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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Textil y sintético
Importado
Caucho único
Seguro ajuste de captura de encaje sistema
Metatomical Plantilla de EVA
Non-marking suela de goma
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 Uneek O2-w Rose de mujer KEEN Uneek Anemone/ 12758 Bright Rose 987d5fc

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 Uneek O2-w Rose de mujer KEEN Uneek Anemone/ 12758 Bright Rose 987d5fc

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

Chaco Women s Z1 Classic Athletic Sandal Trellis Teal Sandalia Archie Fisherman del Dr. Scholl Monumento Nubuck Sandalia NIKE Kawa Slide para mujer Iris oscuro / blanco / aluminio Sandalias Swiftwater Mesh de Crocs para mujer Negro Sandalia Terrell Ivy Post Sport Merrell para mujer Sodalita Drew Womens Cascade Denim Blue Nubuck Chaco Women s Zx3 Classic Athletic Sandal Lima gris , BZees Sandalia con cuña de beso para mujer Negro , Earth Origins Sophie Azul marroquí , Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink Sandalia Drew Angela para mujer Bone Smooth Leather Sandalia KEEN Women s Uneek Slide-w Aqua Sea / Turquesa Pastel , Sandalia Drew Angela para mujer Dusty Multi Metallic Leather Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro , Wolky Comfort Jewel Negro 2 , Sandalia Terrell Post II Femenina de Merrell Tierra oscura CLARKS Sandalia plana Brizo Ravena para mujer Gris , Chaco Mujer Zcloud X2 Sandalia Deportiva Kelp Knit Sandalia original Teva Women s Original Miramar Fade Dark Purple / Multi Sandalia deportiva W Sirra de Teva para mujer Negro , Skechers Performance Women s The Go 400 Radiance Flip Flop Azul marino / blanco , CLARKS Sandalia de pescador de melón Tuvia para mujer Broncearse OluKai Kulapa Kai Buganvillas / negro , Sandalia deportiva Tetra Cloud de Chaco para mujer Jengibre especias Sandalia de mujer Drew Alana Blanco negro Sandalia KEEN Clearwater CNX para mujer Gárgola / azul nórdico , OluKai Kulapa Kai Contton Candy / Black Chaco Zcloud 2 Sport Sandal para hombre Pirámide Verde Sandalia Yucatan ECCO Mujer Chili Red / Concrete / Black
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 Converse Chuck Taylor All Star Seasonal Color Hola Limón amargo NIKE Mujeres Air Zoom Pegasus 33 Blanco / Pink Blast / Electric Green / Black Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Coral vivo / negro / blanco Munsell Zapatillas de running Saucony para mujer Stretch N Go Breeze Negro , Nike Hombres Air Zoom Pegasus 33 Lobo gris / negro / gris oscuro / azul Calzado de running New Balance para mujer Vazee 2090 Rosa / amarillo , Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Hortensias / Plata Metálica / Cool Gray , Zapatillas de running Trail Trail Nomad para mujer Saucony Azul marino , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla del océano / Cirrus Blue / College Navy Zapatillas de running Newton Motion Motion S para mujer rojo , SoftWalk Women s Topeka Flat Sabio , Sandalia universal W Ysidro de mujer Teva Gris pardo , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Cobre rosa / sal marina / chicle Sandalia de tobillo con correa Gomera Finn Comfort para mujer Niebla tóxica Sandalia de vestido de mujer Patty de Mephisto Negro seda / gris oscuro Python , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Lima / Azul PUMA Women s Fierce Rope Pleats Wn Sneaker Paradise Pink-paradise Pink Zapato de golf Biom G2 para mujer ECCO Blanco / Danubio Zapatillas de golf Lexington Marc Joseph New York para mujer Cielo Black Toalla de plataforma ECCO Women s Touch Plateau Platform Slide Rose Dust , Sandalia de cuero Merrell Dwaykair Seaway para mujer Cielo azul , Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Cuero negro Zapatos de alpargatas Toms Classic Tan Shibori Dots para mujer FRYE Bota de tobillo corto con botones Melissa para mujer Ice Washed Antique Pull Up Leather-77908 Themost Womens Pointed Toe Sexy Impreso Stretchy Stiletto High Heels Over The Knee Muslo Botas altas Leopardo , Corral Botas bordado de hueso de mujer & Swarovski Snip Toe C3178 Hueso Sam Edelman Loraine Loaine para mujer Blanco brillante Mocasines de vestir de punta estrecha para mujer de Me Too Women Equipaje , Zapatos antideslizantes clásicos de lona de mujer de TOMS Negro sobre negro , CLARKS Women s CloudSteppers Sillian Jetay Flat Gris claro

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 Uneek O2-w Rose de mujer KEEN Uneek Anemone/ 12758 Bright Rose 987d5fc - 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 Uneek O2-w Rose de mujer KEEN Uneek Anemone/ 12758 Bright Rose 987d5fc 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.

Sandalias y toboganes deportivos
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.