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.

Zapatillas Blue/ para correr ASICS Gel-Venture 6 Island para mujer Negro/ Island Blue/ Pink da8521d




Malla ^ Sintético
Importado
Dar usted mismo una ventaja cuando te atadas el GEL-Venture & reg; ¡6 zapatillas para correr!
Predecesor: GEL-Venture 5.
Soporte Tipo: Neutral a underpronation (Supinación).
Amortiguación: Ligero, flexible respuesta.
Superficie: Trail.Differential: No provisto. malla y sintético materiales superiores. Lengua de peluche y collar. forro de tela ofrece un gran zapato sensación. plantilla de espuma. GEL & reg; el sistema absorbe choque impactante y promueve un transición más suave a midstance.Trail-specific características de la suela orejetas invertidas para cuesta arriba óptima y tracción cuesta abajo en terreno variado. Muy durable AHAR y reg; suela de goma. Importado. Peso: 9 oz
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


Zapatillas Blue/ para correr ASICS Gel-Venture 6 Island para mujer Negro/ Island Blue/ Pink da8521d - 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

Zapatillas Blue/ para correr ASICS Gel-Venture 6 Island para mujer Negro/ Island Blue/ Pink da8521d

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.


Zapatillas Blue/ para correr ASICS Gel-Venture 6 Island para mujer Negro/ Island Blue/ Pink da8521d

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 running ASICS Gel-Nimbus 18 para mujer Negro / Plata / Carbono Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Negro / Flash Coral / Negro , Zapatillas de running New Balance Urgev2 para mujer Silver Mink / Alpha Pink , Brooks Womens Levitate Azul / Menta / Plata , Zapatillas de running Kinvara 8 Saucony para mujer Azul marino Púrpura Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Garza / agua , adidas Yeezy Boost 700 Gris sólido / Tiza Blanco con núcleo negro Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-Gris Oscuro Zapatillas de running New Balance para mujer 490v4 Silver Mink / Spectral , 361 Zapatillas de running para mujer 361-Enjector Blanco negro Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul Zapato de running ASICS Gel-Quantum 360 cm para mujer Corydalis Azul / Carbono / Aluminio Zapatillas de running NIKE Runallday para mujer Fresh Mint / White / Wolf Gray , Supra Martine Run Skate Shoe para hombre Gris blanco , Zapatillas de running adidas Alphabounce Em W para mujer Gris Cinco / Gris Dos / Blanco Zapatillas de running NIKE Lunarglide 9 para mujer Negro / Blanco / Antracita Zapatillas de running New Balance Prism V2 para mujer Sea Spray / Water Vapor , Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Negro , NIKE Women s Revolution 4 Running Shoe Pure Platinum / Sunset Pulse - Wolf Gray Skechers Sport Skech Air Run High Fashion Sneaker Adornos de malla negra / rosa fuerte Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Corte Púrpura / Volt / Violeta , New Balance Women s w940v3 Running Shoe Azul Zapatillas de deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Light Bone Wolf Grey Mushroom 009 Zapatillas de running A7100w Apex para mujer Armada Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Dark Mulberry / Black Rose , Zapatillas de running adidas Alphabounce Rc W para mujer Gris claro Heather / Gris claro / Gris medio , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul , Zapatillas de running NIKE Flyknit Racer NIKE para hombre Oreo-negro / 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 Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Estratos Sandalia deportiva Chaco Zvolv X2 para mujer Fleck Pink Sandalias KEEN Mujer Whisper-w Sterling Blue / Vestido Azul Zapatillas para correr Saucony Women s Cohesion 10 Gris azul , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Refleja Plata , Salomon Women s Speedcross 4 W Trail Runner Deep Peacock Blue / Lime Punch./ Jugo de jugo , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Verde Palm / Negro / Vapor Verde , Zapatillas de running NIKE Airstream Sequent 3 para mujer Blanco / puro platino , Zapatillas de running NIKE Womens Air Presto Negro / gris oscuro , Sandalia de Richetta de Vince Camuto para mujer Negro Vionic Splendid Lou- Sandalia con correa para mujer Negro , Sandalias Donald J Pliner de mujer con cremallera metálica en relieve Fiji Light Pewter Distressed Metall , Zapatillas de entrenamiento cruzado Realflex Train 4.0 de Reebok para mujer Faro nocturno / Eléctrico Azul / Icono Rosa / Blanco , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Negro / blanco / puro platino , DAWGS Womens Arch Support Sandalias Loudmouth Z Miami Slice Sandalia UGG Royale Slide para mujer Rosa neón Sandalia Buxom Slide de Charles David para mujer Blanco , 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) S2806 Cobre y rojo y blanco Calzado de ballet Freeform FF01 Femenino de Capezio Negro , 50 sombras de blanco Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5008- Satén blanco , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2828- Satén dorado satinado Bota de campo Ovation Flex Sport para mujer Negro Ariat Womens Devon Pro VX Paddock Negro , Zapatillas para correr Nike Womens Air Zoom Pegasus 35 Elemental Rose / Barely Rose-vintage Wine , Bomba de vestido Daisie de mujer Steve Madden Patente Dark Blush Bota de mujer FRYE Natalie Short Lug Engineer Black Soft Oiled Suede CLARKS Mujer Maypearl Tulsa tobillera Cuero Tan oscuro SOREL Women s Out n About Moccasins Cantera TOMS - Slip-On de lienzo clásico para hombre Negro , Zapato femenino Keli de Alegria Romance azul

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.

Zapatillas Blue/ para correr ASICS Gel-Venture 6 Island para mujer Negro/ Island Blue/ Pink da8521d - 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 Zapatillas Blue/ para correr ASICS Gel-Venture 6 Island para mujer Negro/ Island Blue/ Pink da8521d 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 8012502
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16816513
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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.