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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Cuero
Importado
Cuero suela
Cuero forro
Cuero único
Cadena detalles del enlace
Abarcando correa del dedo del pie
Corto talón mide 0.5
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

Sandalia Beau Beau Slide Kate rosa Spade New York para mujer Spade Oro rosa e4d33c9

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.


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

Sandalia de diapositiva de Delilah de mujer de Jack Rogers Black1 , KAANAS Sandalias planas de alpargatas tejidas de Martinica para mujer Gris , Sandalia Dolce Vita de mujer Kaira Slide Ante amarillo , Sandalias clásicas para mujer del Dr. Scholl s Shoes Saddle Snake Print Plataforma CLARKS Women s Parram Waltz Cuero negro Sandalia Chloe Slide de KENDALL + KYLIE Violeta , Vince Camuto Sandalias Ellanna Slide para mujer Urban Lux , Sandalias Hush Puppies para mujer Dachshund Slide Cuero marrón , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla SCHUTZ Sandalia Yaslin Slide para mujer Negro , Sandalia Kace para mujer Calvin Klein Blanco suave Sandalias con plataforma de cuña Tory Burch Patty Platform, Cuoio , Birkenstock Mules para mujeres Blanco Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero negro Sandalia Daisy Pave Aqua Slide de Marc Jacobs para mujer Blanco , Sandalia Dolce Vita Jene Slide para mujer Tela blanca a rayas BareTraps Sandalia para mujer Flossey Slide champán , Sandalia de la diapositiva de Truesdale de las mujeres de Steve Madden Ante negro Sandalia Cato Dolce Vita para mujer Cuero blanco roto , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro Zapatilla Ascot de hombre UGG Negro / China Tea , Sandalia Rokkit Crystal Slide de FitFlop para mujer Azul (Supernavy) , Sandalia Jamaika de Finn Comfort para mujer Beige , Sandalias planas Cammie de Jessica Simpson para mujer Desnudo Blush Sandalia para mujer FRYE Ally Criss Cross Slide Blanco , Kenneth Cole New York Sandalia para mujer Janie Slide Multi Sandalia para mujer Castlero Slide de ALDO Armada , La sandalia Finley Dome Stud Slide para mujer Azul brillante , Sandalia de vestir Dansko Dee mujer Negro Full Grain ,
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 viaje Protic W3910 Vista Walker para mujeres Taupe liso , Zapatillas de running Zoot W Ultra Tempo para mujer Amarillo de seguridad / Remolacha / Negro , Sandalia NIKE Benassi JDI para mujer Negro / Vivid Pink Chaco Women s Z1 Classic Athletic Sandal Trellis Teal Corkys Rock Mujer Slip On Amber Multi , OluKai Ohana Dalia / Negro , Brooks Womens Cascadia 13 Tinta / azul marino / rosa , Reebok Women s Ridgerider 2.0 Trail Runner Negro / Rosa Solar / Plata / Estaño Sanda Verra Teva para mujer Surf Negro / Multi Zapatillas de deporte Adidas Sport Performance para mujer Terrex Agravic, gris, textil, goma, 9.5 M - , Zapatillas de senderismo Merrell Moab 2 GTX para mujer Negro , Skechers Women s Go 4-14918 Calzado para caminar Carbón Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Mora Naturalizer Reilly Espadrille Sandal de las mujeres Azul , Sandalia plana Nd54 Stella para mujer El Naturalista Negro 2 , Toalla de vestir Dulce para mujer Touch Ups Silver Shimmer , Finn Comfort Women s Ikebukuro Oxford Marron Crash / Sensorial , Zapatillas de golf Skechers Performance para mujer Go Golf Elite Canvas Carbón Zapatillas DC Danni TX para mujeres Blanco , Zapato de skate DC Wnt Spartan High WC para mujer café , Zapatillas de golf adidas Climacool Knit Cblack / D, mujer, de adidas Energy Blue Sandalias planas Softey para mujer Steve Madden Rosa Multi , Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Leopard Satin & Ultra Gold Trim Cole Staan WP de Cole Haan Women Marca Brown Pikolinos Womens Porto W6J-5804 brandy Bota vaquera Aquila para mujeres muy volátil Gris pardo , Sandalias de verano XYD Fringe Slide Slip On Comfortable Slippers Zapatillas de hombre sólido holgazán Easy Walking Negro , Trotters Liz Loafer para mujer Blanco Trotters Liz Loafer para mujer Rosado Mocasín de gamuza Kilty de Minnetonka para mujer Gris ,

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 Sandalia Beau Beau Slide Kate rosa Spade New York para mujer Spade Oro rosa e4d33c9 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 1581505
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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.