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 trail de running trail loto Salomon Women s Speedcross 3 Burdeos/ rosa fuerte/ rosa de loto 2232025




Sintético
Importado
Caucho suela
Precision-fit corredor de pista con transpirable de protección superior con bungee sistema de cordones
Acolchado lengua y cuello
10 mm de talón a punta soltar
Agresivo huella
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 trail de running trail loto Salomon Women s Speedcross 3 Burdeos/ rosa fuerte/ rosa de loto 2232025 - 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 trail de running trail loto Salomon Women s Speedcross 3 Burdeos/ rosa fuerte/ rosa de loto 2232025

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 trail de running trail loto Salomon Women s Speedcross 3 Burdeos/ rosa fuerte/ rosa de loto 2232025

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 impermeables Columbia para mujeres Trail Redmond Guijarro / iluminado por el sol Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral Mizuno Women s & rsquo; s Wave Daichi 2 (W) Trail Running Shoes Negro (negro / negro / rosa diva) , Zapatillas de trail running XA Pro 3D W para mujer Salomon Luz Onix / Luz Onix / Igloo Azul , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Gris / rosa Correpasillos Merrell para mujer 3 Trail Runner báltico , Merrell Siren Edge Hiker para mujer Azul , Merrell Siren Edge Hiker para mujer Aluminio Zapatillas de trail running Salomon para mujer Ellipse 3 Aero W USA Cantera , Zapatillas de trail WT610V5 para mujer New Balance Negro / rosa Entrenador ECCO Biom Fjuel Racer Cross para mujeres Negro , Zapatillas de running Trail Trail Nomad para mujer Saucony Rosado morado , Calzado de running NIKE Zoom All Out Low para hombre NIKE Azul Inov-8 Womens Trailtalon 250 Neon rosa / negro / verde azulado , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Negro / Fácil Verde / Blanco , New Balance Ww1400v1 Trail Walking Boot-w para mujer Gris , Zapatillas SCARPA Neutron 2 Trail Running para mujer Resplandor gris / azul Merrell Women s Mqm Flex Vapor , Zapatillas de trail running Salomon Women s Speedcross 3 Luz Onix / Topaz Azul / Nube Oscura Zapatillas de running mujer Salomon Wings Flyte 2 W para trail running Lotus rosa / rosa fuerte / Citrus-x ASICS Gel-FujiAttack 5 Trail Runner para mujer Aqua Splash / Diva Pink / Azul índigo New Balance WT510RS3 Trail Running Shoes para mujer Negro , Zapatillas de running de tierra firme para entrenamiento de campo Primus Swimrun FG de Vivobarefoot para mujer Negro / Naranja , Zapatillas de trail running Saucony para mujer Grid Excursion Tr10 Negro / Menta Zapatillas de trail running Hiero V2 para mujer New Balance Espacio ultraterrestre / negro / amanecer blanqueado , Calzado de running XA Amphib W Trail para mujer Salomon Potente púrpura Bota de excursionismo SCARPA Neutron GTX para trail running para mujer Clematis / Coral Red , Zapatillas de trail running WT910V2 New Balance para mujer Gris / rosa , Zapatillas de running mujer Salomon Wings Flyte 2 W para trail running Slateblue / Light Onix / Teal Blue F ,
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 Zapatillas de running Saucony para mujer Hurricane ISO 2 Azul / Plata / Citron Zapatillas de running ZanteV2 Breathe de New Balance para mujer Bn2 , Chaco Women s Z1 Classic Athletic Sandal Daga , Zapato casual bajo NIKE Lupinek Flyknit para hombre Cargo Khaki / Mica Green Zapato acuático de fitness Speedo Hydroforce Xt para mujer Cuero gris , Sandalia Barroll T Correa de Natural Soul para mujer Saddle Tan , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul , Sandalia KEEN Women s Kaci de Grano completo Slip-On-w Concha de tortuga / mantillo , Drew Womens Element Nubuck negro , Zapato deportivo artesanal de mujer superfeet Addy Bálsamo / Blanco Easy Spirit Women s Traveltime Mule Azul marino-azul marino oscuro , Adornos Go Step Lite de Skechers Performance para mujeres Negro Zapatillas de entrenamiento NIKE Zoom Fit para mujer gris Entrenador cruzado de malla 77V2 para mujer New Balance Negro / Plata , Zapatos de mujer de neopreno clásico Pedors Negro Vans Unisex Old Skool Zapatillas clásicas de skate Lienzo cordobés Vans Auténtico New York Yankees / Navy Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Armada , Sandalias Hush Puppies para mujer Dachshund Slide Cuero Ivory , 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) 6030 Stardust de oro y cuero , Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Leopard White , Kaitlyn Pan Block Heel Microsuede Botas sobre la rodilla ajustadas ejercito verde , Botas Qupid para mujer Barnes-99a Negro , J.Renee Womens Bilboa Rosa Pastel Multi Trotters para mujer Dea Slingback Pump Negro Negro , Bernie Mev Womens Tara Cano Gris / rojo , Bota de senderismo impermeable X Alp MTN GTX para hombre Salomon Black, Beluga, Lime Punch Zapatillas de deporte de mujer Alegria Essence Icy Blooms , CLARKS Mujer Kinzie Light Loafer Flat borgoña Clásicos de base Slip-on (Tm) para hombres de Vans PANEL NEGRO NEGRO

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 trail de running trail loto Salomon Women s Speedcross 3 Burdeos/ rosa fuerte/ rosa de loto 2232025 - 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 trail de running trail loto Salomon Women s Speedcross 3 Burdeos/ rosa fuerte/ rosa de loto 2232025 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 2468897
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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.