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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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


Wolky Rio Suede Sandal Crash Denim Denim Crash Suede 14d3842 - 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

Wolky Rio Suede Sandal Crash Denim Denim Crash Suede 14d3842

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.


Wolky Rio Suede Sandal Crash Denim Denim Crash Suede 14d3842

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 deportiva Tetra Cloud de Chaco para mujer Madras Teal , Sandalia de Gladiador de Piedra Lunar de Ahnu para Mujer Agave Green , Chaco Women s Z1 Classic Athletic Sandal Verdure Eclipse , Birkenstock Cleo - 1008773 Negro , Teva Terra-Float Active Lace - Mujeres Negro Sandalia universal W Sanborn de Teva para mujer Nuez Sandalia Drew Angela para mujer Negro Sandalias KEEN Venice II-W para mujer Aceite de visón / Tierra oscura , Sandalia deportiva ECCO Terra 3S para mujer Negro / Teaberry Sandalia Yucatan ECCO Mujer Morillo / Puerto / Negro Chaco Women s Z2 Classic Athletic Sandal Mandarín de mimbre Propet Women s TravelActiv Ss Sandal Púrpura / negro Sandalia UEek Flat-W de mujer KEEN Cuervo azul Sandalia de susurro de mujer Keen Malaquita / Abedul plateado Sandalia de susurro de mujer Keen Imán / Sangría , Teva ALP Premier - Mujeres Negro , Sandalia Teva W Terra-Float Nova de Teva para mujer Tacion Aqua Sandalia deportiva ECCO Cruise para mujer Gris cálido , Chaco Mujer Zcloud Sport Sandal Agrupado azul Chaco Mujer Z Eddy X1 Sport Sandalia Cipher Gold , Chaco Z / Canyon 2 Negro acolchado , Orthofeet 872 Women s Comfort Diabetic Extra Depth Sandal , BZees Sandalia Deportiva Jive para Mujer Negro , ALMA NATURAL Sandalia de pescador Beatrice para mujer Rosa KEEN Hombres Newport H2 Sandal Negro / Gris piedra Chaco Women s ZX2 Classic Athletic Sandal Crest Citrus Columbia Women s Barraca & shy; Sunlight Athletic Sandal Fósil, Natural , Sandalia deportiva Coleman 2.Zerogrand Slide de Cole Haan, cuero negro Oro Metálico-argento Metálico Specchio , Chancleta Eezay adidas para mujer Shock Pink / Collegiate Royal / Aero Blue
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 Adidas Women s Adilette CF + Logo W Sandalia deslizante Ash Blue / Aero Blue / Black , Sandalia Drew Angela para mujer Bone Smooth Leather Zapato Casual de Cuero Cortez Clásico NIKE para Mujer Blanco / universitario Red-varsity Royal Converse Chuck Taylor All Star Seasonal Color Hola Blanco Negro Blanco Zapatillas de running New Balance 1340v3 para mujer Blanco / morado Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Fiyi Zapato de running WE495 para mujer New Balance Dirigir Easy Spirit Mujer Ezcool Mule Azul marino Combo de gamuza , Drew Womens Element Denim Blue Nubuck Sandalia de gladiador sintético Xema de Nine West para mujer Negro / Multi CLARKS Mujer Gracelin Mara Flat Armada , Sandalia de mujer UGG Holly Fusion Coral , OLUKAI Womens Eheu Thong Sandal Arcilla / Java oscuro , Zapato para caminar de devoción Ryka para mujeres Negro / púrpura oscuro / azul claro , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco La Plume Diapositiva de estiramiento de la mujer Beige kensie Sandalia Dominic Slide para mujer Negro Sandalias planas para piscina Walking Cradles para mujer marrón Zapatillas de volante adidas Bounce W de Crazyflight para mujer Tinta noble / oro metálico / tiza púrpura , Zapatillas de deporte de la danza de la práctica de las mujeres para hombre Suela dividida VFSN016EB Cómodo -Muy bien {Paquete de 4} Púrpura Zapato de tacón flexible EZ Flex 2 \ Broncearse , CLARKS Womens Un Cruise Lace (Wave) Sneaker Combi de cuero blanco / arena nubuck Zapatilla Grace de MUK LUKS para mujer Armada ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ 7011 Cobre Bota Deia para mujer de TOMS Canela Suede / Tribal Sandalia Birkenstock de cuero Florida para mujer Cuero engrasado Wolky Comfort Lace Up Shoes Holandés Cuero engrasado de coñac Zapatos con cordones Daphne de Bernie Mev para mujer Blanco , Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer borgoña

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.

Wolky Rio Suede Sandal Crash Denim Denim Crash Suede 14d3842 - 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 Wolky Rio Suede Sandal Crash Denim Denim Crash Suede 14d3842 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.