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.

Sintético

Importado

Sintético suela

Plataforma mide aproximadamente 0.5 pulgadas pulgadas

Goma suela

Rose cocodrilo de metal dorado marca

Patente detalle brillante de la raya

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

Lacoste s Women sintético & rsquo; s Fraisier Slides Blanco/ rsquo; rosa sintético db44399Updated 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.

Lacoste s Women sintético & rsquo; s Fraisier Slides Blanco/ rsquo; rosa sintético db44399

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 Mason para mujer Naturalizer Slide Blanco Sandalia para mujer Kate Spade New York Coby para mujer Blanco Calvin Klein Dalana Slide Sandalia para mujer Negro , Sam Edelman Women s Slide Sandalia Oro / cuero desnudo metálico / cuero de gamuza Sandalia Cato Dolce Vita para mujer Cuero grabado en relieve de serpiente Kenneth Cole New York Mujeres Osmond Fringe Pom Detail Slide Sandalia Blue Denim Alpargata FRYE Femenina Ally Deco Stud Criss Cross Sandalia Slide Dusty Rose Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno rosa , Sandalia de vestir Dansko Dee mujer Caramel / Sand Full Grain , Skechers para mujeres Reggae Fest-Marlin-Fisherman Open Back Mule Ajuste relajado y espuma de memoria a / C Zapato de agua Carbón ALMA NATURAL Sandalia Kiki Slide para mujer Pulir Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sam Edelman Women s Mares Slide Sandal Brocade floral multicolor negro Sandalia Kiki-Nt Slide de Loeffler Randall para mujer Buff / rosa , BareTraps Women s Jaylyn Slide Sandal Castaño Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Blush Ropa de cama Sandalia Taryn Rose de mujer Iris Knit Slide Mezclilla Sam Edelman Women s Bay Slide Sandal Cuero metálico pulido de yute , Sam Edelman Sandalia Bernice Slide para mujer Nuevo Nude Leopard Sandalia Cato Dolce Vita para mujer Bordado blanco / negro , Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno azul , Sandalia Quasida Slide de Flexus by Spring Step para mujer Navy Multi , Sam Edelman Women s Bay Slide Sandal Denim Blue / New Blue Vaquero Saddle Leather , Sandalia de cuero Merrell Dwaykair Seaway para mujer Negro Sam Edelman Sandalia Bernice Slide para mujer Candy Red Vaquero Saddle Leather , Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Cuero negro de acción KAANAS Santa Cruz Raffia Pom Pool Sandalia plana deslizante Gris Lauren by Ralph Lauren Sandalia mujer Davan Slide Cemento ,

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 Lanzamiento de Brooks Womens 4 Negro / Fucsia Púrpura / Azul deslumbrante , Saucony Women s Ride 10 Running Shoe Gunmetal / rosa ASICS La sacudida de las mujeres Carbono / Murasaki , Calzado de running ft. Perpetua ft de memoria Fila Women s Castlerock / Aruba azul / blanco , Skechers Women s Go Trail 2 - Zapatillas de senderismo con agarre Chocolate , Zapatillas de trail running Salomon Speedcross 4 para hombre Nube oscura , Salomon Women s Speedcross 4 W Trail Runner Artic Blue , Zapatillas para correr Nike Flex Experience RN 5 Cool Grey / Black-Gym Rojo-Blanco , Camiseta adidas Swift Run W para mujer Core Black / Core Black / White , Zapatillas de running Columbia Montrail Mountain Masochist III para mujer Bounty Blue, Black Zapatillas de trail running Salomon Speedcross 4 para hombre Ibis del Atlántico Norte / Negro / Escarlata , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Rosa roja Zapato acuático Merrell Capra Rapid Sieve para mujer Púrpura , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro plata Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Piso de mujeres de Coral Pies extra anchos Diabético ortopédico Zapatos atléticos Sneakers para caminar Negro Sam Edelman Women s Slide Sandalia Plata , Sandalias de cuero de Viv Viv de TOMS Silver Specchio Zapatillas para caminar Propell Mujer Travellite MJ Gris pardo Hush Puppies Men s Gus Oxford Broncearse Drew Shoe Women s Tulip Comfort Blanco Zapatillas para caminar Propell Mujer Travellite MJ Blanco Zapatilla de deporte KAANAS Santa Fe Fashion Skate Slip-on Casual Sneaker Color de malva , Sandalia Kya Slide de Donald J Pliner para mujer Magenta , Sandalia Mason para mujer Naturalizer Slide Blanco , Sandalia de la diapositiva Indi Kate Spade New York para mujer Grosgrain negro / blanco Bota de montar Legivia de Call It Spring para mujer Gris , Sandalia Birkenstock de cuero Florida para mujer Negro Birko-flor Botines Wander de mujer de NAOT Brushed Seal Brown Suede , Dr. Martens Women s Estrela Oxford Negro , Rocket Dog Mujeres Destin Orchard Hanger Cotton Oxford Aceituna ,

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 Lacoste s Women sintético & rsquo; s Fraisier Slides Blanco/ rsquo; rosa sintético db44399 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