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.

Skechers Sneaker Women B07353Q13V s Appeal s Flex New Image Sneaker Negro 7d9607f




Sintético
Importado
Sintético único
Siente el viento fresco a través de tu cara desde el estacionamiento mucho en el elegante SKECHERS & reg; - Nueva imagen.
Suave tela tejida y parte superior sintética materiales.
Slip-on diseño.
Talón pull loop.
Ligeramente lengua acolchada y cuello.Transpirable forro de tela ofrece un gran zapato sentir. De longitud completa Plantilla de espuma viscoelástica. Peso ligero Entresuela FlexSole 2.0 proporciona flexibilidad y absorción de choque. Flexible suela de goma suministros duraderos traction.Imported.Measurements: Peso: 6 oz Mediciones del producto fueron tomadas usando tamaño 8, ancho B - Medio. tenga en cuenta que las mediciones puede variar por tamaño
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


Skechers Sneaker Women B07353Q13V s Appeal s Flex New Image Sneaker Negro 7d9607f - 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

Skechers Sneaker Women B07353Q13V s Appeal s Flex New Image Sneaker Negro 7d9607f

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.


Skechers Sneaker Women B07353Q13V s Appeal s Flex New Image Sneaker Negro 7d9607f

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

Easy Spirit Mujer Ezcool Mule Blanco Easy Spirit Solana Oxford para mujeres Nuez Toni Pons Womens Noa Multi Cordelia , Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on gris , Easy Spirit Solana Oxford para mujeres Negro Nubuck multi , Zapato informal Fulcrum de Fila para mujer Blanco / Wht / Mslvr Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco , Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Multi , Zapatillas para caminar ProGrid Integrity ST2 para mujer Saucony Gris oscuro Hush Puppies Power Walker Sneaker para mujer Cuero de abedul Hush Puppies Mujeres Power Walker Ii Loafer Cuero negro , Sandalia Juno Slide de Vionic Women s Rest Oscuridad , Zapatillas para caminar Go Flex 2-14992 de Skechers Performance para mujer Gris pardo , MOZO Zapatillas para mujer The Maven Sneaker Lunar Rock , Sandalias de mujer Ros Hommerson Mellow Arena , Zapatos de mujer RYKA Sky Walking Gris / Coral OLUKAI Nohea Nubuck - Mujeres Dark Java / Tan , Walking Cradles Womens Henna Azul marino Nubuck Estampado , Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Plano extra ancho Feet Breeze Womens Zapatos para caminar , Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro , Atletismo femenino de Apex Blanco / Vinca , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata , Easy Spirit Mujer Ezcool Mule Azul marino Combo de gamuza , Calzado de senderismo Go Joy 15602 de Skechers para mujer Negro / rosa caliente , Drew Shoe Women s Tulip Comfort Brown Croc Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Azul Zapatilla deportiva Skechers Go Walk 4-Attuned Negro , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco 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 Women s Air Zoom Structure 20 Racer Pink / White-pure Platinum Revs New Classic Masaje de reflexología Sandalias para hombres & Mujer. Mira. Diseño. Terapia. Cojín Comfort Con plantilla de masaje & Absorción de impactos Único. marrón , Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Blanco , KEEN Hombres Newport H2 Sandal Imán / Capuchina NIKE Mujeres Downshifter 7 Gris 361 Trail Runner Overstep para mujeres Castlerock / Green , Drew Shoe Women s Paradise II Velco Slip-on Ternero negro , Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Negro , Easy Spirit Women s Motion Lace up Oxford Blanco , Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos castaña Zapatillas para correr W630V5 para mujer New Balance Gris oscuro , Zapatillas para correr New Balance Rushv3 para mujer Poisonberry / Black Zapato deportivo ASICS Gel-Cumulus 18 para mujer Blanco / azul índigo / lavanda , G.H. marrón Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Grey Knit Walking Cradles Womens Henna Nubuck estampado rojo Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Blanco Zapatillas de golf ECCO Women s Casual Hybrid 2 Oyester , ALMA NATURAL Sandalia Kiki Slide para mujer Color de malva Kea de las mujeres de Flexus Ante rojo Sandalias planas con cordones Pinch Montauk de Cole Haan para mujer Impresión tropical de la primavera de la palma Birkenstock Arizona Sandalia de ante con suela suave Ante Mocha , Dr. Scholl s Women s Blink Slide Sandal Elegante azul marino , LOVELY BEAUTY Lady s Ballroom fabricmps Zapatillas de baile Negro , Zapatilla Glamour de mujer verde Daniel Black Kidskin , Bomba de vestido Pointe para mujer Easy Street Patente desnuda , W X-ALP ENDURO IV Negro / Paloma , Bota de media caña Georgia GB00179 Barracuda Gold , Zapatillas VI-S para fitness y yoga Vibram para mujer Rosa oscuro , Trotters Lucy Ballet para mujer Coñac ,

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.

Skechers Sneaker Women B07353Q13V s Appeal s Flex New Image Sneaker Negro 7d9607f - 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 Skechers Sneaker Women B07353Q13V s Appeal s Flex New Image Sneaker Negro 7d9607f 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 3146888
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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.