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.

Brooks Womens Ghost 9 Silver/ Blue Blue Womens Atoll Brooks/ Lime Punch e265b42




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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


Brooks Womens Ghost 9 Silver/ Blue Blue Womens Atoll Brooks/ Lime Punch e265b42 - 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

Brooks Womens Ghost 9 Silver/ Blue Blue Womens Atoll Brooks/ Lime Punch e265b42

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.


Brooks Womens Ghost 9 Silver/ Blue Blue Womens Atoll Brooks/ Lime Punch e265b42

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 NIKE Men s Free 5.0 Tr Fit 5 Fabric Brght Crmsn / Prm Rd-atmc Pnk-wh , Brooks Women s Adrenaline GTS 17 Carbón metálico / negro / oro rosa , Short NIKE Tempo para mujer Armada Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal , Zapatillas de running Kinvara 8 Saucony para mujer Azul marino Púrpura , Zapatillas de running NIKE RN 2017 para mujer gratis Binario azul / oscuro cielo azul-obsidiana Zapatillas de running NIKE Air Zoom Vomero 13 para mujer Negro / Blanco / Antracita , Zapatillas para correr NIKE Womens Sock Dart Negro / Negro-azul Laguna-blanco , Zapatillas de running adidas Ultraboost W para mujer Still Breeze / Still Breeze / Black , NIKE Men s Flex 2017 RN Pure Platinum / Black / Wolf Gray / Cool Gray , Zapatillas de running Altra Women s Intuition 4 Negro Zapatillas de running adidas Women s Edge Lux w Cristal blanco Choque blanco Rojo Amortiguación 635v2 New Balance para mujer Plata , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Blanco / Gris Oscuro Zapatillas de running NIKE Zooms Winflo 4 para mujer Antracita / Gris Oscuro-negro , Zapatillas de running Nike Zoom para mujer Hasta / Mtlc Red Bronze / Green Glow Zapatillas de montaña de montaña Akyra La Sportiva para mujer Esmeralda / menta , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / oscuro, pasas de rosa, mortal , Zapatillas de deporte Nike Roshe Run para mujer Deep Granet / Black-gym Red-vry Berry NIKE Women s Tanjun Negro / Negro / Blanco , New Balance Women s w990v4 Running Shoe Komen Pink NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / White , Zapatillas de running Mizuno Wave para mujer Wave Horizon Dark Shadow / Clownfish Zapatillas para correr Saucony Freedom Iso Unisex borgoña , Zapatillas de running New Balance Urgev2 para mujer Silver Mink / Alpha Pink Zapatillas de running NIKE Air Max Torch 4 para mujer Negro / Plata Metálico / Rosa Flash , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Menta / Plata / Azul
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 Womens Air Zoom Pegasus 32 Pink Pow / Blk / Brly Vlt / Ghst Grn (12) - Converse Chuck Taylor All Star Seasonal Color Hola Amarillo fresco Zapatillas de running Merrell para mujer All Out Crush Tough Mudder Trail naranja , Zapato deportivo ASICS Gel-Excite 4 para mujer Diva Azul / Blanco / Sol Sbicca Women s Salem Sandal Gris Sandalia de cuña de Easy Street para mujer Whisky , Drew Shoe Athena para mujeres Blanco azul , Sandalia Teva Encanta para mujer Negro Toni Pons Womens Sol-S Marina Susi Easy Spirit Women s Traveltime Mule Patente negra / plateada Vivien Flat de Bella Vita para mujeres Blush Kid Suede Leather Calzado deportivo G-Defy Gamma-Ray para mujer Gravity Defyer Gris, rojo Entrenador cruzado Minimus 20v7 de New Balance para mujeres Negro Zapatillas para mujer Del Rey TOMS Patente negra Zapatillas de skate DC Mujeres Heathrow SE Purple Rain / Dusty Purple Sandalia FitFlop para mujer Lulu Shimmersuede Slide Estaño Sandalia de tacón dulce de mujer Freebird Blanco , Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Blush Ropa de cama , Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA7004 Talón de 2.5 pulgadas + Cepillo plegable Haz Oro , Calzado Capezio Manhattan para personajes, Negro Botas de camuflaje Ariat Women s Challenge Contour Field Ternero negro , Bomba de vestido LifeStride Sevyn para mujer Lux Navy 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) 1625- Granito dorado , Glerups modelo unisex Gr Slipper Carbón , Bomba Stillson para mujer de Sam Edelman Pink Mauve Kid Suede Leather RoseG Top de antelina con cordones planos de la plataforma de las mujeres zapatos Botas Gris , Zapatillas de deporte lavables Walker Propet para mujer Blanco Blondo Womens Liam impermeable Bootie Grey Suede , New Balance NB RCVRY Shoe Casual para mujer Rosado , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Toffee Suede Cupsole

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.

Brooks Womens Ghost 9 Silver/ Blue Blue Womens Atoll Brooks/ Lime Punch e265b42 - 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 Brooks Womens Ghost 9 Silver/ Blue Blue Womens Atoll Brooks/ Lime Punch e265b42 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

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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.