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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Hecho en EE. UU. O importado
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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


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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

Adicción a/ a Brooks Adicción 12 Antracita/ Bluefish/ Silver 979df78

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.


Adicción a/ a Brooks Adicción 12 Antracita/ Bluefish/ Silver 979df78

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 running adidas Alphabounce Rc W para mujer Gris claro Heather / Light Solid Gray / Medium Solid Gray Zapatillas para correr Mizuno Women s Wave Paradox 3 Negro eléctrico verde , Zapatillas de running Saucony Women s Mirage 4 Verde negro Calzado de running NIKE Kids Roshe One (GS) Piedra de hueso claro / blanco / adoquín Zapatillas de running New Balance 520v3 para mujer Artic Fox / Black Calzado de running casual para mujer New Balance WL555, para mujeres Menta de plata Zapatillas de running NIKE Lunar Skyelux para mujer Aluminio / Binario Azul / Azul Mediano , Zapatillas de running Wstro para mujer New Balance Visón plateado Lanzamiento de Brooks Womens 4 Antracita / cerámica / sacador de lima , Zapatillas de running ASICS Gel-Kayano 25 para mujer, blanco / blanco, 6.5 B (M) US - , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow Zapatillas para correr NIKE Kaishi 2.0 para mujer Wolf Gray / Pink Black / White , Skechers Sport Skech Air Run High Fashion Sneaker Adornos de malla negra / rosa fuerte , Amortiguación 635v2 New Balance para mujer Gris Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink Black Bright Mango 600 , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , NIKE Men s Roshe Two Running Shoe University Red / White / Track Red Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel Zapato de running ASICS Alpine XT para mujer Gris Pizarra / Gris Pizarra , Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul , Zapatillas de running definitivas Ryka para mujer Frost Grey / English Rose / Chrome Silver Skechers Performance Women s Go Run 400 Obstruct Azul marino / Aqua Zapatillas de running adidas Ultraboost W para mujer Misterio Rojo / Misterio Rojo / Rosa Táctil Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento , NIKE Women s Tanjun Iced Jade Aurora verde blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Gris pardo / Vino de puerto-solar rojo , NIKE Men s Benassi Just Do It Athletic Sandal Midnight Navy / Windchill ,
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 Altra Women s Intuition 4 Ocean / Teal , Nfinity Vengeance Cheer Shoe (par) Blanco Zapatillas de running Adidas Performance Ultra s Boost M para hombre Claro / Gris / Claro / Gris / Gris Medio , Zapatillas de running AMPLICA para mujer ASICS Gris Medio / Plata / Aruba Azul Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento , Zapatillas sin cordones Flex Merit para mujer Merrell Rosa roja Zapatillas de trail running Salomon Speedcross 4 para hombre Ibis del Atlántico Norte / Negro / Escarlata MONTAÑA BLANCA Mujer Marvy Peep-Toe Mule Negro Zapato para caminar Metrolyte para mujer ASICS Azul índigo / Orquídea / Plata , Good-memories 2018 verano mujer zapatillas plano antideslizante interior informal peludo playa sandalias Frost Black Sperry Womens Fathom Negro , Skechers para el trabajo de la mujer Albie Walking Shoe Blanco DAWGS Womens Arch Support Z Sandals Tréboles , Tecnología Vionic Men s Orthaheel White / Navy Walker Gris , Zapatillas Ryka Dynamic 2.5 Cross-Trainer para mujer Blanco negro , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Pure Platinum / Night Purple / Wolf Gray New Balance 711v3 Heather Cross Trainer para mujeres Blanco negro , Zapatillas de skate DC SE Pure High-Top TX SE de mujer Gris / gris / gris , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Negro / Estiramiento , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro , Sandalia Slide de Elina de Franco Sarto para mujeres Platino , Ariat Womens Scout Paddock Negro , Noble Outfitters Womens Muds Stay Cool 6 Deep Turquoise , Rieker Women s, 62854 Slide Sandal Azul , Bomba de cuña Yaralla J.Renee para mujer Negro , TOMS - Slip-On de lienzo clásico para hombre Gray Tribal Wool / shearling , Zapatilla Glamour de mujer verde Daniel Gold Kidskin Zapatillas de deporte Apollo Chukka de hombre nativas Regatta Blue / Shell White / Shell White Rubber SOREL Women s Out n About Moccasins Nori / Nectar Mocasines casuales Alpargata Deconstructed de TOMS para mujer Saffron Suede

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.

Adicción a/ a Brooks Adicción 12 Antracita/ Bluefish/ Silver 979df78 - 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 Adicción a/ a Brooks Adicción 12 Antracita/ Bluefish/ Silver 979df78 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

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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.