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.

Mocasines 18621 de para vestir de punta estrecha para vestir mujer de Me Too Women Blue Champagne Denim 0aaa52f




Denim
Hecho en EE. UU. o importados
goma suela
Medidas de talón aproximadamente 0.5 pulgadas
Completo tu mirada en estos versátil Me Too & trade; pisos.
Cayó cuero o metálico superior.
Fácil estilo sin cordones.
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


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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

Mocasines 18621 de para vestir de punta estrecha para vestir mujer de Me Too Women Blue Champagne Denim 0aaa52f

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.


Mocasines 18621 de para vestir de punta estrecha para vestir mujer de Me Too Women Blue Champagne Denim 0aaa52f

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

Propet Women s Tour Walker Strap Sneaker Negro Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Negro / Bronce Metálico , Clásicos de base Slip-on (Tm) para hombres de Vans Carbón , Mocasines de vestir de punta estrecha para mujer de Me Too Women Blue Champagne Denim ENMAYER Mujeres Moda retro negra Mocasines con punta redonda Resbalón en piso al aire libre Zapatillas con peludo y hebilla Rosa # 1 , Zapato femenino Keli de Alegria Birdland TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia Zapatillas Slip-On Roamer SAS para Mujer Sabio , Clásicos de base Slip-on (Tm) para hombres de Vans Negro / Marshmallow , Zapatillas de barco Bluefish Sperry Top-Sider para mujer Lino de dos tonos , Easy Spirit Women s Traveltime Mule Cuero blanco / azul claro TOMS Clásicos de mujer Lienzo Turquesa , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Azul marino Slub Chambray Cupsole , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / blanco Toms Womens Classics Black / Black Canvas 10002472 Armada , TOMS Clásicos de mujer Negro Mocasines casuales Alpargata Deconstructed de TOMS para mujer Ante negro Trotters Liz Loafer para mujer Estaño Trask Womens Ali Peltre de gamuza italiana , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Chambray azul , TOMS - Slip-On de lienzo clásico para hombre Azul eléctrico , Finn Comfort Women s 2552-014099 Nappa negro , Trotters Liz Loafer para mujer Hueso CLARKS Women s CloudSteppers Sillian Jetay Flat Tela gris aterciopelada CLARKS Cheney Madi Slip-on Loafer para mujer Cuero marrón oscuro Scrunch Zapato femenino Keli de Alegria Desenfoque de belleza , Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer Blanco Zapato femenino Keli de Alegria Quarry Crackle Skechers Women s Reggae Fest Willows Flat Malla negra ,
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 Juvenate para mujer Chalk Blue / Loyal Blue , Zapatillas de running W775V2 para mujer New Balance Espacio exterior adidas outdoor para mujer Terrex Swift R2 GTX Grey Three / Gray Two / Chalk Coral Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Vestido Blues / Neutral Gray , adidas Terrex Agravic GTX Shoe Trail para mujer Running 11 Tactile Pink-Haze Coral-White - Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Saúco , Zapatillas de running NIKE Womens Zoom Rival D 9 Trail Fútbol Azul / Azul Fox-ice Blue , Zapatillas de trail running Columbia Montrail para mujer Bajada III Ti Gray Steel, Coastal Blue , NIKE Women s Revolution 4 Running Shoe Negro antracita blanco , J.Renee Womens Soncino Blanco / Kidskin Zapatillas para caminar New Balance 85v1 para mujer Verde gris , Zapatillas Erika Propet para mujer Hueso Sandalia Vanessa de mujer Alegria Mosaico de peltre , Sandalia Bella Vita IMO-Italia para mujer Cuero negro Sam Edelman Womens Barry Piel de serpiente Royal Natural Print Zapatillas de viaje Proather para mujer II Malla blanca Zapatilla Ascot de hombre UGG Pewter / Sahara Drew Smiles Slip para mujer Negro / Microdot Adidas Adipower Weightlift Cross Trainer Trace Cargo / Trace Cargo , Zapato Dantera, para hombre, L Low Waxed Court Blanco , Zapatillas DC Mujeres Chelsea TX SE Sneaker Azul / azul / blanco Sandalia Buxom Slide de Charles David para mujer Plata , Charles by Charles David Sandalia de cuña Brooke para mujer Microsuede gris claro Sandalia Kai Dolce Vita para mujer Cuero Ivory CLARKS Calla Rose Pump para mujer Crema de charol , Bota de plataforma Ranger-301 para mujer Demonia Pleaser Black Pu TOMS - Slip-On de lienzo clásico para hombre Faded Rose Microfiber Monocromo , Merrell Solo Aluminio , Steve Madden Womens Kyra Blanco Multi , Easy Spirit Women s Traveltime Mule Blanco 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.

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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 Mocasines 18621 de para vestir de punta estrecha para vestir mujer de Me Too Women Blue Champagne Denim 0aaa52f 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.

Mocasines y Slip-Ons
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 281212
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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.