluKai Ohana Sea Glass / Black , Sandalia Kea Vent mujer Columbia Marrón oscuro, Hawk Earth Origins Sophie Sedona Brown , Sandalia Terrell II de Merrell para mujer Negro Sandalia de cuña Jubilee Propet para mujer Denim Multi BZees by Naturalizer Sandalia Dream para mujer Negro ,

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 Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Skechers Performance On The Go 400 Vivacity Flip Flop Azul marino / blanco , Skechers Women s Go Ride 7 Running Shoe Blanco negro Zapatillas de running New Balance Urgev2 para mujer Negro / Champagne Metálico , Zapatillas Columbia Trail Trail WMNS de Plains Ridge para mujer Silla de montar / pluma CrossKix 2.0 Athletic Water Shoes - Hombres y mujeres Chaqueta amarilla Sandalia Teagan de mujer Adrianna Papell Negro , Calzado de running NIKE RN 2017 2017 para mujer Black / Hot Punch / Aurora Green Zapatos de mujer nacidos de Tidore Azul apenado Zapatillas deportivas para caminar Spira CloudWalker para mujer con resortes Blanco negro , Bernie Mev Women s Gummies Charm Calzado para caminar Hielo Sandalia Vionic Shelley Slide para mujer menta Zapato de entrenamiento ASICS Gel Acclaim para mujer Blanco / Carbón / Rosa Entrenador cruzado Hydrosphere Ryka para mujeres Negro / gris helada , Zapatillas de deporte adidas by Stella McCartney Ultraboost Parley Sneakers de adidas by Stella McCartney Res. Alta Azul S18 / Núcleo Blanco / Oscuro Callistos 07 adidas Baseline K Sneaker Blanco / Blanco / Coral , Entrenador cruzado New Foam Crush V1 de New Balance para mujer Silver Mink / Pigment Zapato de skate Aris Propet para mujer Negro , Vans Auténtico Black Denim / True White Zapatillas de golf ECCO Women s Casual Hybrid 2 Negro Calvin Klein Dalana Slide Sandalia para mujer Rojo carmesí , Bota de campo soberano de Mountain Horse marrón , 3N2 Mofo Turf Trainer Blanco Zapatillas Ivy de mujer TOMS Jersey de punto blanco y negro , Bomba Candra para mujer Bandolino Negro sintético , Hush Puppies Womens Claudia Catelyn Boot Ante rojo oscuro , Bota de mujer FRYE Melissa Button Short 2 Negro Botín Eli para mujer Bella Vita Almond Kid Suede Bota de mujer L19 de Arcopedico Reptil negro Dr. Martens Mens Pascal Boot Multi

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 deportiva Tirra para para mujer Teva Tirra Negro/ B00FS0CHS0 Blanco Multi 2c9870f 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