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 B00PRO25BQ de Zapatillas running running La Sportiva para mujer Trailido Bushido Fiordo 097fa86




tela y sintético
AWARD-WINNING - Diseñado para rendimiento inmejorable en terreno técnico, Bushido de La Sportiva Zapatillas de trail running se ha ganado el Selección del editor premio de Trail La revista Runner
LIGHTWEIGHT - Diseño de bajo perfil pesa solo 8.80 oz (248g), haciendo es un zapato de ir a para larga distancia fuera de carretera; Construcción de control STB utiliza un TPU marco que se envuelve bajo pie para proporcionar máxima estabilidad y receptividad sin peso adicional
ALTO PROTECCIÓN DE RENDIMIENTO - Tacos de suela tener bordes redondeados y se superponen entresuela para agregar tracción y mejorado estabilidad en off-camber terreno; EVA comprimido protector de roca en el antepié / TPU Shank ayuda a prevenir moretones
TRANSPIRABLE - Parte superior AirMesh, inserciones laterales de malla, y Stretch AirMesh trabalenguas juntos para mantener los pies secos y ayuda a prevenir ampollas; construcción antideslizante se adapta al pie como un calcetín sin causando excesiva presión
ESTABILIDAD Y CONTROL - Dual-Density FriXion XT V-Groove con freno de impacto El sistema proporciona excelente estabilidad en descensos empinados, senderos resbaladizos; navegar rocas, raíces y otros obstáculos con el agresivo, pegajoso de goma único
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 B00PRO25BQ de Zapatillas running running La Sportiva para mujer Trailido Bushido Fiordo 097fa86 - 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 B00PRO25BQ de Zapatillas running running La Sportiva para mujer Trailido Bushido Fiordo 097fa86

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 B00PRO25BQ de Zapatillas running running La Sportiva para mujer Trailido Bushido Fiordo 097fa86

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 running Altra Lone Peak 3 Mid Neo - Mujer Negro Zapatillas de running Gobiv2 para mujer New Balance Semillas con Grove Merrell Women s Bare Access Flex Trail Runner Hielo Zapatillas de trail running WTSUMV1 para mujer New Balance Guayaba Calzado de trail running Inov-8 para mujer Terraclaw 250 Azul / Berry / Negro , Zapatillas de trail running Merrell para mujer Bare Access Arc 4 Negro / Aventurina , Columbia Rogue F.K.T. Steam / Dark Raspberry , Zapatillas de trail WT10V4 para mujer New Balance Libélula , Inov8 Roclite 290 Off Road Shoes & Workout Visor Bundle Verde azulado / negro / blanco Zapatillas para correr Saucony Women s Cohesion 10 Gris / verde azulado / cidra Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Rosa táctil, negro, naranja fácil Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Gris / rosa , Corredor de malla Primus FG L Mesh Trail de Vivobarefoot para mujer Negro / Carbón , Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Blanco / Antracita / Lobo Gris / Sigilo Zapatillas de running New Trail Hierro V3 Fresh Foam Trail para mujer Gris UB-ONEMIX Unisex Original Walking Shoes Transpirable Suela de lengua elástica Damping Sneakers Rosado , Arc teryx Womens Norvan LD GTX Dahlia / Lumen Lime Zapatillas de trail running Inov-8 Roclite 290 (W) para mujer Coral rojo Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Acero Zapatillas de trail running Ryka Women s Intent XT 2 SR Negro / plata cromo , Salomon Women s Speedcross 4 W Trail Runner Blue Jay / Black / Infrared Corredor Trail ECCO Terracruz II de Mujeres Negro Negro , Zapatillas de running NIKE Zoom All Out Low para hombre Obisidian / Paramount Blue-black , Zapatillas de trail running Salomon Women s Speedcross 3 Luz Onix / Topaz Azul / Nube Oscura Zapatillas de running NIKE LD Runner LW para mujer Negro / Volt-glacier Blue , Zapatillas de running adidas Women s Rockadia W Trail Carbono / Acero sin procesar / Trace Scarlet , Zapatillas de trail running Saucony para mujer Grid Excursion Tr10 Negro / Menta Zapato ATA 3 Challenger de mujer ONE ONE de HOKA Marina de medianoche adidas Performance Women s Incision W Trail Runner Easy Mint / Black / White ,
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 KEEN Mujer Whisper-w Higo de verano / Crabapple , Sandalia KEEN Women s Venice H2 Midnight Navy / Coral caliente , Sandalia Savannah de Ryka para mujer Gris blanco , NIKE Women s Tanjun Marrón nocturno , Skechers Women s Go Trail 2 - Zapatillas de senderismo con agarre Negro , Zapatillas NIKE Women s Free Run 2017 negro / azul de hidrógeno / gris oscuro-lobo gris Negro / Hidrógeno / Azul Gris Oscuro , Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Gris , Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker W para mujer Gray Five / Chalk White / Easy Coral Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro , Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento Drew Smiles Slip para mujer Negro / Microdot Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Gris / Franela Zapato clásico de mujer Spira para caminar con correa EZ Blanco , Skechers Womens Microburst - Perfect Note Ballet Flat Negro / Carbón , Calzado para caminar CUSH + WA365v1 New Balance para mujer Gris Tierra Dalia Negro , Sam Edelman sandalia Gio Slide para mujer Ante negro Zapatillas de entrenamiento Nike Women s Free 5.0 TR Fit 5 blanco azul Skechers Women s Go 4-14918 Calzado para caminar Blanco negro , Zapatilla para mujer Dr. Comfort Katy Turquesa Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Negro ECCO Golf Womens Biom G 2 Gratis Concrete / Silver Pink Marc Jacobs Women s Aurora Mule Negro La Plume Womens Jessica rojo UGG Mujer W Joni Lude Gray Bota Ecuestre Xandy LifeStride para mujer Negro Zapatillas de tela Raissa de Nine West para mujer Luz de oro , Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse Clásicos de base Slip-on (Tm) para hombres de Vans Black Off White Checker , Zapatillas Slip-On Roamer SAS para Mujer castaña ,

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 B00PRO25BQ de Zapatillas running running La Sportiva para mujer Trailido Bushido Fiordo 097fa86 - 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 B00PRO25BQ de Zapatillas running running La Sportiva para mujer Trailido Bushido Fiordo 097fa86 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

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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.