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 running mujer/ HOKA Passion HOKA ONE ONE Clifton 4 Cerezas Jubileo/ Purple Passion 1439fe5




malla
Importado
Suela de goma
multicolor
puntera redonda
amortiguación
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 running mujer/ HOKA Passion HOKA ONE ONE Clifton 4 Cerezas Jubileo/ Purple Passion 1439fe5 - 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 running mujer/ HOKA Passion HOKA ONE ONE Clifton 4 Cerezas Jubileo/ Purple Passion 1439fe5

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 running mujer/ HOKA Passion HOKA ONE ONE Clifton 4 Cerezas Jubileo/ Purple Passion 1439fe5

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

Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Glicina / Rosa Violeta Zapatillas de calle YKTM Performance Women Boost 4.0 Street Running Shoe Road Negro Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino Blanco / Blanco-puro , NIKE Women s Revolution 4 Running Shoe Blue Recall / Pure Platinum / Purple Slate Zapatillas para correr NIKE Air Max 90 SE para mujer Saucony Women s Guide 10 Zapatillas para correr Azul claro / azul , Zapatillas de running PUMA para mujer Ignite Omphalodes / Coral caliente / Ultramar , Zapatillas de running NIKE Lunartempo 2 para mujer Azul , Zapatillas de running adidas Adizero Boston 6 W para mujer Easy Blue Tactile Blue Energy Blue S , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris claro Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow , Calzado de running NIKE Kids Roshe One (GS) Midnight Navy Blanco , Zapatillas de running Flash para mujer New Balance Blanco / Alfa Rosa / Negro , Zapatillas de running con amortiguación W560v7 para mujer New Balance Armada NIKE Men s Benassi Just Do It Athletic Sandal Negro Saucony Women s Ride 10 zapatillas para correr Blanco azul , Zapatillas de running adidas Originals para mujer Ultraboost W Aero Blue S, Aero Blue S, Noble Indigo S , Saucony Women s Ride 10 Running Shoe Gunmetal / rosa Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima , New Balance W880gb6 para mujeres Gris azul Zapatillas de running Mizuno Wave para mujer Inspire 10 Blanco , Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple Zapatillas de entrenamiento cruzado Trainflex para mujer Reebok Colegial azul marino / blanco / rosa estelar / plata metalizada / gris , Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino , Zapatillas de running Saucony para mujer Hurricane ISO 3 Gris / Rosa / Blanco Skechers Performance Women s Go Run 400 Obstruct Negro , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Carbón / Blanco / Flash Coral , NIKE Huarache Run Ultra Fashion Sneaker para mujer Cool Gray / Anthracite-black-white Zapatillas de running NIKE Lunar Skyelux para mujer Purple Shade / Bright Mango ,
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 NIKE Women s WMNS Zoom Fly, negro / blanco Provence púrpura / negro Zapatillas de running NIKE Unisex Flyknit Racer Neo Turquesa / Brillante Glaciar Crimson Hielo , Sandalia de deslizamiento ultra Comfort NIKE para mujer Negro / Blanco / Negro Zapatillas de running NIKE Air Max Flair para hombre Turquesa negra Zapatillas de trail running Merrell para mujer Bare Access Arc 4 Negro / Aventurina , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar 212black , Zapatillas de deporte con diseño hidroeléctrico de mujer W Ahiga V3 de Helly Hansen Blanco apagado / rosa de Shell , Easy Spirit Women s Motion Lace up Oxford Armada , Sandalia CLARKS Wave Grip para mujer Gris Claro Nubuck Birkenstock Unisex Arizona Essentials Sandalias EVA Rosa Eva Sandalias de mujer Vionic Perk Carmel Backstrap Sandal Serpiente natural , Sandalia Bella Vita IMO-Italia para mujer Cuero negro , Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Gris , Birkenstock Blue Footbed-Arch Soporte plano Azul , Diapositiva Encanta Teva para mujer Broncearse , SOREL Womens Ella Lace-Up Natural , Calzado de running adidas Pureboost X Element para mujer Blanco / Gris Dos / Verde , Chaco Mujer Fallon Sandal Flamenco Zapatillas PUMA Tazon 6 Wide Wn para mujer Puma White-island Paradise-puma Silver Trotters Mujer Simmy Mary Jane Flat rojo ASICS Tiger Unisex Gel-Lyte V Sanze Negro / Negro 1 Supra Scissor Skate Shoe Hueso blanco Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Rosado , Zapato de golf perforado Biom Hybrid 2 ECCO para mujer Blanco / Granito Verde , Calzado de golf Skechers Women s Go Eagle Major Blanco / Multi BareTraps Women s Beverly Slide Sandal Negro Zapatilla Nakiska SOREL para mujer Esquisto Tierra Pegaso Cuero de grano completo de almendra , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Taupe Gus y Jaq , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer 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.

Zapatillas de running mujer/ HOKA Passion HOKA ONE ONE Clifton 4 Cerezas Jubileo/ Purple Passion 1439fe5 - 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 running mujer/ HOKA Passion HOKA ONE ONE Clifton 4 Cerezas Jubileo/ Purple Passion 1439fe5 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.