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 NIKE Femenil Flex 2016 Flex/ Rn Running Negro 2016/ Blanco/ Gris Fresco 8271a11




Textil
Importado
Caucho único
Co-moldeado entresuela entrega un viaje suave
Tri-star diseño de suela se flexiona como su correr
Flywire los cables ofrecen una dinámico, de apoyo encajar a través de la mediopié
Mínimo el diseño del talón proporciona un más natural ajuste
El talón es redondeado para un suave transición en la inicial aterrizar
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 NIKE Femenil Flex 2016 Flex/ Rn Running Negro 2016/ Blanco/ Gris Fresco 8271a11 - 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 NIKE Femenil Flex 2016 Flex/ Rn Running Negro 2016/ Blanco/ Gris Fresco 8271a11

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 NIKE Femenil Flex 2016 Flex/ Rn Running Negro 2016/ Blanco/ Gris Fresco 8271a11

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

Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , Under Armour Women s RailFit 1 Antracita (100) / Acero , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Medium Heather gris / Medium Gray Heather / Dark Heather gris , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial Borgoña / Cardenal / Misterio Rojo Zapatos descalzos CVT-Hemp de mujer Vibram FiveFingers y paquete Premium Toesock Negro Under Armour Women s HOVR Phantom NC Gris Nublado (107) / Blanco Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Hyper Magenta / Royal Pulse NIKE Men s Lunar Fingertrap Cross Trainer Negro Negro Under Armour Women s HOVR Phantom NC Merlot (500) / Blanco , Zapatillas de running NIKE Juvenate para mujer Negro / Blanco / Negro , Zapatillas NIKE Air Huarache Run SE (GS) Girls Fashion 904538 Blanco Blanco Zapatillas NIKE Women s Revolution 3 para correr Gris oscuro / blanco / negro Under Armour Speedform Slingwrap para mujer Glacier Grey (100) / Acero Mizuno Wave Rider 21 , Under Armour Women s Speedform Slingride 1.1 Gris azul , Zapatillas para correr Under Armour UA Flow Criss Cross para mujer Negro / Blanco / Blanco Zapatillas de running NIKE Unisex Flyknit Racer Fireberry / Black-racer Pink , Zapatillas NIKE Women s Revolution 3 para correr Gris Lobo / Aluminio / Negro / Azul Tinte Sandalias OLKAKAI Ohana - Hombre Carbón de leña / carbón NIKE Mujeres Air Zoom Pegasus 33 Dark Purple Dust / Loyal Blue / Pink Blast / White NIKE Mujeres Air Zoom Pegasus 33 Negro / Gris oscuro / Sigilo / Plata metálica Zapatillas de deporte Rockport XCS Urban Gear para caminar de Rockport Cuero marrón , Calzado casual tejido NIKE NIKE para hombre Azul costero / blanco negro apagado Zapatillas de deporte casual Gazelle de adidas para hombre Verde , Zapatillas de running adidas Adizero Tj / Pv Núcleo Negro, blanco, naranja , Zapatillas de running Spira Phoenix para mujer con resortes Chr / Pnk , Zapatillas de running NIKE Unisex Flyknit Racer Universidad Red Black 608 , Zapatillas Nike Women s Flex Trainer 5 Blanco / plata metalizada / puro platino Under Armour Women s RailFit 1 Negro (001) / 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 NIKE para mujer Lunarepic Low Flyknit 2 Wolf Grey / Black-cool Grey-pure Platinum Zapatillas de running W490v3 New Balance para mujer Negro / púrpura Zapatillas de running NIKE Air Max Motion LW Print para mujer Dark Purple Dust / Sail / Black Zapatillas de running Mizuno Women s Wave Horizon 2 Rosa Glo / Blanco Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Baja Azul Zapatillas para correr Topo Athletic Runventure 2 - Mujer Frambuesa / Negro , Supra Martine Run Skate Shoe para hombre Negro-real Zapatillas de running La Sportiva Helios 2.0 para mujer - SS18 Verde , La Sportiva Ultra Raptor Mountain Running Shoe - Mujer Esmeralda / menta , Hush Puppies Men s Gus Oxford Cuero blanco deportivo Merrell Womens Barrado Cuero negro Sandalia Misty Fisherman de Ryka para mujer Turquesa / Amarillo New Balance WL574 Summer Waves Running Shoe para mujer Azul Ártico / Libélula Zapatillas de running Spira Scorpius Stability para mujer Negro Negro , Bomba de cuña Lansing para mujer Trotters Ante negro , MAIERNISI JESSI Clásico de mujeres para hombres Estiramiento de dos correas Sandalias de tacón bajo (Las mujeres necesitan 1.5-2 Tamaño abajo) Blanco , Zapato de mujer Jessica Simpson Fitt Walking Estrella fugaz Entrenador cruzado 611v1 de New Balance para mujeres Carbón de leña / Metallics Deflector de gravedad Alivio del dolor comprobado Desafío G de las mujeres Mighty Walk - Mejor Zapatos para el talón Dolor, dolor en el pie y fascitis plantar gris , Sandalia para mujer Kate Spade New York Coby para mujer Plata 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) 6006- Oro Pu y Sparkle , Ariat Womens Volant S Tall Zip Tall Riding Negro , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Llovizna yute metálico gris FLY London Yuzi798fly Pump para mujer Mousse negro Dualyz Fit Unisex 100% lana Zapato con suela extraíble Gris claro / negro Haflinger Unisex Journey Gris , CLARKS Keesha Rosa Dress Pump para mujer Cuero negro , Bomba de vestido exquisita Aerosoles mujeres Ante negro , Bomba de vestido Pointe para mujer Easy Street Textura negra / negra , Ziera Alley Women s Lace Up Negro / Stingray ,

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 NIKE Femenil Flex 2016 Flex/ Rn Running Negro 2016/ Blanco/ Gris Fresco 8271a11 - 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 NIKE Femenil Flex 2016 Flex/ Rn Running Negro 2016/ Blanco/ Gris Fresco 8271a11 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 506316
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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.