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.

TOMS Lienzo - Suela la de lino de lino clásica, cómoda - y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia d7f7171




tejido y sintético
Hecho en EE. UU. o importado
Con cada par de zapatos usted compra, TOMS & reg; dará un nuevo par de zapatos para un niño necesitado One for One & reg;.
El TOMS & reg; Los clásicos mantendrán eres cálido y tostado cada día frío!
Parte superior viene en una variedad de materiales. declarado en color nombre.)
Características puntada clásica del dedo del pie detallando.
Elástico V-panel para fácil encendido y apagado.Plush faux shearling forro.Suede cuero plantilla proporciona amortiguación ligera.Arch insertar para agregar apoyo. Flexible, de una sola pieza, caucho mezclado suela.Importado. Mediciones: Peso: 5 oz Mediciones del producto fueron tomadas usando tamaño 12, ancho D - Medio. tenga en cuenta que las mediciones puede variar según el tamaño. Peso de calzado es basado en un solo artículo, no un par
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


TOMS Lienzo - Suela la de lino de lino clásica, cómoda - y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia d7f7171 - 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

TOMS Lienzo - Suela la de lino de lino clásica, cómoda - y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia d7f7171

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.


TOMS Lienzo - Suela la de lino de lino clásica, cómoda - y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia d7f7171

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

Golf de mujeres y rsquo; s Peso ligero genuino Cuero Lexington Rendimiento de golf Marc Joseph NY Zapatos de moda Tan granulado con cremosa patentada Kilt , CLARKS Jocolin Vista Boat Shoe para mujer Deep Fuchsia Synthetic Trotters Liz Loafer para mujer Marrón oscuro acolchado en relieve , Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Miel , Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Mazarine azul / blanco Mocasín con cordones ocasionales de cuero KEESKY para mujer Zapatos sin cordones planos Negro a Drew Shoe Women s London II Slip On Textil Casual Flats Negro , La emperatriz de las mujeres de Skechers se ve bien Rosa topo TOMS Clásicos de mujer Gris , Toms Womens Classics Black / Black Canvas 10002472 Negro sobre negro TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Dark Navy / Drizzle Grey Slub Chambray , Mocasín de gamuza Kilty de Minnetonka para mujer Tela de gamuza marrón / Arizona , Skechers You de Women s One Slip-on Shoe Negro , Arcopedico Womens Lolita Slip-On Gris pardo Drew Shoe Women s London II Slip On Textil Casual Flats Negro , TOMS Clásicos de mujer Chambray azul , Arcopedico Womens Lolita Slip-On Gris , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Negro / Negro Camo CLARKS Women s CloudSteppers Sillian Jetay Flat Black Perf Textile , TOMS - Slip-On de lienzo clásico para hombre Lona de caoba roja Easy Spirit Women s Traveltime Mule Blanco Multi TOMS - Slip-On de lienzo clásico para hombre Negro , Zapatillas Skechers Women s Microburst One hasta Fashion Sneaker Plata blanca , Sam Edelman Loraine Loaine para mujer Amarillo / Multi Gingham , Toms Womens Classics Black / Black Canvas 10002472 Negro sobre negro , Prop & eacute; t Zapatillas de deporte TravelActiv Slip-On Fashion Sneaker Plata TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Puntada azul marino de gamuza / manta TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Ballena multi acuarela TOMS - Slip-On de lienzo clásico para hombre Red Stripe University ,
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 Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente , Zapato de running ASICS GEL-Venture 5 para mujer Indigo Blue / Pink Glow / Living Coral , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Sandalias para mujer Teva Women s W Hurricane XLT2 Kerne Blue / Multi Zapatillas antideslizantes Skechers for Work para mujer Eldred Blanco , Sandalia KEEN Women s Exo-W Uneek Vapor / Paloma , Sandalia original Teva Women s Original Pirámide azul Zapatillas Altra AFW1755F Mujeres Lone Peak 3.5 Negro Zapatillas de agua ZipWalker para mujer Speedo Rosa / negro , Sandalia Drew Cascade para mujer Nubuck marrón Zapatillas de running NIKE Roshe Two para mujer Negro / rosa , Chancleta Yoga Sling 2 Sanuk para mujer Frambuesa Lanai Blanket Zapato de entrenamiento cruzado RYKA Women s Influence Negro / rosa Adidas Mujeres Aleki X Cross Tainer Bold Blue / Dark Indigo / Black , Vans Unisex Old Skool Zapatillas clásicas de skate Borgoña / True White Zapatillas de skate Vans Unisex Authentic Platform Cadmio Verde 7295 , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Armada Calzado de mujer de fila Cress Walking Blanco, Highrise, Goma , Zapatillas de deporte Adidas Adipure Sport para mujer Rosado Sandalia Isla Slide de KENDALL + KYLIE Plata , Sandalia del vestido de la inversión de LifeStride de las mujeres Lux Navy BareTraps Women s Jaylyn Slide Sandal Negro Sandalia para mujer Layne Slide de Pour La Victoire Natural Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Terciopelo burdeos (pedido personalizado, 4-6 semanas de tiempo de procesMocasines casuales Alpargata Deconstructed de TOMS para mujer Gris 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2713- Ultra Silver TOMS Kids 10009918 Alpargata-K Birch Heather Jersey / Faux Shearling Sandalia de gamuza Poppy de TOMS para mujer Cuero miel Zapatillas Slip-On Roamer SAS para Mujer Sabio

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.

TOMS Lienzo - Suela la de lino de lino clásica, cómoda - y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia d7f7171 - 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 TOMS Lienzo - Suela la de lino de lino clásica, cómoda - y de ajuste fácil para mujer Lienzo blanco antiguo de la herencia d7f7171 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.

Mocasines y Slip-Ons
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 4663117
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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.