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.

Zapatilla deportiva con Mexico Tiger cordones Slip-On Onitsuka Tiger Mexico Ash 66 Ash Rose/ Ash Rose 6a233bb




Cuero
Importado
Esto es el México 66 Slip On.
Es presenta un similar mira como lo normal encajes México 66.
Los cordones se han eliminado y reemplazado con una banda elástica para los \ mira.
Hecho de una mayoría de materiales tejidos con un poco de cuero y gamuza en el Superior.
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


Zapatilla deportiva con Mexico Tiger cordones Slip-On Onitsuka Tiger Mexico Ash 66 Ash Rose/ Ash Rose 6a233bb - 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

Zapatilla deportiva con Mexico Tiger cordones Slip-On Onitsuka Tiger Mexico Ash 66 Ash Rose/ Ash Rose 6a233bb

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.


Zapatilla deportiva con Mexico Tiger cordones Slip-On Onitsuka Tiger Mexico Ash 66 Ash Rose/ Ash Rose 6a233bb

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 bajas con cordones All Star Canvas de Converse Chuck Taylor rojo , Zapatillas de tenis rígidas 996v3 de New Balance para mujeres Pigmento Zapatillas Mujer adidas by Stella McCartney Ultraboost X para mujer Piedra / Núcleo Blanco / Cáscara de huevo / Gris / Smc , Zapatillas de tenis adidas Women s Barricade 2018 W Blanco / escarlata / escarlata Zapatillas de tenis adidas Adizero Defiant Bounce W, de mujer, adidas Blanco / escarlata / núcleo negro Zapatillas de tenis acolchadas 786v2 de New Balance para mujeres Blanco azul , Zapatillas Mujer adidas by Stella McCartney Ultraboost X para mujer Piedra / núcleo blanco / cáscara de huevo gris , Botas de tobillo Pallaphoenix K Palladium para mujer Rosado , APL: Athletic Propulsion Labs Zapatillas de running mujer Techloom Phantom Plata Metálica / Gris Cósmico , Botines PUMA Puma x Rihanna para mujer Puma Black / Puma White / Puma White , Teva Hydro-Life Ballet de mujeres Teva Negro Zapatillas de tenis New Court 896v2 Hard Court para mujer Negro / Seafoam Zapatillas de tenis NIKE Womens Zoom Vapor X Lt. Blue Fury / Multi-color / Bleached Aqua Zapatillas de tenis Adidas Adizero Club w, para mujer Tiza Coral / Blanco / Azul Tinte , Zapato Dantera, para hombre, L Low Waxed Court Blanco APL: zapatillas de deporte Athletic Propulsion Labs para mujer Techloom Pro Black / 24k Melange Zapatillas de deporte de moda Gola para mujer Pastel Azul / Blanco hueso , Zapatillas de tenis adidas Barricade Court para mujer Blanco / piel de limón / blanco Phenma femenino Phenom Satin EP Puma Black / Puma Black , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco azul Zapatillas de tenis 696v3 New Balance para mujer Oxblood / Coral vivo Zapato Dantera, para hombre, L Low Waxed Court Pastos blancos / verdes / Blanco Zapatillas adidas Kids Cloudfoam Race K Sneaker Negro / amarillo solar Zapatillas Hailee Ryka para mujer Negro / plata / ciruela , Zapatillas de tenis NIKE Womens Zoom Vapor X Negro / Solar Rojo-blanco ASICS Tiger Mujer Gel-Kayano Trainer Knit Abedul / abedul Zapatillas de tenis adidas Ligra 5 W para mujer Tinta noble / oro metálico / tiza púrpura , Zapatilla de deporte de gamuza bala para mujer Gola Pale Gray / blanco roto , Columbia Womens Santiam Flip Oxford Tan / Violeta intenso
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 Sandalia de cuña Jubilee Propet para mujer Corteza múltiple , Sandalia Merrell de mujer con gladiador de poste marrón Zapatillas de running adidas Originals Tubular Defiant PK W Originals para mujer Blanco Ftw / Running Blanco Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Indigo vintage / Reflexión Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres Blanco ASICS Gel-Nimbus 20 para mujer, azul porcelana / azul blanco Porcelana Azul / Blanco / Asics Azul , Zapatillas de running ISO para mujeres de la guía Saucony Blanco azul , Zapatillas de running Skechers GOtrail para mujer Taupe / rosa , Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / Blanco-gris oscuro , Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Gris crudo, Gris crudo, Gris ceniza Atletismo femenino de Apex Negro Zapatillas de running definitivas Ryka para mujer Gris blanco , NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco , Zapatillas de running mujer HOKA ONE ONE Clifton 4 Aquifer / Vintage Indigo , Sandalia de mujer Venus de la tierra Sonrojo Sandalia de cuña Daytona SoftWalk para mujer Pewter / Black , Adidas Adizero Adios 3 Blanco Botín Rubin para mujer Sam Edelman Gamuza de Caramelo Dorado Zapato para caminar Ryka Women s Sky Gris / Rosa , Zapatillas de entrenamiento Reebok Crossfit Lite Lo TR para mujer China Rojo / Electro Rosa / Acero Orthofeet 877 Women s Comfort Diabetic Therapeutic Extra Profundidad Calzado Lycra Velcro Zapatillas de tenis Clay Court 996 V3 para mujer New Balance Deep Ozone Blue / Ozone Blue Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / blanco Calzado de golf Halo de Callaway para mujeres Negro Stuart Weitzman Landslid femenino Jeans ante , Sandalia Florence Slide de Vionic Women s Rest Negro Calzas de la dinastía Blanco negro , Zapatilla Glamour de mujer verde Daniel Bone Kidskin Zapatos de tacones urbanos para mujer UNA-01S Botas de tacón alto de mezclilla sobre rodilla hasta el muslo Lt Blue K1 Eddie Oxford para mujer bajo Negro

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.

Zapatilla deportiva con Mexico Tiger cordones Slip-On Onitsuka Tiger Mexico Ash 66 Ash Rose/ Ash Rose 6a233bb - 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 Zapatilla deportiva con Mexico Tiger cordones Slip-On Onitsuka Tiger Mexico Ash 66 Ash Rose/ Ash Rose 6a233bb 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.

Tenis y deportes de raqueta
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 194889
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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.