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.

Sendero de la Mujer Salomon/ Marin la W Trail Runner Sendero Deep Peacock Blue/ Ceramic/ Lime Punch. dd51d10




3D Stretch Air Malla
Importada
Caucho suela
Un zapato eso te detiene apretado y no lo hará dejar ir, con reforzado Tecnología Sensifit.
A pila de tecnología para mantenerte en marcha Sin sentimiento los golpes. el camino hacia una cinta transportadora, ¡continúa!
A punta reforzada tapa y mayor las orejetas hacen esto calza tu largo compañero del juego del término.
Salomon 2 años limitados garantía
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


Sendero de la Mujer Salomon/ Marin la W Trail Runner Sendero Deep Peacock Blue/ Ceramic/ Lime Punch. dd51d10 - 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

Sendero de la Mujer Salomon/ Marin la W Trail Runner Sendero Deep Peacock Blue/ Ceramic/ Lime Punch. dd51d10

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.


Sendero de la Mujer Salomon/ Marin la W Trail Runner Sendero Deep Peacock Blue/ Ceramic/ Lime Punch. dd51d10

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 trail running New Foam Gobi Neutral New Balance para mujer Guayaba / Negro Columbia Montrail Women s Fluidflex X.S.R. Camelia Roja, Melonade Merrell Women s Mqm Flex Monumento Colchoneta de amortiguación 620v2 New Balance para mujer Silver Mink / Gunmetal UB-ONEMIX Cool Sports Shoes Running Malla transpirable Zapatillas de deporte con cordones Tamaño EU 35-45 Rosa Rosa Zapatillas de trail WT10V4 para mujer New Balance Negro , Entrenador ECCO Biom Fjuel Racer Cross para mujeres Negro ECCO Sport Womens Terrawalk Titanio / Titanio Calzado de running Inov-8 Roclite 305 para mujer Rosado Zapatillas de running Mizuno Women s Wave Kazan Trail Púrpura / rosa Zapatillas de deporte trail running Hiero V2 para mujer New Balance Typhoon / Supercell / Alpha Pink , Zapatos Salomon Sense Pro Max Running Trail para mujer Fair Aqua, Crown Blue, Sulphur Spring Salomon Women s X-Mission 3 W-w Atlántico Norte Zapatillas Altra AFW1755F Mujeres Lone Peak 3.5 Lima Zapatillas de trail running Arc teryx Norvan VT GTX - Mujer Liberty / Arcturus Zapatillas adidas Outdoor Terrex Trailmaker para mujer Easy Orange, Black, Tactile Pink Zapatillas de running ONEMIX para hombre y mujer Zapatillas de deporte para correr Blanco negro , Calzado de trail running Salomon para mujer Ellipse 2 Mid LTR GTX W Dirigir Zapatillas de running Zante V4 para mujer New Balance Blue Iris / Pigment , Zapatillas New Balance para mujer Fresh Foam Gobi Trail Running Blanco / oro rosa Calzado de running Columbia Montrail Caldorado III para mujer Nappa Green, joya de la corona Reebok - Ridgerider Trail 3.0 Sneaker para mujer Polvo Gris / Smoky Taupe / Urban Grey / Desert Dust Salomon Women s X-Mission 3 W-w Laguna profunda , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Gris / Dark Shale / Easy Coral , Zapatillas para correr Saucony Women s Cohesion 10 Gris / verde azulado / cidra , Zapatillas de running para mujer adidas outdoor Kanadia 8.1 W Acero sin pulir / Blanco apagado / Rosa real Zapatillas de trail running WT10v1 Minimus para mujer New Balance Azul negro , Zapatillas de running La Sportiva para mujer Trailido Bushido Gris / menta , Zapatillas de trail running Salomon Women s Sense Mantra 3 W Lotus rosa / blanco / verde abuelita ,
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 para correr Nike Women s Free Rn Flyknit 2017 TRABAJA AZUL / OSCURO OSCURO-CLORO AZUL NIKE Men s Roshe Run Midnight Navy / Negro-Blanco , Calzado deportivo de punto Chelsea con memoria de Fila para mujer Blanco / Blanco / Highrise Zapato de running ASICS GEL-Kayano 22 para mujer Indigo Blue / Pink Glow / Pistachio , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Diva Pink / Black / Iceland Blue , Corredor de Trail Toggle Aspina para mujer de ECCO Gris cálido / gris cálido Zapatillas para correr ASICS Gel-Venture 6 para mujer Negro / Verde hielo / Naranja , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Shock Purple / Night / Easy Coral , Zapatillas de trail running X-Mission 3 CS W de Salomon para mujer Pasión Púrpura / Coral Punch / Cristal , Skechers Women s Walk Walk Joy-Invite Sneaker gris , Sandalia adidas Duramo Slide Poder azul / blanco / azul de energía , Zapatillas de running NIKE Zoom para mujer Negro / Blanco Gris Oscuro PUMA BTS x Collaboration Court Star (366202) Zapato de agua Easy Spirit para mujer First Walker Armada Zapatillas Cross-Trainer CrossFit Speed ​​Tr para mujer Reebok Blanco / Negro / Azul Beam / Fire Spark / Pure Silver , Sandalias de mujer de Britton Leather Open Toe con punta abierta marrón , Z / 2 Classic Pizza Zapatillas Vibram VI-s Cross-Trainer para mujer Nightshade / Violet Me Too OLYMPIA1 Zapato plano de cuero para mujer Rose Gold Metallic , Jack Rogers Women s Lauren Sandal Negro , Sandalia Evelia Slide de Bandolino para mujer Tela negra , Sandalia de cuero Lucili de Nine West para mujer Cuero blanco , Vince Camuto Sandalias Ellanna Slide para mujer Negro Zapato de voleibol Boost W Volley Response 2 de adidas Performance para mujer Collegiate Royal / Metallic Silver / Black Zapatillas de salón Issabella de Bloch para mujer Negro , Noble Outfitters Mens todo alrededor del dedo del pie cuadrado resistente Negro , Bota alta de mujer Christine de Franco Sarto Negro 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- Gold Sparklenet (Orden personalizado, 4-6 semanas de tiempo de proceso) , Bota Timberland para mujer Somers Falls Chelsea Marrón medio grano completo Clásicos de base Slip-on (Tm) para hombres de Vans Armada ,

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.

Sendero de la Mujer Salomon/ Marin la W Trail Runner Sendero Deep Peacock Blue/ Ceramic/ Lime Punch. dd51d10 - 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 Sendero de la Mujer Salomon/ Marin la W Trail Runner Sendero Deep Peacock Blue/ Ceramic/ Lime Punch. dd51d10 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.

Trail Running
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 2877722
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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.