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.

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Amortiguado y receptivo, 0-drop trail running zapatos
Alto caucho de rendimiento único
Relajado talón de elastano para un más natural sentir
Mejor para trail running
High suela de tracción
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 Runventure para correr Topo Athletic Mujer Runventure 2 - Mujer Topo Gris/ amarillo 43b7038 - 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 Runventure para correr Topo Athletic Mujer Runventure 2 - Mujer Topo Gris/ amarillo 43b7038

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 Runventure para correr Topo Athletic Mujer Runventure 2 - Mujer Topo Gris/ amarillo 43b7038

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

HOKA ONE ONE Zapatillas de running Challenger ATR 4 para mujer Negro / Azalea , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Cool Gray / Anthracite-volt / Black-black Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Saúco , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Lobo Gris / Blanco / Hematita Metálica , Zapatillas para correr Mizuno Women s Wave Paradox 3 Capri-rosa Zapatillas de running NIKE RN 2017 para mujer gratis azul , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ecru Tint / Ash Pearl / Ash Pearl , Zapatillas de running NIKE Air Max Zero Premium para hombre Pure Platinum / Wolf Gray , Zapatillas de running Reebok Women s Runner Mt Calavera gris / Asteroide Polvo / Niebla / Blanco / Negro / Peltre Zapatillas de trail running Topo Athletic MT2 - Hombre Azul marino / gris Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple Calzado de running Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle , WMNS Zoom Fly Vasto gris Zapatillas de running NIKE Huarache Run SE para mujer Azul Zapatillas de running A7100w Apex para mujer Armada Zapatillas de running ASICS Gel-Nimbus 19 para mujer Flash Coral / Dark Purple / White , Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird Zapato deportivo ASICS Gel-Cumulus 18 para mujer Azul Indigo / Plata / Rosa Brillo , NIKE Women s WMNS Air Max 270, Blanco / Blanco-Blanco Blanco / Blanco-blanco Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino , Calzado de running NIKE RN 2017 2017 para mujer Solar Red / Black / University Red Calzado deportivo New Balance para mujer Vazee FuelCell V1 Saúco / Plata Zapatillas para correr New Balance Rushv3 para mujer Trueno Negro Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Negro Zapatillas de running Wstro para mujer New Balance Negro / gris / Lime Glow , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle , Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Blanco Zapatillas de running Skechers Go Run Ultra Road 2 para mujer Carbón , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Microchip / Blanco / Carbón metálico ,
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 Nike Kids Air Max Zero Essential GS River Rock / Black-CobbleStone PUMA Men s Roma Basic Sneaker Blanco / Peacoat / avena Sandalia de espalda de mujer Merrell Women s Town Freesia , Sandalia KEEN Clearwater CNX para mujer Laguna / Vapor , Zapatillas de running NIKE Lunartempo 2 para mujer Cloro azul / blanco , Nike Hombres Air Zoom Pegasus 33 Concord / Blanco-negro-eléctrico Verde Zapatillas de agua Merrell para mujer All Out Blaze Aero Sport Beige / Caqui , Camiseta sin mangas con espalda descubierta para mujer NIKE Breathe Púrpura Calzado de senderismo Go Joy 15603 de Skechers para mujer Armada , Deflector de gravedad Alivio del dolor comprobado Desafío G de las mujeres Mighty Walk - Mejor Zapatos para el talón Dolor, dolor en el pie y fascitis plantar gris Timberland Womens Los Angeles Wind Slingback Rose Gold Metallic , Sandalia ECCO Women s Offroad Lite Moon Rock / Gravel , Sandalias de cuero de Viv Viv de TOMS Pom natural del cáñamo Linea Paolo Felicia | Cuero perforado blanco , Skechers Women s Go Walk Joy-15615 Sneaker Azul marino / Multi , Zapatillas Hailee Ryka para mujer Negro / Plata / Ciruela Prop & eacute; t Zapatillas para caminar TravelFit para mujer Púrpura , FootJoy New Womens FJ emBody zapatos de golf de liquidación - Elija el tamaño! Blanco , Sandalias de mujer Ros Hommerson Mellow Negro , Sandalia Ryana Marilyn Slide para mujer Ciruela / blanco , Sandalias Jerusalem Sandalia Nuri Slide rojo Zapatillas de volante de mujer adidas Crazyflight X W de adidas Tiza Púrpura / Oro Metálico / Tinta Noble , BlueBell Blue Bell Zapatos hechos a mano Salón de baile de mujeres Concurso de Boda Salsa Zapatos de baile estilo Matilda 3.5 \ Plata Laredo Women s Miss Kate Western Boot Marrón / verde azulado ScubaPro Unisex 5mm Delta Boots Alto 5mm , Frye Womens Nora Stud Zip Short Arena Zapatillas de deporte Moc de Acorn Women s Spa Support Negro , Sorel Women s Danica Bootie Tabaco / negro , Bota Dafne Thierry Rabotin para mujer Negro Betunia Loafer de mujer Aerosoles Serpiente negra

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 Runventure para correr Topo Athletic Mujer Runventure 2 - Mujer Topo Gris/ amarillo 43b7038 - 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 Runventure para correr Topo Athletic Mujer Runventure 2 - Mujer Topo Gris/ amarillo 43b7038 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 1149387
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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.