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.

Calzado B06XS99N3G de running ancho NIKE para mujer 7 Experience 7 running de NIKE Negro/ Blanco - Blanco 0c8e9e7




Sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Hexagonal ranuras de flexión en la suela promueve flexibilidad natural.
Heathered tejer ofrece peso ligero comodidad y transpirabilidad
Redondeado talón promueve un rango natural de movimiento
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


Calzado B06XS99N3G de running ancho NIKE para mujer 7 Experience 7 running de NIKE Negro/ Blanco - Blanco 0c8e9e7 - 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

Calzado B06XS99N3G de running ancho NIKE para mujer 7 Experience 7 running de NIKE Negro/ Blanco - Blanco 0c8e9e7

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.


Calzado B06XS99N3G de running ancho NIKE para mujer 7 Experience 7 running de NIKE Negro/ Blanco - Blanco 0c8e9e7

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

Under Armour Women s Drift 2 Gris Nublado (941) / Blanco , Zapatillas de running adidas Alphabounce Beyond W, para mujer Gray Four / Carbon / Dark Solid Gray , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul Zapatillas de running Salomon Women s Sense Link Lotus Pink / Papaya / Lucite Green Zapatillas de running Mizuno Wave Rider 20 para mujer Baton Rouge / Blanco Zapatillas de running Classic-M Vibram para hombre Negro Calzado de running para mujer New Balance Cypher Mulberry / Angora , Zapato deportivo ASICS Gel-Excite 4 para mujer Azul índigo / azul / orquídea Zapatillas deportivas NIKE Lunar Apparent para hombre Cool Grey / Deep Royal Blue Zapatillas de running NIKE para mujer Lunarstelos Golpe caliente / negro / blanco / azul medio , NIKE Men s Roshe Two Running Shoe Negro / Blanco-antracita-blanco NIKE Mens N98 Tribute Track Jacket Aunque rojo / blanco / blanco , Supra Martine Run Skate Shoe para hombre Gris blanco , Zapatillas Reebok para mujer Trilux Run PNT Sneaker Prnt - Negro / Blanco , Zapatillas de running Skechers Go Run Ultra Road 2 para mujer Rosado PUMA Men s Roma Basic Sneaker Blanco / azul Clematis , Zapatillas de running NIKE Air Max Thea Ultra PRM para mujer Cumbre blanca / Cumbre blanca Cumbre blanca , Zapato de running GEL-Fortitude 7 para mujer ASICS Azul índigo / plata / ciruela , NIKE Air Huarache Run Ultra Blanco / Blanco-Blanco-Negro (Mujer) , Zapatillas de running adidas Ultraboost W para mujer Ceniza Perla / Ceniza Perla / Ceniza Perla , Zapatillas running para mujer Saucony Liteform para mujer borgoña ASICS Metrolyte II Shoe Women s Walking Negro / Plata / Blanco Zapatillas de running Saucony PowerGrid para mujeres Gris / Té / Rosa Under Armour Men s Speedform Intake Rhino gris / azul infinito / blanco , Zapatillas de running adidas Alphabounce Beyond para mujer Gris / gris / naranja claro , Zapatilla Skechers para mujer Go Run 600-Obtain Frambuesa , Zapatillas de running Mizuno Wave para mujer Inspire 14 Acuario / Blanco Zapatillas de running ASICS GEL-Storm 2 para mujer Negro / Onyx / Esmeralda , Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / 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 Chaco Mujer ZX1 Classic Sport Sandalia Negro , Zapatillas de running NIKE Air Huarache City para mujer Vela / Luz Menta-lt Racer Blue , NIKE Women s Tanjun Paramount Blue White Lava Glow , Earth Origins Sophie Azul marroquí Sandalia Merrell de mujer con gladiador de poste Negro Zapatillas de deporte Rasko para mujer Reebok Acero / orquídea celestial / melocotón eléctrico / plata / blanco / negro , Amalfi de Rangoni Women s Candito Sport Sandal Grafite / Black Nubuck Liz Skechers Women s Go Trail 2 - Zapatillas de senderismo con agarre Chocolate Correpasillos Merrell para mujer 3 Trail Runner Turbulencia , Zapatillas de running V-Trek Black / Gum Trail Vibram para mujer Negro / goma de mascar OLUKAI Nohea Nubuck - Mujeres Dark Java / Tan , Tacón de punta abierta Vionic Women s Perk Blaire Serpiente Merlot , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Blanco negro Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Azul marino / gris / blanco Zapatillas de deporte Skechers Sport Women s Empire Inside Look Blanco negro , Calzado de senderismo Skechers Go s 15601 para mujer Negro , Sandalias de mujer Ros Hommerson Mellow rojo , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Clásicos de base Slip-on (Tm) para hombres de Vans Negro / Marshmallow Sandalias de tacón alto con hebilla de Onex para mujer Negro / Plata , Sandalia Dolce Vita con diadema para mujer Tejido negro Sandalia de la diapositiva de junio de las mujeres de Propet Broncearse Sandalia Heather Slide de mujer Vía Spiga Midnight Suede Sandalia Sonia Slide de Dolce Vita para mujer Elástico de plata Sandalias planas para piscina Walking Cradles para mujer Abrigo blanco de cachemira / corcho , Aria Unisex Concord Chaps Negro Cobb Hill Rockport - Aubrey Dress Pump para mujer Azul / Multi Bomba de vestido Pointe para mujer Easy Street Patente roja Zapato de plataforma de lame las mujeres de Iron Fist menta Trotters Liz Loafer para mujer Lila ,

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.

Calzado B06XS99N3G de running ancho NIKE para mujer 7 Experience 7 running de NIKE Negro/ Blanco - Blanco 0c8e9e7 - 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 Calzado B06XS99N3G de running ancho NIKE para mujer 7 Experience 7 running de NIKE Negro/ Blanco - Blanco 0c8e9e7 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2831
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15617499
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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.