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.

Carlos by Carlos Carlos Santana by Sandalia Adalyn para 19954 mujer Brulée 2662290




Gamuza
Importada
Suela sintética
Sandalia
Detalle de flores
Cierre de hebilla
Producto importado
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


Carlos by Carlos Carlos Santana by Sandalia Adalyn para 19954 mujer Brulée 2662290 - 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

Carlos by Carlos Carlos Santana by Sandalia Adalyn para 19954 mujer Brulée 2662290

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.


Carlos by Carlos Carlos Santana by Sandalia Adalyn para 19954 mujer Brulée 2662290

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

KEEN Kids Seacamp Ii CNX Water Shoe Blue Depths / Gargoyle Sandalia de vestir elegante 401 de las mujeres Fabulicious Claro Sandalias de cuña Cupid para mujer de NAOT Oily Dune Nubuck, Camel Braid Zapatos OLUKAI Pehuea - Mujer Pale Gray / Carboncillo , Aerosoles botines de alta frecuencia del tobillo de las mujeres Ante negro , CLARKS Womens Un Karely Hail Cuero blanco Sandalia de vestir de la ciudad de Aerosoles mediados de mujeres Azul CLARKS Mujer Caddell Shine Sandal Nubuck sintético de arena Sandalia Gora Yoga Sanuk para mujer Natural , Sandalia de cuña de Easy Street para mujer Champagne Snake Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Azul / Denim , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro CLARKS Womens Wynnmere Avah Nobuck gris cálido Sandalias planas de mujer Grasshoppers Estaño Zapato de malla Nohea de OLUKAI - Hombre Dark Java / Rojo , Sbicca Women s Salem Sandal marrón , Sandalias Paulie T-Strap de mujer Vionic champán Sandalias Joanie II de mujer SOREL Harina de avena , Naughty Monkey Women s Amare Sandal Blanco Sandalias de tacón de soñador Easy Street para mujer Pewter Metallic , Chancleta Yoga Sling 2 Sanuk para mujer Lluvia de América , Sandalia plana Alanis de Miz Mooz para mujer Negro Sandalia Skechers Women s on-Go 600 Blanco negro , Sandalias de mujer Wolky Jeans Blue Multi Color Fantasy , Sandalia de gladiador de sandalias de destello para mujer de calzado ecológico de mujer de ECCO café Sandalia de tira de Ghillie Bre-Italy Bella Vita para mujer Cuero dorado Tierra Orígenes Shane Sedona Brown Sandalia de mujer de Steve Madden Cuero amarillo Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Estaño ,
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 de running ASICS GT-1000 6 para mujer Porcelana Azul / Humo Azul / Blanco Zapatillas de running Under Armour para mujer Toccoa Soldadura (300) / Gris opaco , NIKE Men s Roshe Run Negro Negro , Sandalia Drew Element para mujer Nubuck naranja , Sandalia de deslizamiento ultra Comfort NIKE para mujer Mango brillante / negro , Zapatillas de running Topo Athletic Runventure Trail - Mujer Vino / gris , Easy Spirit Solana Oxford para mujeres Nuez , David Tate Wish Sandalia para mujer Negro Diapositivas del anillo del tobillo Palermo de Cocobelle para mujer marrón CLARKS Womens Wynnmere Avah Nobuck gris cálido Sandalias de vestir West Hamptons de Jack Rogers para mujer Oro rosa Linea Paolo Felicia | Cuero perforado blanco Zapatillas Isla de las mujeres Softinos Blanco , ASICS Womens Weldon X Gris piedra / gris medio / Limelight Vans Auténtico (Ojales iridiscentes) Negro , Calzado Lakai Limited para hombre Bristol Ante negro / blanco Star Wars Jedi Obi-Wan Kenobi Cosplay Shoes Botas marrones por encargo , Zapato de skate Wnt High rebote DC para mujer Armada Zapatillas de golf Adicross para mujer de adidas Blanco / azul claro / real Sandalia de la diapositiva de Florene de las mujeres de Lucky Brand Anacardo asado , Carlos by Carlos Santana Sandalia Gilmore Slide para mujer Oro Calzado de ballet Freeform FF01 Femenino de Capezio Negro Ariat Womens Vortex Tall Riding Ternero negro , Zapatilla deslizante Loki II para mujer BEARPAW Nuez dura Zapatilla de lona lavada sin rueda W de Teva Women True Black Old Gringo Womens Bianca Vino Rockport Women s Total Motion Abelle Laceup Oxford Perla de bronce Trotters Liz Loafer para mujer rojo , Sanuk Women s Donna Paige Flat Negro Clásicos de base Slip-on (Tm) para hombres de Vans Clásico Negro / 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.

Carlos by Carlos Carlos Santana by Sandalia Adalyn para 19954 mujer Brulée 2662290 - 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 Carlos by Carlos Carlos Santana by Sandalia Adalyn para 19954 mujer Brulée 2662290 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
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 4621277
5582600
11289463
13624275
15426073
17476232
17915278
18913560
18946453
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.