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.

Funtasma Oxford de Pleaser Men cebra s Jazz-02 Platform Oxford Piel de de cebra cbf82ad




Sintético
Sintético
Hecho por el hombre único
Incluye: Piel de charol blanca Zapatos plataforma
Plataforma mide aproximadamente 11/2
Hombres s Mocasín con costura Detalles
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


Funtasma Oxford de Pleaser Men cebra s Jazz-02 Platform Oxford Piel de de cebra cbf82ad - 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

Funtasma Oxford de Pleaser Men cebra s Jazz-02 Platform Oxford Piel de de cebra cbf82ad

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.


Funtasma Oxford de Pleaser Men cebra s Jazz-02 Platform Oxford Piel de de cebra cbf82ad

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

Zapatillas de deporte lavables Walker Propet para mujer Sr Royal Blue / White Easy Devitt Oxford Flat para mujer marrón , Converse Chuck Taylor All Star Core Ox Carbón BigPix todo-terreno WP Oxford de las mujeres de Cole Haan Cuero café helado , Propet Walkers Evolution Oxford lavable de las mujeres Gris claro / azul claro Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue café Zapatillas de deporte lavables Walker Propet para mujer Sr Black Suede Easy Spirit Cacia Mary Jane para mujer Negro Steve Madden Womens Kyra Gray Multi OTBT Womens Lunar Sneaker Paloma gris , Zapatilla de deporte sin cordones sin mangas lavada Chillax de Keds para mujer Carbón , Colección de artistas Franco Sarto Autumn Women s Oxford Cuero de la caja negra Katy Perry Women s The Kailee Oxford Desnudo , ED Ellen DeGeneres Womens Ellert Peony Pink / Primrose Oiled Calf Suede Drew Shoe Women s Hope cuero terapéutico moda Oxfords Denim Lea / Nubuck , Ivanka Trump Women s Tropica6 Ballet Flat Blue Denim Hush Puppies Women s Adler Zelder Oxford Negro CLARKS Tilmont Ivy Oxford para mujeres Black Combi Zapato Comet para mujer OTBT Cálido rosa Zapato para caminar con láser Mephisto para mujer Warm Gray Bucksoft / Beige Vionic Womens Elevated Becca Wedge con cordones Gris Nine West de mujer Garroy Leather Oxford Flat Cuero blanco Dr. Martens Unisex 1461 Oxford Caramelo de azúcar con mantequilla , Zapatilla de deporte Cre216 / bvs Demonia para mujer Gamuza vegana negra , Zapatillas para mujer Del Rey TOMS Black Cherry Poly Canvas Dunham Men s Stephen-Dun Oxford Negro , Zapato de punta de ala 3989 de Dr. Martens para mujer Rojo cereza , Arcopedico Femenino Francesca Oxford Armada , Hush Puppies Womens Cesky Perf Oxford Rosa Claro Nubuck ,
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 NIKE Women s Air Force 1 Mid 07 Le , ASICS Gel-Cumulus 19 femenino, Aruba azul / Aruba azul / azulejo turco Aruba azul / azul de Aruba / azulejo turco NIKE Women s Tanjun Black Metallic Gold , Under Armour Women s HOVR Phantom NC Merlot (500) / Blanco Zapatillas ultraligeras La Sportiva Helios 2.0 para mujer Fiordo Zapatillas de running New Balance Gobi v2 Fresh Foam Trail para mujer Rosa / negro Zapatillas de running adidas Aerobounce Pr W para mujer Aero Blue / Aero Blue / Noble Indigo NIKE Men s Benassi Just Do It Athletic Sandal Blanco / Variedad Royal White , Zapatillas de running Saucony para mujer Hurricane ISO 2 Negro / Carbón / Coral Zapatillas de senderismo Zen Pro para hombre SCARPA Carbón de leña / tónico Zapatillas de running New Balance 1340v3 para mujer Blanco / morado Carlos by Carlos Santana Sandalia Adalyn para mujer Brulée Me Too Brielle Slip On para mujer Patente de granito , Sandalias de vestir de piña de mujer de Jack Rogers Oro negro Hush Puppies Men s Gus Oxford Marron oscuro , New Balance 711v3 Heather Cross Trainer para mujeres Blanco negro Zapatillas de skate DC Women s Villain SE Slip-on Cenizas azules / blanco Zapatillas de deporte Xelero Genesis Comfort Comfort Therapeutic Extra Deep Sneaker Leather / Mesh Lace-up Blanco azul , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / de peltre Sandalias Emmerly Slide de Vince Camuto para mujer Puro , BareTraps Women s Cherilyn Slide Sandal Negro , Zapato de personaje de luz de posición de estudiante 650 de Capezio Negro , Belleville 700V WP Black Combat Flight Boots para hombre Negro , Bota de encaje FRYE Cece Tassel para mujer Elefante Calzado de trabajo Rockport Work Top Shore RK600 para mujer Negro , Bota de mujer Patricia Nash Sabrina Negro Easy Spirit Women s Motion Lace up Oxford Blanco Calzado para mujer Anywear Women s Mimi Health Care & Food Service Negro , Bikini de corbata Harper Kiltie Tie de mujer de Sebago Nubuck marrón oscuro OTBT Womens Lunar Sneaker Paloma 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.

Funtasma Oxford de Pleaser Men cebra s Jazz-02 Platform Oxford Piel de de cebra cbf82ad - 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 Funtasma Oxford de Pleaser Men cebra s Jazz-02 Platform Oxford Piel de de cebra cbf82ad 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.

Oxfords
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1304651
6681698
11063846
11782515
13336384
16055635
16223584
17314571
18660305
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.