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.

Saucony Women s Guide Guide 8 Running Shoe Rosa Blanco/ Women Crepúsculo/ Rosa d41f74d




Textil
Importado
Caucho único
Running zapato con malla y sintético superior con acolchado lengua y cuello
PWRGRID Amortiguación integral
SRC Zona de impacto
Durable iBR + suela de mediopié pods
Traction suela con uso inteligente Compuesto XT-900
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


Saucony Women s Guide Guide 8 Running Shoe Rosa Blanco/ Women Crepúsculo/ Rosa d41f74d - 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

Saucony Women s Guide Guide 8 Running Shoe Rosa Blanco/ Women Crepúsculo/ Rosa d41f74d

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.


Saucony Women s Guide Guide 8 Running Shoe Rosa Blanco/ Women Crepúsculo/ Rosa d41f74d

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 ancho NIKE para mujer Experience 7 de NIKE Negro / Blanco - Blanco , Zapatillas de running New Balance Boracay V3 para mujer Supercell / Ozone Blue / Bleached Sunrise , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Blanco / Gris Oscuro , Nike Men s Air Presto Essential Negro / Tour Amarillo / Universidad Rojo / Blanco Zapatillas de running New Balance 790v6 para mujer Black / Poison Berry , Zapatillas de running Vibram para mujer, negras, 39 EU / 8 M ES - , Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Volt / Blue Lagoon / Blanco Zapatillas para correr Saucony Xodus Iso 2 para mujer Berry Coral , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / antracita Zapatillas Reebok Women s Plus Lite HB Track Heather Black - Black / Asteroid Dust / Rbk Rubber Gum / Chalk Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Phantom / Black Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Gamma Blue / White / Rio Teal Zapatillas de running Saucony para mujer Omni 16 Gris azul Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Racer Pink / Vastgrey Camiseta adidas Swift Run W para mujer Core Black / Core Black / White , Zapatillas de running TOPO FLI-Lyte 2 para mujer Azul rosado , Nike Men s Air Presto Essential Gris , Zapatillas de running FuzeX PR para mujer de ASICS Pink Glow / White / Pink Ribbon , Under Armour Threadborne Fortis para mujer Gris Nublado (103) / Blanco , Zapatillas de running NIKE Huarache Run SE para mujer Blanco / negro-blanco / negro , NIKE Women s Wmns Air Vapormax Flyknit 2, negro / blanco-gris oscuro Negro / Blanco-gris oscuro , Zapatillas para correr Saucony Koa Tr para mujer Gris azul , Zapatillas de running NIKE para mujer Lunarepic Flyknit Shield Negro / Metálico Plata-oscuro Gris-sigiloso Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey Saucony Women s Guide 10 Zapatillas para correr Azul claro / azul Zapatillas de running New Balance 1340v3 para mujer Gris oscuro Zapatillas de running Saucony para mujer Stretch N Go Breeze borgoña , Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Puerto / Aqua
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 NIKE Juvenate para mujer Cool Grey / Cool Grey / Sunset Glow / White , Zapatilla Slip-On Hydro-Life W de Teva para mujer Gris oscuro Sandalia deportiva Tirra para mujer Teva Zaca Dark Purple / Gold Zapatillas de running adidas Alphabounce Rc W para mujer Core Black / White / Utility Black , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Sandalia Kea Vent mujer Columbia Fósil, Cervatillo Calzado de running New Balance para mujer Vazee 2090 Azul marino / amarillo , Columbia Montrail Women s Trans Alps F.K.T. Jade Lime, Splash Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Calzado de natación paso-naval Go Skechers Performance para mujer Gris pardo Zapatillas de neopreno para caminar Pedors Womens Classic MAX Negro LifeStride Womens Corinne Blanco Zapatillas para caminar Skechers Go Step Lite-Wander para mujer Negro Drew Womens Ariana Dusty Gold , Easy Spirit Women s Traveltime Mule Azul marino / ante blanco , Drew Shoe La alegría de las mujeres Hueso , Skechers Performance Women s Go Step Modish Negro , Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Azul Indigo / Plata / Azul Regatta Supra Vaider LC Sneaker Amber Gold / White , Vans Auténtico (Camo) Olivo / Marshmallow Militar Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Casino / Blanco / Negro , Zapatillas de golf Minimus WP para mujer New Balance Gris / menta Zapatillas de golf Ignite Blaze Sport Disc PUMA para mujer Peacoat / Blanco Sandalia para mujer PUMA Popcats Slide Negro Negro Cocos de Matisse Women s Gaucho Boot Tan / rosa Shellys London Dilys Oxford para mujeres Plata , Apex Women s Ambulator Biomechanical Negro La Sportiva Boulder X Shoe - Mujer Azul hielo , Drew Shoe Women s Roma Oxfords Cuero negro / microfibra , TOMS Clásicos de mujer Ceniza ,

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.

Saucony Women s Guide Guide 8 Running Shoe Rosa Blanco/ Women Crepúsculo/ Rosa d41f74d - 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 Saucony Women s Guide Guide 8 Running Shoe Rosa Blanco/ Women Crepúsculo/ Rosa d41f74d 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

Thirteen elementary straightedge 3255974
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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.