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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Ajuste seguro, cómodo y parte superior transpirable gracias a lo innovador Construcción de 3 capas sistema
Trail La suela Rocker promueve exterior natural talón a la punta del pie apoyo mientras corriendo
Agresivo asas combinadas con Caucho FriXion y Trail Bite talón proporcionar todo terreno tracción y máximo apretón
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 La de montaña Slate de Jadegreen montaña Akyra La Sportiva para mujer Slate/ Jadegreen a72e700 - 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 La de montaña Slate de Jadegreen montaña Akyra La Sportiva para mujer Slate/ Jadegreen a72e700

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 La de montaña Slate de Jadegreen montaña Akyra La Sportiva para mujer Slate/ Jadegreen a72e700

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

Calzado de running NIKE RN 2017 2017 para mujer Negro / Fuchsia Glow / Hyper Magenta Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo , Under Armour Women s HOVR Sonic NC Grafito / Plata Metálica / Grafito Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa Zapatillas de running New Balance 520v3 para mujer Negro Calzado de running New Balance Foam Zante v2 para mujer New Balance Gris / rosa Calzado de running Kindred para mujer Ryka Gris / rosa Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird Zapatillas para correr inteligentes para mujer Runtopia Reach 2.0, equipadas con chip de seguimiento de datos Ash Black Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de running NIKE Air Max Flair para hombre Wolf Gray / Cool Grey-black , PUMA Men s Roma Basic Sneaker Blanco / Blanco / Goma , HOKA ONE ONE Bondi 4 Mujeres Corso azul / coral de neón , Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Turq / Black-volt-rio , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , Zapatillas de running ISO para mujeres de la guía Saucony Pizarra / Melocotón Zapatillas de running adidas Performance Women s Lite Slip-On Marrón / marrón / rosa , Zapatillas de running adidas Ultraboost W para mujer Blanco / blanco / blanco cristal Zapatillas de running NIKE Zooms Winflo 4 para mujer Ink / Deadly Pink-fire Pink , Zapatillas para correr New Balance Rushv3 para mujer Poisonberry / Black , Amortiguación 635v2 New Balance para mujer Negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Colegio Navy / Black Concord , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running Kineta Relay para mujer Saucony Gris blanco Brooks Womens Mazama Diva Rosa / Antracita / Bluefish Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / 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 ADRIENNE VITTADINI Sandalia plana Celie para mujer Negro Zapatillas de running Questr Cc W de mujer adidas Aero Blue S, Aero Blue S, Colegiata Marina , Zapatillas de running Spira Scorpius Stability para mujer Negro Negro Sandalia Yucatan ECCO Mujer Abedul , Sandalia KEEN Women s Uneek Slide-w Aqua Sea / Turquesa Pastel Zapatillas NIKE Femenil Flex 2016 Rn Running Negro / Punch caliente / Dark Mushroom / Sail , Zapatillas de trail running Vivobarefoot para mujer Trail Freak todoterreno Lima / Naranja Corredor Trail SCARPA Atom WMN Trail Running Trail para mujer Laguna / Lima , ALMA NATURAL Ballet femenino femenino Tela de microfibra negra , Zapato de caminar WW928HW2 para mujer New Balance Blanco , Sandalia Damara correa de mujer ECCO Negro casual Zapatillas de viaje Proather para mujer II Malla azul claro Zapato de agua Easy Spirit para mujer First Walker Azul-menta Zapato para caminar con cordones Go Flex Ability de Skechers Performance para mujer Gris / rosa fuerte , Good-memories 2018 verano mujer zapatillas plano antideslizante interior informal peludo playa sandalias Vino tinto Earth Origins Tova Polvo Zapatillas para caminar Travelite para mujer Propet Gris pardo , Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Negro , Zapatillas para caminar Travelite para mujer Propet Negro , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Baya Converse Chuck Taylor All Star Core Ox Armada , Skechers Sport Women s Burst City Scene Zapatillas de moda Natural , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Azul marino / Stv Navy Sandalia Dolce Vita de mujer Kaira Slide Cuero Ivory Plataforma CLARKS Leisa Grace para mujeres Cuero rojo Sandalia Slip con nudo de mujer Soludos Flat Fucsia , Bota vaquera Durango DCRD180 para mujeres marrón Zapatillas de ciclismo para mujer Sidi Alba Road Bicycle Blanco Bota de mujer FRYE Dara Chelsea Negro PUMA Women s Match Animal Wn s Shoe Puma Black Natural V

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 La de montaña Slate de Jadegreen montaña Akyra La Sportiva para mujer Slate/ Jadegreen a72e700 - 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 La de montaña Slate de Jadegreen montaña Akyra La Sportiva para mujer Slate/ Jadegreen a72e700 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.