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
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Ligero, malla transpirable superior entrega mayor ventilación con película adherida en el mediopié
bordado superposición agrega estructura y soporte
EVA plantilla para añadido comodidad paso a paso
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


Under/ Armour Women s s Under Thrill 3 Apollo Grey (962)/ Blanco bb8770c - 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

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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,

Calzado de running NIKE Kids Roshe One (GS) Blanco / puro platino / blanco , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Dahlia roja , Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro , NIKE Women s Tanjun Coral Stardust / Sail NIKE Women s WMNS M2K Tekno, Negro / Negro-Phantom-Volt Negro / Black-Black-Phantom-volt New Balance Women s w990v4 Running Shoe Armada , Saucony Women s Guide 8 Running Shoe Azul / azul marino / amarillo , Zapatillas de running NIKE para mujer Epic React Flyknit Negro / gris oscuro Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blanco Blanco Zapatillas de running Zoot para mujer Del Mar Fruta de la pasión / Deep Purple / Mand , PUMA Men s Roma Basic Sneaker Blanco / azul palacio , Zapatillas de running New Balance 520v3 para mujer Artic Fox / Black , Zapatillas de running Reebok Zpump Fusion 2.0 para mujer Carbón / Negro / Icono Rosa / Blanco ASICS Metrolyte II Shoe Women s Walking Hielo medio gris / gris medio / albaricoque , Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa Zapatillas de running Gobiv2 para mujer New Balance Semillas con Grove , Zapato de running Saucony Zealot ISO 2 para mujer Coral / púrpura / algodón , Zapatillas de running de malla cruzada Nova Vam Cruz V1 para mujer New Balance Reflexión / Blanco NIKE Men s Benassi Just Do It Athletic Sandal Negro / Juego Rojo / Blanco , Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink Black Bright Mango 600 Zapatillas de running Comfort Ride 575v2 New Balance para mujer Azul gris / Coral Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Multi-Color / Multi-Color , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul Mediano / Negro-aluminio-electro Verde , Zapatillas de running Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño Zapatillas Reebok Women s Runner 2.0 MT Track Negro / carbón / ácido rosa / blanco , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco , Zapato deportivo ASICS Gel-Excite 4 para mujer Uva / Plata / Begonia Rosa
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 Zapato de running ASICS Gel-Quantum 360 cm para mujer Blanco / Nieve / Flash Coral , Sandalias de mujer con tirantes Peep Toe de Bordello by Pleaser para mujer Fucsia Chaco Women s Zx3 Classic Athletic Sandal Lluvia de frambuesa Zapatillas de running NIKE Air Max Thea para mujer Laguna Blanca / Azul / Verde Fantasma , Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Acero Zapatillas de deporte NIKE Metcon 2 tobillo-alta moda mujer Pure Platinum / Hyper Purple Amortiguación 460v2 New Balance para mujeres Gris , Zapatillas de running New Balance para mujer 590v3 Negro / Castlerock Apex Emmy Slip para mujer Broncearse , Easy Spirit - Zapatillas Faisal2 para mujer gris Sandalia de Gladiador Conchlusion de mujer Aerosoles Serpiente negra Sandalia Nora Ryka para mujer Negro / Menta / Azul Sandalia Rosa mujer TOMS Ante de abedul , Sandalia de la paz de Easy Spirit Women Oro Skechers Women s Go Walk Joy-Paradise Sneaker Gris / azul Sandalia de enseñanza Grecian de cuero para adultos, T8900 Negro , Zapato de entrenamiento cruzado 577v3 New Balance para mujer Gris / rosa , Zapatillas para caminar Travelite para mujer Propet Negro Ryka Womens Divya Blanco / plata cromada , Lakai Limited Calzado para hombre Daly Black Camo Textile Supra mujer Skytop Sneaker White Oil Slick Supra Vaider Skate Shoe Negro / Blanco 023 Sandalias planas Softey para mujer Steve Madden Color de malva , Sandalias planas para piscina Walking Cradles para mujer Envoltura Metálica Multi / Cork , Sandalia de diapositivas de ocio para mujer de Seychelles Ante amarillo , Zapato de voleibol de control de corte de gel femenino ASICS Uva / azul marino / blanco Zapatillas de ciclismo de carretera Giro Womens Empire ACC Negro , Zapatillas Ixia de Easy Spirit para mujer Negro Cole Haan Women s 2. Zerogrand Stitchlite Ox Oxford Destello CLARKS Mujer Ashland Bubble Slip-On Charol de color borgoña ,

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 Under/ Armour Women s s Under Thrill 3 Apollo Grey (962)/ Blanco bb8770c 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

Thirteen elementary straightedge 1508784
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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.