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.

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Sintético / Vegano
Goma suela
extraíble plantilla. Cuña de talón U4icX crea más amortiguador
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 negro de running Mizuno Rider Wave Rider 20 para B00EEUWKXC mujer Púrpura/ negro 882c6fd - 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 negro de running Mizuno Rider Wave Rider 20 para B00EEUWKXC mujer Púrpura/ negro 882c6fd

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.


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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 adidas Originals Tubular Defiant PK W Originals para mujer Blanco Ftw / Running Blanco Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Azul Cloro / Antracita / Gris Fresco NIKE Airs para mujer Zoom Pegasus 34 Low Top para correr con cordones Sneaker negro , Zapatillas de running New Balance Zante v4 Fresh Foam para mujer Azul , Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Zapato de running ASICS Gel-Quantum 360 cm para mujer Corydalis Azul / Carbono / Aluminio Calzado de running de aproximación Philadelphia para mujer Plata metalizada / Bluefish / Fiery Coral , Saucony Triumph ISO 3 Running Sneaker para mujer Gris morado Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul , Calzado deportivo New Balance para mujer Vazee FuelCell V1 Saúco / Plata , New Balance Women s w990v4 Running Shoe Armada , Zapatillas de running Endurant para mujer ASICS Aluminio / Plata / Acuario , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Microchip / Blanco / Carbón metálico , Brooks Womens Ghost 9 Negro / Sparkling Grape / Ceramic , Zapatillas de running adidas Adistar Boost ESM para mujer Naranja / Blanco , Zapatillas para correr ASICS Gel-DS Trainer 22 para mujer Azul / morado / negro / coral , Zapatillas de running Mizuno Wave para mujer Inspire 13 Barbados Cereza / Negro Nike Men s Air Presto Essential University Red, University Red Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha ASICS Metrolyte II Shoe Women s Walking Negro / Plata / Blanco , Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul Calzado de running NIKE Kids Roshe One (GS) Gris , NIKE Men s Flex 2017 RN Negro / Mtlc Cool Gray-light Carbon , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Desert Moss / Desert Moss , Zapato de correr ideal para rejilla de mujer Saucony Blanco negro Zapatillas de running Mizuno Wave para mujer Wave Horizon Azulejo azul-amarillo , Zapato de running COMUTORA para mujer ASICS Negro / Negro / Blanco , Zapatillas para correr NIKE Women s Air Zoom Structure 21 Provence Purple / Green Glow Zapatillas de entrenamiento Nike Womens Metcon 3 Persian Violet / Black-cool Gray
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 ASICS Gel-Cumulus 19 para hombre Primer rojo / negro / plata , Zapatillas de running NIKE Lunartempo 2 para mujer Nbl Rd / White Brght Crmsn Pnk B NIKE Kids gratis Rn (grande) Gamma azul / negro / blanco Zapatillas de running Nike Air Max Fury para mujer Negro / Blanco-negro , Chaco Mujer Zcloud 2 Athletic Sandal Berry Anemone , Sandalia Drew Angela para mujer Dusty Multi Metallic ASICS Gel-Fujiendurance Trail Runner para mujer Indigo Blue / Aqua Splash / Diva Blue Zapatillas de running Nike Zoom All Out Low para mujer Pure Platinum / Chrome-glacier Blue NIKE Men s Roshe Two Running Shoe Wolf Gray / White / Volt Zapatillas de running ISO para mujeres de la guía Saucony Gris / menta , Hush Puppies Power Walker Sneaker para mujer Gris pardo Apex Women s Ambulator Biomechanical Negro Converse Chuck Taylor All Star Core Ox Armada HEAD Zapatillas de tenis Sprint Pro 2.0 para mujer, blancas e irisadas- (274128WHID-S18) Blanco e iridiscente , Zapatillas de tenis 696v3 New Balance para mujer Blanco / Alpha Pink Zapatillas de golf ECOM Hybrid Hydromax II para mujer ECCO Concreto / Mineral , Zapatillas de golf adidas Climacool Knit Cblack / D, mujer, de adidas Negro Sandalia Jula de Donald J Pliner para mujer Amapola , Zapatillas de deporte Vans Old Skool para mujer Mole / True White , Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres 9622 - 2.5 \ Zapato plegable Heel + Paquete de cepillo Malla Satin-109 Negra Adtec - Bota de trabajo marrón de 9 \ marrón Woolrich Women s Lodge Bootie Ii Slipper castaña Puf lindo de Patricia Green para mujer Negro , Bombas de charol de mujer Modemoven, tacones peep toe, sandalias Slingback, zapatos de noche, stilettos lindos Peach Pale Pink Calvin Klein Women s Gayle Pump Blush Gold , Bomba de vestido exquisita Aerosoles mujeres Cuero Tan oscuro , Mephisto Women s Nykita Oxford Black Velours Zapatillas de deporte Easy Spirit AP1 Sport Blanco Horze Jodhpur con cremallera en la espalda marrón , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Blue Slub Chambray

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.

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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 negro de running Mizuno Rider Wave Rider 20 para B00EEUWKXC mujer Púrpura/ negro 882c6fd 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.