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.

Zapatillas de running New negro Zapatillas Trail Balance Gobi v2 Fresh Foam Trail para mujer Rosa/ negro 2b6da3f




Textil y sintético
Importado
Sintético único
Eje mide aproximadamente parte baja del arco
Fresco Entresuela de espuma
AT Suela TREAD
Toe proteger
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


Zapatillas de running New negro Zapatillas Trail Balance Gobi v2 Fresh Foam Trail para mujer Rosa/ negro 2b6da3f - 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

Zapatillas de running New negro Zapatillas Trail Balance Gobi v2 Fresh Foam Trail para mujer Rosa/ negro 2b6da3f

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.


Zapatillas de running New negro Zapatillas Trail Balance Gobi v2 Fresh Foam Trail para mujer Rosa/ negro 2b6da3f

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

Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Imán / Rojo verdadero , Topo MT-2 Trail Running Shoes & Workout Headband Bundle Negro / Hielo Zapatillas de running Icebug para mujer Acceleritas4 RB9X Traction Orquídea / Veneno Zapato casual bajo NIKE Lupinek Flyknit para hombre Vela / Negro / Antracita Merrell Women s Mqm Flex Gris negro Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Coral Fácil / Coral Onix / Haze Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker W para mujer Gray Five / Chalk White / Easy Coral Zapatillas de trail running WT10v1 Minimus para mujer New Balance Negro Inov-8 Roclite 295-U Trail Runner Gris / azul marino / azul Zapatillas de trail running Olympus 2 de Altra para mujer Orquídea / Gris New Balance WT510RS3 Trail Running Shoes para mujer Gris oscuro Zapatillas de montaña La Sportiva Tempesta GTX para mujer Stone Blue / Mint Zapatillas de running adidas Terrex Agravic trail running para mujer Core Green / Black / Easy Orange - 7.5 - Zapatillas de running Trail Trail Trail Trail Trail Trail Trail N1 para mujer de Pearl Izumi Shadow Gray / Meadow Mauve Columbia Montrail Women s Trans Alps F.K.T. Jade Lime, Splash , Correpasillos Merrell para mujer 3 Trail Runner báltico UB-ONEMIX Cool Sports Shoes Running Malla transpirable Zapatillas de deporte con cordones Tamaño EU 35-45 Blanco verde Zapatillas de trail running Trail Trail Trail N3 para mujer Pearl iZUMi Vino Púrpura / Clementina , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Reflexión Salomon Women s Speedcross 4 CS W Trail Runner Mallard azul Corredor SCARPA Neutron Wmn Running Shoe Trail Runner para mujer Atolón , Zapatillas de trail running Saucony para mujer Grid Excursion Tr10 Negro / Menta , Zapatillas Altra Women s Superior 3.5 Sneaker Rosado Correpasillos Merrell para mujer 3 Trail Runner Punch ácido Zapatillas de trail running WT10v1 Minimus para mujer New Balance Negro Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Acero , Zapatillas de running La Sportiva para mujer Trailido Bushido Fiordo Zapatillas de running New Trail Hierro V3 Fresh Foam Trail para mujer Gris Zapatillas de senderismo V753WW085 para mujer Apex 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 Converse Chuck Taylor All Star Seasonal Color Hola Cosmos rosa / caballito de mar Adidas Women s Adilette CF + Logo W Sandalia deslizante Ash Blue / Aero Blue / Black , Brooks Womens Ariel 18 Gris / gris / rosa Zapatillas de running adidas Aerobounce Pr W para mujer Núcleo Negro, Plata Met., Ftwr Blanco , Converse Chuck Taylor All Star Seasonal Color Hola Nocturno Rojo / Azul , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Sun Blow / Coral Pink Chancleta Eezay adidas para mujer Shock Pink / Collegiate Royal / Aero Blue , Sandalia Capri universal Teva para mujer Chocolate perlado Zapatillas de running Nike Air Zoom Elite 8 para mujer Blanco / Negro-volt-hiper Naranja Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Trotters Mujer Simmy Mary Jane Flat Sabio , Zapatilla de deporte con raya de mujer Onemix, zapatos para caminar para correr Rosado azul , Sandalia deportiva Marc Jacobs Women s Somewhere Raspberry Multi , Zapatillas para caminar Propell Mujer Travellite MJ Plata Zapatillas de deporte GEL-Fit Sana 2 de ASICS para mujer Campana azul / azul violeta / lima Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Ultra Magenta-puma Blanco Calzado de senderismo Skechers Go s 15601 para mujer Blanco negro , Correa para mujer Earth Slip-On Suede metálico estampado plateado claro DC Youth Rebound Skate Shoes marrón Sandalia Victoria Slide de mujer SCHUTZ Perla , Sandalia para mujer Odosi City Slide de Soludos Denim mediano Betsey Johnson Women s Slide Sandalia Guinga roja Skechers para mujeres Reggae Fest-Marlin-Fisherman Open Back Mule Ajuste relajado y espuma de memoria a / C Zapato de agua Natural , UGG Danes Danes Slide Sandal Horchata Twisted X Ladies Turquoise Lace Driving Mocs Bombardeo Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Gris , Botín Abiha SCHUTZ para mujer Otoño , Me Too Womens zena14 Suede Almond Toe Tobillera Botas de moda Simba Suede , Bota de campo Ovation Flex Sport para mujer Negro , Zapatillas de fútbol adidas Performance Women s Ace 17.4 FG W Blanco / Negro / Core Red S

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.

Zapatillas de running New negro Zapatillas Trail Balance Gobi v2 Fresh Foam Trail para mujer Rosa/ negro 2b6da3f - 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 Zapatillas de running New negro Zapatillas Trail Balance Gobi v2 Fresh Foam Trail para mujer Rosa/ negro 2b6da3f 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.

Trail Running
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2182710
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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.