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.

Rookite de 325 Trail Runner de Gris Runner mujeres Inov-8 Gris oscuro a8f6d02




Textil y sintético
Importado
Caucho único
Siguiente generación Suela Roclite, presenta nuestro Tri-C compuesto de goma y multidireccional calas en forma de garra, proporcionando un incomparable agarre sobre impredecible terreno y el lanzamiento rápido de escombros.
Aumentado soporte y comodidad: El corte de arranque ofrece aumento de tobillo soporte, mientras superior superdurable materiales combinados con una caída de 8 mm asegurar una excelente comodidad en el correr.
Acolchado paseo: dos piezas PowerFlow entrega 10% mejor choque absorción y 15% mejor energía retorno de lo normal suelas.
El nuevo Adapterweb adaptadores met-cradle a lo natural movimiento e hinchazón del pie en movimiento; El sistema X-Lock retiene el talón en su lugar y ofrece aumento apoyo.
Segunda generación Meta-Shank es un placa de roca que protege el pie de rocas afiladas, mientras que un fortalecido puntera de goma protege los dedos de los pies de obstructiva escombros.
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


Rookite de 325 Trail Runner de Gris Runner mujeres Inov-8 Gris oscuro a8f6d02 - 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

Rookite de 325 Trail Runner de Gris Runner mujeres Inov-8 Gris oscuro a8f6d02

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.


Rookite de 325 Trail Runner de Gris Runner mujeres Inov-8 Gris oscuro a8f6d02

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 trail running X-Mission 3 CS W de Salomon para mujer Pasión Púrpura / Coral Punch / Cristal , Columbia Montrail Mountain Masochist IV Outdry Trail Running Shoe para mujer Grafito, camelia roja Zapatillas de running La Sportiva para mujer Trailido Bushido Plum Apple Green , MT-onemix Women Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Blanquecino , Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Azul marino / rojo / blanco , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Gris negro ASICS Gel-Kahana 8 Trail Runner femenino Gris medio / carbono / Limelight , Zapatillas para correr Inov8 Women s Parkclaw 275 GTX Negro / Coral Zapato de mujer Columbia de Redmond Trail Boulder, azul cielo , Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Negro , Salomon Women s X-Mission 3 W-w Atlántico Norte , Salomon Xa Pro 3D GTX W Trail Runner Tahitian Tide / Eggshell Blue / Estanque Reflectante La Sportiva Women s Wildcat 2.0 GTX Trail Running Shoe Gris / rojo , Corredor de trail Speed-Wak W-W de mujeres Salomon Nube Oscura / Luz Onix / Azul Burbuja , Saucony Women s Xodus Iso Trail Runner Denim / negro / rosa Zapatillas de trail running Saucony para mujer Grid Excursion Tr10 Gris / Azul / Coral Zapatillas de trail running WT10v1 Minimus para mujer New Balance Gris azul Salomon Women s Speedcross 4 W Trail Runner Cantera Zapatillas de trail running Salomon Women s Sense Mantra 3 W Pasión Púrpura / Púrpura Cósmica / Granny Green Altra Women s Torin 2.5 Trail Runner Gris / frambuesa Zapatillas para correr de velocidad vertical Vasquez para mujer y paquete de gorro de punto Ébano / blanco , Zapatillas de mujer Salomon X Ultra 3 W para excursionismo Sombra Merrell Siren Edge Hiker para mujer Loza , Zapatillas de running Zante V4 para mujer New Balance Blue Iris / Pigment Zapatillas de running NIKE Zoom All Out Low para hombre Negro / Plata Metálico / Concord Xero Shoes TerraFlex Trail Running Senderismo Zapato - Minimalista Zero-Drop Ligero Barefoot-Inspired - Mujer Negro Zapatillas de trail running Salomon para mujer Techamphibian 3 W Clima tormentoso Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Imán / Rojo verdadero Zapatillas Trail para caminar New Balance 779v1 para mujer Gris oscuro ,
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 Sandalias Newport H2 de KEEN para mujer zen Sandalia deportiva Nolita para mujer Anne Klein Tela negra Skechers Performance On The Go 400 Vivacity Flip Flop Negro , Zapato de agua Tetrex Rapid Crest para mujer Merrell Negro Drew Shoe Athena para mujeres Plata blanca , Sam Edelman Women s Slide Sandalia Oro / cuero desnudo metálico / cuero de gamuza , CLARKS Sandalia Saylie Jade para mujer Cuero marrón caído Sandalia de mujer David Tate Violet Negro Apis Mt. Negro , Easy Spirit Women s Traveltime Mule gris , Me Too Women s Lightning 2 Pump Nappa Metálica Rosa Dorada , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Silver Mink / Pigment , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Verde Converse Chuck Taylor All Star Core Ox Sodalita Azul / Blanco Under Armour Womens UA Remix Blanco / acero / acero , Reebok Crossfit Nano 7.0 Track Shoe para mujer Rojo / negro / blanco primigenio , Reebok Crossfit Nano 7.0 Track Shoe para mujer Nube gris / morado / blanco Vans Auténtico Plata , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Black Pro Purple Ridgerock Lig , Zapatillas de tenis New Court 896v2 Hard Court para mujer Naval Blanca Zapatos de golf de ocio FootJoy para mujer Gris / rosa-m Zapatillas de golf adidas Bounce para mujer, adidas, para mujer Ftwr Blanco / Plata Met./Semi Frozen Yellow Fabric Sam Edelman Women s Slide Sandalia Yute / Caramelo Dorado , Sandalia de viaje Via Spiga Women s Porter Pavo real gamuza , Sandalia Dolce Vita Delana Slide para mujer Cuero rojo 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1680- Cuero plateado , Zapatilla de ballet metálico con suela completa B902 de Dance Class para mujer Oro Pantuflas Kendale de mujer Foamtreads Dusty Rose Pantuflas Lorna de Katy Perry para mujer Negro , Pantuflas Mahabis Classic para hombre gris Gris oscuro / Negro Skien

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.

Rookite de 325 Trail Runner de Gris Runner mujeres Inov-8 Gris oscuro a8f6d02 - 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 Rookite de 325 Trail Runner de Gris Runner mujeres Inov-8 Gris oscuro a8f6d02 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

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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.