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.

Ryka 19148 Women s Cross Destiny Cross Trainer Trainer Gris/ Rosa b038d26




Sintético
Importado
Sintético único
Eje mide aproximadamente parte baja del arco
6.4 oz.
memoria Forro de espuma en el forro.
Compresión entresuela de EVA moldeada.
Segunda densidad moldeada EVA con pivote Punto.
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


Ryka 19148 Women s Cross Destiny Cross Trainer Trainer Gris/ Rosa b038d26 - 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

Ryka 19148 Women s Cross Destiny Cross Trainer Trainer Gris/ Rosa b038d26

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.


Ryka 19148 Women s Cross Destiny Cross Trainer Trainer Gris/ Rosa b038d26

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

Entrenador Yui Cross de Yuki Gel-Fit para mujer de ASICS Negro / Regatta azul / plata Entrenador campeón Gusto Cross Trainer Malla blanca Zapato de entrenamiento minimal 40v1 New Balance para mujer Vintage Indigo / Vivid Tangerine NIKE Condición de Zoom para Mujer TR Cross Trainer Rosado , Plataforma de satén mono de ante PUMA para mujer Cesta Blanco Negro 1689 , Reebok Crossfit Nano 7.0 Track Shoe para mujer Lila / Plata , Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Blanco azul , Zapatillas de entrenamiento NIKE Women s Air Bella Azul marino / Blanco-Wolf gris New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Porcelana / cardo profundo , Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Blanco azul , Zapato de entrenamiento cruzado GEL-190 TR para mujer ASICS Blanco / Vinca / Marina de medianoche Zapatillas deportivas Superga Womens 2750 Cotu Canvas Blanco , Zapato de entrenamiento cruzado GEL-190 TR para mujer ASICS Blanco / Vinca / Marina de medianoche Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Black-paradise Pink , Under Armour Women s Precision Mid Negro (002) / Negro Zapatillas de entrenamiento cruzado Froma FM de PUMA Tazon 6 para hombre Periscope-puma Silver , Zapatillas Superga 2750 Cotu para mujer Azul medio New Balance 711v3 Disney Cross Trainer para mujeres Puntos negros / blancos / rojos Zapatillas deportivas Skechers para mujer Syn Elite Fashion Zapatillas deportivas Negro , Nike Tech Xtreme Cadet Glove Pure Platinum / Blanco / Volt / Black , Plataforma de satén mono de ante PUMA para mujer Cesta Blanco Negro 1689 New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Negro oscuro , Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Llovizna Gum Grey , Zapatillas DC Mujeres Chelsea TX SE Sneaker Azul oscuro Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro Vans Auténtico Azul Clásicos de base Slip-on (Tm) para hombres de Vans Black / Formula One Red Check Clásicos de base Slip-on (Tm) para hombres de Vans Blanco (lienzo) Vans Unisex Old Skool Zapatillas clásicas de skate Mezcla de Blues / Material de Vestir
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 de running Reebok Print 2.0 para mujer Gable Gray / Fire Spark / Blanco / Pure Silver , Sandalia Teva Women s W Hurricane Xlt2 Sport Lago Black / Gray , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Negro / Fantasma Zapatillas de running adidas Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco Sandalias KEEN Mujer Whisper-w Coral azul / Báltico , Zapatillas para correr Nike Women s Free Rn Distance 2 Hyper Violet / White Dark Iris Zapatillas de running New Balance 520v3 para mujer Gris Calzado de running XA Amphib W Trail para mujer Salomon Clima tormentoso Zapatillas para correr Salomon S / Lab Speed ​​2 para hombre Negro / Racing Rojo / Blanco , Reebok Women s Ridgerider 2.0 Trail Runner Negro / Rosa Solar / Plata / Estaño Zapatillas sin cordones Flex Merit para mujer Merrell Armada Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer No teñido, blanco, blanco de tiza , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Gris / plata , Coach Women & iexcl; & macr; s Cassidy Metallic Tumbled Sandals, Style A01627 Colorete Rieker womens Slipper oldgold / fango-silver Ice Multi , Zapato casual Brooks Mujer Addiction Walker V-Strap 2E Blanco , KEEN Women s Uneek 8MM Sandal Corazón Púrpura / Niebla de Lavanda , Sandalia de hecho de mujer de Pelle Moda Grafito nubuck Easy Street Adagio Wedge Slide para mujeres Patente / estiramiento rojo , Sam Edelman Geren Sandalia para mujer Candy Red , Zapatos de mujer RYKA Sky Walking Cool Mist Gray / Orchid Bouquet Zapatilla para mujer Dr. Comfort Katy Verde y Turquesa Sandalias planas para piscina Walking Cradles para mujer marrón , Sandalias Jerusalem Boaz Sandalia para mujer marrón , Sandalias de tacón alto con hebilla de Onex para mujer Negro / Plata Sandalia Beau Slide Kate Spade New York para mujer Oro Zapatillas Nd57 Pleasant Ocean / Stella para mujer El Naturalista Oceano Mocasines con suela de piel de oveja Minnetonka para mujer - 3341 Broncearse Bota de trabajo con punta de acero resistente al agua EH PO 11 BRN de John Deere para hombre marrón Blundstone Super 550 Series Boot Gris acero ,

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.

Ryka 19148 Women s Cross Destiny Cross Trainer Trainer Gris/ Rosa b038d26 - 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 Ryka 19148 Women s Cross Destiny Cross Trainer Trainer Gris/ Rosa b038d26 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.

Fitness y entrenamiento cruzado
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1094597
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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.