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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Sintético
Importado
Suela de goma
Ligero y cómodo
Plantilla acolchada
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

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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,

Prop & eacute; t Womens Aurora Slide Negro , Sam Edelman Women s Bay Slide Sandal Marfil negro / negro Sandalias Jerusalem Sandalia Joanna para mujer Blanco , Sandalias Brie con sandalias Kate Spade New York para mujer Denim Suede , Sandalia de la diapositiva de Marlowe de las mujeres chinas del lavadero Nude Suede , Sandalia Helina Slide de mujer kensie Floral negro Sandalia Dolce Vita Jene Slide para mujer Light Blush Satin Sandalia Rockdale Lanea Woven Slide para mujer Latte tejido , Sandalia Dian J Pliner Mujer Ginie Slide Plata La sandalia Finley Dome Stud Slide para mujer Azul brillante Sandalia de diadema Florita Carlos de Carlos Santana Blanco Sandalia HEATHER2 Slide de Via Spiga para mujer Pelo negro de Pony Sam Edelman Sandalia Bernice Slide para mujer Candy Red Vaquero Saddle Leather , Sandalia Mason para mujer Naturalizer Slide Negro Sandalia Carlos de Carlos by Carlos Santana para mujer Blanco BareTraps Women s Rebecca Slide Sandal Blanco Sandalia plana Dionne de Easy Street para mujer Armada Sam Edelman Sandalia Bernice Slide para mujer Cuero negro Koolaburra de UGG Women s W raychel Slide Sandal Sonrojo Sam Edelman Sandalia Glenn para mujer Cuero metalizado opaco oscurecido en oro fundido Sandalia Sloan Slide de Naturalizer para mujer Negro , Sandalias planas Cammie de Jessica Simpson para mujer Desnudo Blush Sandalias planas Nori de Easy Street para mujer Equipaje Sam Edelman Women s Slide Sandalia Ante negro Aveline Slide de Ted Baker para mujeres Blue Harmony , Sandalia plana deslizante trenzada metálica Soludos para mujer Nude / Pale Gold , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Brillo dorado , Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno turquesa Sandalia de diapositivas para mujer Daydream Shoes de Dr. Scholl s Shoes Frost Grey Mesh
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 Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Gris , NIKE Women s Tanjun Verde Zapatillas ultraligeras La Sportiva Helios 2.0 para mujer Fiordo Salomon Women s Speedcross 4 CS W Trail Runner Azul , Mujer adidas Originals Ultraboost St Carbono / Blanco / Tiza Azul , Zapato de agua NRS Crush - para mujeres Gunmetal Aerosoles botines de alta frecuencia del tobillo de las mujeres Gamuza naranja Zapatos del Dr. Scholl s Women s Powers Sandal Negro Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul , Easy Spirit Solana Oxford para mujeres Negro , Bota de senderismo Gore-Tex de media caña alta para mujer Merrell Moab Sedona Sage Zapatillas informales para mujer Chuck Taylor All Star Madison Ox de Converse Gimnasio Rojo / Dorado / Blanco Birkenstock Blue Footbed-Arch Soporte plano Azul (deporte) , Sandalias con hebilla de botón Ridgeport para mujer Rockport Bronce Sam Edelman sandalia Gio Slide para mujer Ante negro , L & rsquo; Artiste por Spring Step Sandalia plana Lizzie para mujer Natural Ahnu Women s Yoga Flex Pure Atlantis , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Crema / Combo K-Swiss Tubos de mujer Infinity CMF Cross Trainer Patriot Blue / Raspberry Wine , Zapatillas de golf adidas W Adistar Lite Boa Ftwwht de adidas para mujer Rosado Sandalias planas de piscina Walking Cradles para mujer Envoltura Metálica Multi / Cork Sandalia de diapositivas Henlee de Dolce Vita para mujer Tela roja Dr. Scholl s Women s Blink Slide Sandal Negro Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra Blanco , Bomba de vestido de pasión Easy Street para mujer Patente negra , FRYE Bota corta Renee Seam para mujer Charcoal-72064 Me Too Arena Sandalia para mujer Black Goat Spore , Zapatillas clásicas Vionic para mujer Walker Blanco / rosa , Zapatillas deportivas con cordones Easy Sport para mujer de Easy Spirit Broncearse Botas Adele BEARPAW para mujer Negro

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.

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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 Caucho Vince Vince Westcoast para Vince mujer Vince Caucho magenta 0a6710e 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.

Diapositivas
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.