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.

Zapatillas/ de 23 running ASICS Gel-Kayano 23 para mujer Silver B00XKOZRYW/ Pink Glow/ Parachute Purple f6d5013




FluidFit / Mesh
Importado
Caucho único
Mpact Sistema de orientación (I.G.S) (Corriendo) : Diseño de ASICS filosofía que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegue del pie
FluidRide: FluidRide proporciona la combinación definitiva de rebote y amortiguación propiedades con peso reducido y excepcional durabilidad
FluidFit: ASICS FluidFit tecnología superior combina multi-direccional malla elástica con refuerzos de estiramiento que se adaptan a la pie de atleta, creando un verdadero personalizado como un guante ajuste
Clutch de talón Tecnología de sistema: Talón exoesqueleto contador proporciona soporte mejorado y crea mejoras entorno de ajuste del talón..Con mejor ajuste, el nueva forma no solo sostiene tu talón mejor, sostiene todo el pie mejor al proporcionar puntera superior.
Retropié y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo
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/ de 23 running ASICS Gel-Kayano 23 para mujer Silver B00XKOZRYW/ Pink Glow/ Parachute Purple f6d5013 - 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/ de 23 running ASICS Gel-Kayano 23 para mujer Silver B00XKOZRYW/ Pink Glow/ Parachute Purple f6d5013

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/ de 23 running ASICS Gel-Kayano 23 para mujer Silver B00XKOZRYW/ Pink Glow/ Parachute Purple f6d5013

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 ft. Perpetua ft de memoria Fila Women s Castlerock, blanco, negro , Zapatillas de running NIKE Lunartempo 2 para mujer Púrpura , Zapatillas de running Mizuno Wave para mujer Wave Horizon Azulejo azul-amarillo Zapatillas para correr Nike Women s Free Rn Distance 2 Blue Lagoon / Black-industrial Blue Zapato de correr ideal para rejilla de mujer Saucony Gris blanco , Zapatillas de running NIKE para mujer Air Huarache Run Ultra Elemental Rose / Wolf Gray / White Zapatillas de running NIKE Flyknit Racer NIKE para hombre Oreo-negro / blanco , Zapatillas de running NIKE Lunarglide 9 para mujer Negro / blanco / gris oscuro / gris lobo , Calzado de running casual para mujer New Balance WL555, para mujeres Negro Zapatillas de running Vibram Women s Classic, negras, 41 EU / 9-9.5 M US - , Zapatillas de running NIKE Lunarglide 9 para mujer Gunsmoke / Solar Red / Hot Punch / Vast Grey , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Mtlc Cool Gray Zapatillas de running Saucony Women s Breakthru 3 Berry Coral Calzado deportivo New Balance para mujer Vazee Rush V3 Vivid Coral / Fiji Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey Zapatillas de running ISO para mujeres de la guía Saucony Blanco azul Under Armour Speedform Velociti para mujeres Hielo Lavanda (500) / Vincapervinca , New Balance Women s 555 Lifestyle Fashion Sneaker-Suede / Mesh Zapatillas para correr Gris / gris adidas Performance Ultraboost X para mujer Gris / Gris / Hi-res Azul , Zapatillas de running NIKE para mujer, con altura libre, ango alto Cannon / Green Glow / Hasta Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Acero , Zapato de correr con cambio Gel-Quantum 360 Shift para mujer de ASICS Glaciar Gris / Blanco / Carbono Calzado de running para mujer New Balance Cypher Mulberry / Angora , Zapatillas de running NIKE Air Max Zero Premium para hombre Pure Platinum / Wolf Gray Zapatillas de running Nike Women s Zoom Vomero 9 Mgnt Gray / Hypr Pink-hyper Cobalt-rf Zapato de running ASICS Gel-Quantum 360 cm para mujer Melocotón negro / Onyx Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / Negro-supremo Azul-binario Azul , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Purple / Pure Platinum / Night Purple / Dark Raisin , Zapatillas de running New Balance Boracay V3 para mujer Dark Denim / Sunrise Glow
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 Mizuno Running Women s Wave Creation 19 Castlerock / Pink Glo / Black , Roxy Women s Set Session Zapatillas atléticas para caminar Antracita , Zapatillas de running NIKE Unisex Flyknit Racer Blanco / negro-voltio Zapatillas NIKE para mujer Air Max 1 PRM Running Peltre metálico / Peltre metálico 361 Zapatillas de trail running 361-Overstep 2 para mujer Aguanieve / ébano Zapatillas de trail running Ryka Women s Intent XT 2 SR Blanco / plata cromada Zapatillas de trail running Icebug Women s Mist RBX9 Iris / Grape NIKE Men s Benassi Just Do It Athletic Sandal Oro negro / metálico , Zapatillas de running NIKE Lunar Skyelux para mujer Aluminio / Binario Azul / Azul Mediano , Zapatillas de running Kinvara 8 Saucony para mujer Azul marino Púrpura Zapato de lona KEEN Sienna MJ para mujer Abigarrado , Sandalias de diapositivas casuales Easy Street Womens Conca Fabric Open Toe Natural , Zapatillas de running NIKE Air Max Thea Ultra PRM para mujer Summit White 100 Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco / Negro-juego Royal , Sandalia Terra FI Lite de Teva para mujer Cool Blue , CLARKS Haley Stork Loafer para mujer Negro , Zapatillas de running Questar BYD W de mujer adidas Originals Core Black / Core Black / Haze Coral Zapatos de entrenamiento Nike Free TR 6 para mujer Negro / Mtlc Red Bronze / Summit White Lakai Daly Skate Shoe Ante azul marino , Drew Shoes Toscana Diabético Terapéutico Mujer Profundidad Extra Zapatos de Cuero / Malla Con Cordones Negro / Combo , Botín \ Noir / Antico , Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Azul marino / apagado blanco Sandalias AS-Mona Slide de mujer Ash Oro negro , L & rsquo; Artiste by Spring Step - Sandalia Sumacah-Slide para mujer Camello , Vince Camuto Sandalias Ellanna Slide para mujer Moody Blues Bota western Glitter Breeze de Roper Kids Broncearse , Ariat Womens V Sport Tall Zip Tall Riding Negro Sandalia de cuña alta (90 mm) Soludos para mujer Nuez moscada , Zapatilla W Tasman de mujer UGG Negro , Propet Women s Tour Walker Strap Sneaker Gris pardo ,

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/ de 23 running ASICS Gel-Kayano 23 para mujer Silver B00XKOZRYW/ Pink Glow/ Parachute Purple f6d5013 - 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/ de 23 running ASICS Gel-Kayano 23 para mujer Silver B00XKOZRYW/ Pink Glow/ Parachute Purple f6d5013 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.