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.

mobils by Mephisto - tacón - Sandalias de tacón Platinum Getha para mujer Camel Savana/ Platinum Magic Leather 64b836a




Cuero
Goma suela
Altura del talón: 1 \ Importado
Ajuste: Fiel al tamaño
Plantilla: RELAX extraíble Plantilla
suela: Caucho
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


mobils by Mephisto - tacón - Sandalias de tacón Platinum Getha para mujer Camel Savana/ Platinum Magic Leather 64b836a - 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

mobils by Mephisto - tacón - Sandalias de tacón Platinum Getha para mujer Camel Savana/ Platinum Magic Leather 64b836a

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,

Northside Womens Savannah Sport Closed Toe Sandal Tan / Coral , Chaco Mujer Zcloud 2 Athletic Sandal Trace Eclipse ALMA NATURAL Sandalia de pescador Beatrice para mujer Chocolate , Chaco Mujer Diana Sport Sandalia Pulse Eclipse Sandalia Savannah de Ryka para mujer Coral / gris , Sandalia de mujer Bellini Nickel Turquesa , Sandalia Drew Angela para mujer Bone Smooth Leather Sandalia deportiva Tirra para mujer Teva Trufa de ciruela , Sandalia Yucatan ECCO Mujer Shadow White Earth Origins Sophie Frost Grey Chaco Women s Z1 Classic Athletic Sandal Verdure Eclipse Columbia Women s Barraca & shy; Sunlight Athletic Sandal Tin Grey Steel / Stream , Sandalia para mujer Terrell Ari Wrap de Merrell Resquicio de esperanza , KEEN Hombres Newport H2 Sandal Negro Sandalia deportiva Nolita para mujer Anne Klein Luz Natural / Oro Metálico , Sandalias KEEN Venice II-W para mujer Bleacher / Raven , Sandalia NIKE Benassi JDI para mujer Negro / oro rosa , The North Face Bridgeton Slingback Mujeres Tnf Black / Tnf Black , Chaco Mujer Diana Sport Sandalia Avo florentino Chancleta Eezay adidas para mujer Vapor Gris Blanco / Amarillo Fácil S , Sandalia deportiva Tirra para mujer Teva Zaca Dark Purple / Gold CLARKS Womens Wave Grip Negro sintético Sandalia deportiva Marc Jacobs Women s Somewhere Raspberry Multi Cristal dorado Boda de mujeres de verano sandalias de moda zapatos Novia fiesta zapatos de vestir altos talones damas Sandalias Plata Sandalias de encaje Ros Hommerson para mujer Plata , KEEN Hombres Newport H2 Sandal Negro / Gris piedra CLARKS Sandalia plana Brizo Ravena para mujer Armada Teva Terra-Float Active Lace - Mujeres Chispas de chocolate , Skechers Performance On The Go 400 Vivacity Flip Flop Azul marino / blanco ,
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 Zapato deportivo ASICS Gel-Kayano 24 para mujer Concha Rosa / Blanco / Begonia Rosa , Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Muy uva Earth Origins Sophie Women s Sandal Negro Chaco Mujer ZX1 Classic Sport Sandalia Aguacate estático Zapatillas de running Saucony Women s Triumph ISO 4 Blanco azul , Zapatillas de running NIKE Unisex Flyknit Racer Negro / Blanco-blanco , Zapatillas de correr 3D Salomon X Scream para mujer Lotus rosa / negro / abuelita verde , Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Negro / Blanco / Stealth / Cool Gray Sandalia NIKE Kawa Slide para mujer Hyper Turquoise White Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Racer Pink / Vastgrey Zapatillas de trail running Salomon Women s Sense Mantra 3 W Cosmic Purple / White / Coral Punch Salomon Women s Speedcross 4 W Trail Runner Dark Purple / White / Deep Lake , Zapatillas de running NIKE para mujer Lunarstelos Resplandor medio azul / negro-atardecer , Calzado de espuma fresca New Balance para mujer 1165v1 Púrpura Zapato de running ASICS GT-1000 6 para mujer Gris / blanco / gris , Splendid Feodora Sandalia para mujeres Sonrojo , Zapatillas para mujer Reebok Zprint Her MTM para mujer Negro / Gris Ceniza / Blanco Sandalia casual destello de las mujeres de ECCO Marina Trotters Mujer Simmy Mary Jane Flat Sabio Zapato Cross-Trainer Grafik 2 para mujer Ryka Gris / menta Zapatillas de rebote DC Boys Negro / azul / blanco , Golf de mujeres y rsquo; s Peso ligero genuino Cuero hecho en Huerto de Brasil Street Golf Performance Marc Joseph NY Zapatos de moda Blue Mint Grainy Sandalia para mujer Merrell, 1SIX8 Linna AC + Slide Azalea , Koolaburra de UGG Women s W raychel Slide Sandal Sonrojo , Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7010-negro Zapatilla Teresa Stud de mujer Teresa Sana Oro Bomba de vestido de cuero Fabulist para mujer de AK Anne Klein Negro Sandalia plana Marlina de Calvin Klein para mujer Plata Plataforma de moda de mujer Square Toe Lace up Plaid Creeper Wedge Heel Oxford Negro con blanco Cicadillo Lax Finisher Under Armour para mujeres Blanco (101) / Blanco ,

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.

mobils by Mephisto - tacón - Sandalias de tacón Platinum Getha para mujer Camel Savana/ Platinum Magic Leather 64b836a - 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 mobils by Mephisto - tacón - Sandalias de tacón Platinum Getha para mujer Camel Savana/ Platinum Magic Leather 64b836a 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.