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.

Sandalia Faye Eastland Slide Slide de de mujer de Eastland Broncearse bcfe212




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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


Sandalia Faye Eastland Slide Slide de de mujer de Eastland Broncearse bcfe212 - 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

Sandalia Faye Eastland Slide Slide de de mujer de Eastland Broncearse bcfe212

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.


Sandalia Faye Eastland Slide Slide de de mujer de Eastland Broncearse bcfe212

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

Sandalias planas de piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural Sandalia Rockdale Lanea Woven Slide para mujer Tejido negro , Sandalia Maria Slide de Nanette Lepore para mujer Blanquecino Sandalia Rilee Slide Dolce Vita para mujer Blush Suede , UGG Danes Danes Slide Sandal Horchata , Sandalia de diapositivas de ocio para mujer de Seychelles Negro , Sandalia KAANAS Sausalito Bow Espadrille Slide para mujer Maravilla , Sandalias con tachuelas Gale para mujer Donald J Pliner Armada Sandalia Dolce Vita Celita para mujer Cuero Mocha , Sam Edelman Women s Bay Slide Sandal Saddle Vaquero Saddle Leather Sandalia de mujer Woodstock de Jambu Slide Taupe claro Skechers Cali Women s Beverlee Wedge Sandalia de cuña Multicolor , Sandalia Dolce Vita con diadema para mujer Cuero Ivory Sandalia para mujer J.Renee Maribeth Slide Negro , Sandalias planas de piscina Walking Cradles para mujer Black Matte Snake Print / Cork Wrap Sandalia Dolce Vita Gia-S Slide para mujer Cuero rosa dorado , Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro Sandalia Aerosoles Midday Women Slide Tela de mezclilla , Sandalia para mujer, de color rosa, de Jack Rogers, black1, 10 Medium US - Sandalia para mujer Ivanka Trump Earin Slide Ante rojo , Sandalia plana Kip de Trotters para mujer Negro Sandalia Emilia High Slide de Cole Haan para mujer Ante negro Sandalia Vionic para mujer con tanga Broncearse , Sandalia Sonia Slide de Dolce Vita para mujer Elástico negro Sandalia de la diapositiva Izbremma de la mujer de Lucky Brand Negro Birkenstock Women s Arizona SFB Fashion Leather Espectacular cuero platino Calzado de mujer Calvin Klein Patreece Slide Coñac , Birkenstock Mujer Mayari Birko-Flor Sandal Negro , Sandalia de mujer Frisco Slide de Franco Sarto Lavanda ,
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 KEEN Mujer Whisper-w Vino de uva / vino de uva , Sandalia atlética Salomon Women s Crossamphibian Swift W Higo Zapatillas de running Strobe V1 para mujer New Balance Blanco Zapatillas náuticas para mujer Saucony Originals borgoña Zapatillas de running NIKE Air Huarache para hombre Blanco negro , Zapato de correr ideal para rejilla de mujer Saucony Negro / Gris / Pr , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Trueno , Zapato de correr ideal para rejilla de mujer Saucony Gris blanco Calzado de running para mujer New Balance Cypher Castlerock / White Sandalias EVA Gizeh de mujer Birkenstock Eva negro , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Pinkorange 789 , Zapatillas NIKE Kids Revolution 3 (GS) para correr Dinastía púrpura / plata metálica Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro Sandalia de vestir de la ciudad de Aerosoles mediados de mujeres Negro Sandalia Drew Cascade para mujer Sand Nubuck New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Gris , nativo AP Mercury Liteknit Shell White / Shell White / Natural Rubber Zapatilla de deporte Fierce Bright Cross-Trainer para mujer de PUMA Peacoat / Peacoat , Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Blanco , Zapatillas de baloncesto NIKE Air Python PRM - Hombre, de punta redonda, de cuero Pure Platinum / Pr Pltnm / Pr Pltnm , Sandalia con tiras metálicas Violet Shimmer de Taryn Rose para mujer Poppy Silky Suede Sandalia de la diapositiva de Truesdale de las mujeres de Steve Madden Ante negro , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1612- Black Pu , Bomba de vestimenta Cambredge para mujer Jessica Simpson Desnudo , Zapatilla Ansley para mujer de UGG Melocotón tropical , Bota Premium Timberland para mujer de 6 \ Simplemente Taupe Suede , Bota Deia para mujer de TOMS Cuero negro , U-lite mujer perforada con cordones Wingtip cuero plano Oxford zapatos brogues vintage Negro Sanuk Women s Pair O Dice Flat Negro TOMS Clásicos de mujer Oxford Tan Teepee

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.

Sandalia Faye Eastland Slide Slide de de mujer de Eastland Broncearse bcfe212 - 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 Sandalia Faye Eastland Slide Slide de de mujer de Eastland Broncearse bcfe212 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 432547
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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.