Whistler Alley Mathematics

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.

Mocasín de mujer RIALTO Vette Mocasín Serpiente mujer de negra ad225ae

Hecho en EE. UU. o importados
Caucho único
redefinir tu estilo con elegancia simple.
Fácil deslizarse en mula.
Faux-Crocodile parte superior hecha por el hombre.
Flexible cornada en el lado.
Hecho por el hombre revestimiento. plantilla. suela artificial. Bloquear el talón Altura del tacón: 1 3 & frasl; 4 in Peso: 8 oz Mediciones del producto fueron tomadas usando tamaño 7.5, ancho B - Medio. tenga en cuenta que las mediciones puede variar por tamaño
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

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Recent Changes

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

Mocasín de mujer RIALTO Vette Mocasín Serpiente mujer de negra ad225ae

Updated 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:

The Brachistochrone

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.

Mocasín de mujer RIALTO Vette Mocasín Serpiente mujer de negra ad225ae

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 de cuña Santorini Jambu para mujer Armada Sandalia de cuero de travesía New Balance para mujer Negro FLY London Womens Yamp 836 Sandalias de cuero Negro Sam Edelman Women s Gigi True Black Leather , Sandalias de tacón de soñador Easy Street para mujer Broncearse , NIKE Men s Benassi Just Do It Athletic Sandal Negro / Blanco Noir / Blanc , Skechers Women s, 2nd Take Tied up Sandalias de diapositivas Negro , Sandalia plana Halle de Walking Cradles para mujer Serpiente negra / plateada Josef Seibel Women s Rosalie 27 Fisherman Sandal Negro , Naot Footwear Mujer Verbena Piel Oliva Aceitada Oliva / Cuero de Camel Vintage , Crocs Flip Crocband unisex Volt Green / White , Tamaris Women s Alis 28335 Oxford Flat Armada CLARKS Sandalia de cuero Manilla Lift para mujer Cuero negro Tamaris Women s Alis 28335 Oxford Flat Blanco , Sandalias de vestir de piña de mujer de Jack Rogers Hueso / Oro , Sam Edelman Sandalia de mujer Berke Cuero blanco brillante , CLARKS Haley Stork Loafer para mujer Armada CLARKS Womens Quartz Sandalia Nubuck azul marino Sandalia Barrial Blowfish Mujer Gris SAS Mujeres, Sanibel Thong Sandal Serpiente negra , Tierra Orígenes Shane Sedona Brown , Sandalia de cuero original premier original Teva W de mujer Broncearse , Sandalia Tonya de mujer Trotters Equipaje Sandalia Tamara Trotters para mujeres rojo , JBU by Jambu Women s Wildflower Sandal rojo Aerosoles Womens High Alert Hueso de gamuza Sandalia Barrial Blowfish Mujer Meteorito de bronce / Tan Dyecut Sandalia LifeStride para mujer Eden Soft Blush , Skechers Women s, 2nd Take Tied up Sandalias de diapositivas gris
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 Skechers Women s Go Ride 7 Running Shoe Negro , Reebok Classic Sneaker Sage Mist / Coal / Fierce Fucsia Sandalia deportiva Zvolv X de Chaco para mujer Arco caliente , Teva Winsted Solid - 1017425RASP Rosado Salomon Women s X-Mission 3 W-w Aruba Azul / Azul marino Blazer / Verde Flash , Sandalias de tacón de soñador Easy Street para mujer Armada Zapatillas de running Nike Air Huarache Run para mujer Pure Platinum / Bright Crimsn-leal azul Sandalias de cuero de Viv Viv de TOMS Cáñamo natural / pompones Zapatillas Skechers Go s Lite Lite-Mystic para caminar Negro FitFlop Womens Serene Python Print Slip-On Blanco Urbano Zapatillas de fitness y yoga Vibram VI-B para mujer Rosa oscuro Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Ahumado azul claro adidas Crazytrain Pro W Cross Trainer para mujer Noble Ink / Cyber ​​Gold / Super Purple , Zapato de skate Etnies Jameson Vulc para mujer Borgoña / Tan / Blanco Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on Azul marino / blanco , Sandalias planas Alex de Steve Madden para mujer Leopardo Charles by Charles David Sandalia para mujer Nicki Slide Desnudo Sam Edelman Women s Slide Sandal de mujer Azul / Multi kensie Sandalia Dominic Slide para mujer Negro , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres 1613LEDSS Tacón de 3.5 pulgadas, Cepillo plegable Haz Satén marrón Corral Circle G turquesa grabado Botín Baile vestido de mujer Jessica Simpson Baleenda Patente negra Bomba de Angel II para mujeres estilo suave Elegancia de la Armada Sandalias con cordones Joanie de SOREL para mujer VERDE OLIVA Zapatos de tierra serval Rojo brillante CLARKS Womens Trigenic Flex Esmeralda Mephisto Women s Nykita Oxford Ternero negro Hush Puppies Women s Graham Blucher Oxford Cuero marrón , TOMS Kids 10009918 Alpargata-K Marina Shibori Dots Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Estaño ,

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.


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.

Mocasín de mujer RIALTO Vette Mocasín Serpiente mujer de negra ad225ae - movie-store.online

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 Mocasín de mujer RIALTO Vette Mocasín Serpiente mujer de negra ad225ae it out, and I got a chance to try out this new graphics software.


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 193585
and compass constructions are described and illustrated. The original version, in Word format, can be downloaded and distributed.

Back to Whistler Alley

Last update: March 16, 2016 ... Paul Kunkel whistling@whistleralley.com
For email to reach me, the word geometry must appear in the body of the message.