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.

Calzado de mujer running NIKE 2017 RN 2017 2017 para de mujer Gris 6b37950




Gris
Hecho en Vietnam
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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


Calzado de mujer running NIKE 2017 RN 2017 2017 para de mujer Gris 6b37950 - movie-store.online

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

Calzado de mujer running NIKE 2017 RN 2017 2017 para de mujer Gris 6b37950

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.


Calzado de mujer running NIKE 2017 RN 2017 2017 para de mujer Gris 6b37950

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

Zapatillas de running adidas Aerobounce Pr W para mujer Blanco / Blanco / Core Black , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris negro , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Aruba azul / azul de Aruba / azulejo turco , Zapato de running 840v4 New Balance para mujer Gris oscuro , Zapato de running ASICS Gel-Quantum 360 cm para mujer Negro / Onyx / Rojo caliente Zapatillas de running NIKE Fs Lite 2 para mujer Antracita / Blanco / Negro Zapatillas para correr New Balance Rushv3 para mujer Trueno Negro , Zapatillas de running NIKE Mens Lunarglide 8 Multicolor / multicolor Zapatillas de running adidas Adistar Boost ESM para mujer Taupe / Mint Saucony Women s Ride 10 Running Shoe Gunmetal / rosa adidas outdoor para mujer Terrex Swift R2 GTX Grey Three / Gray Two / Chalk Coral Zapatillas de running Nike Air Huarache Run Ultra Br para mujer Platino Blanco / Blanco-puro Zapatillas de running NIKE Air Zoom Pegasus 33 para mujer Gris oscuro / negro - blanco Zapatillas para correr W630V5 para mujer New Balance Rosado Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul , Zapatillas de running NIKE Air Max Thea para mujer Blanco negro , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco / Universidad Rojo / Negro , HOKA ONE ONE Bondi 4 Mujeres Corso azul / coral de neón Nike Men s Air Presto Essential Velvet Brown / Light Bone , Zapatillas de running ASICS Gel-Noosa Tri 11 para mujer Flash Coral / Spring Bud / Sun , Zapatillas de running Alphabounce HPC w para mujer Adidas Performance Multi Sólido Gris / Calzado Blanco , Zapatillas para correr Saucony Xodus Iso 2 para mujer Berry Coral Zapatillas de running Questr Cc W de mujer adidas Aero Pink S, Aero Pink S, Carbono S Calzado de running Filaity Memory Finity de Fila Negro, blanco, plata metalizado Zapatillas de running Kineta Relay para mujer Saucony Té de menta , Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / espuma de mar , Calzado de running NIKE RN 2017 2017 para mujer Pure Platinum / Black / Wolf Gray / Cool Gray , En la ejecución de calzado de mujer en la nube Todo blanco
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 Terra-Float Livia de Teva para mujer Negro , Chaco Women s ZX2 Classic Athletic Sandal Palm aguacate , Zapatillas de deporte casual Gazelle de adidas para hombre Negro / Correr Blanco , Zapatillas de running ASICS Gel-Kinsei 6 para mujer DIVA AZUL / PLATA / AQUA SPLASH , Zapatillas de running NIKE Roshe One Print para tobillo y tela para mujer Bean Teal / Green Glow Sandalia deportiva ECCO Cruise para mujer Negro Sanda Verra Teva para mujer Negro , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Claro Onix / Luz Onix / Negro , Zapatillas de running New Balance 990v4 para mujer Verde militar / verde militar , La Sportiva Akasha Running Shoe - Mujer Stone Blue / Jade Green , Zapato de running Saucony Zealot ISO 3 para mujer Gray Coral , Zapatillas de running NIKE para mujer Lunarstelos Plata negra / metálica , Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Mora Calzado de caminar para la salud WW840 de New Balance para mujeres Negro Sandalia Freestyle Slide Ii para mujer ECCO Paloma Salvaje , CLARKS Sonar Sandalia para mujer Patente sintética negra Zapato para caminar WW411v2 New Balance para mujer Blanco azul Sandalias Rainbow 301ALTS Single Layer Premier Leather Tan clásico , Zapatillas de running adidas CF Element Race W para mujer Blanco / Gris One / Aero Green , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Candy Pink Suede , Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero marrón , Skechers Women s Walk Walk Lite-15375 Sneaker gris J.Renee Womens Soncino Armada , J.Renee Womens Soncino Cuero metalizado Taupe , Zapato de entrenamiento cruzado Nova Fit Fit para mujer de ASICS Negro / Plata / Rosa Caliente , Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Negro 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) S92309 Trenza dorada y plateada , Bomba de vestido exquisita Aerosoles mujeres champán Zapato casual mujer TOMS Cabrillo Tela blanca / tejido de cesta , TOMS Kids 10009918 Alpargata-K Peacock apenado hoja

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.

Calzado de mujer running NIKE 2017 RN 2017 2017 para de mujer Gris 6b37950 - 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 Calzado de mujer running NIKE 2017 RN 2017 2017 para de mujer Gris 6b37950 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.

Carretera en funcionamiento
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 703172
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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.