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 10 Zapatillas 10 para Zapatillas correr Plata correr/ Berry fe0e956




Malla de ingeniería
Importada
Caucho único
Eje mide aproximadamente 2.5 desde el arco
Desmontable plantilla
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 10 Zapatillas 10 para Zapatillas correr Plata correr/ Berry fe0e956 - 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 10 Zapatillas 10 para Zapatillas correr Plata correr/ Berry fe0e956

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 10 Zapatillas 10 para Zapatillas correr Plata correr/ Berry fe0e956

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 New Balance Urgev2 para mujer Negro / Champagne Metálico Zapatillas de running New Balance para mujer 490v4 Droplet / Silver , Zapatillas de running HOKA ONE ONE Clayton 2 para mujer Negro / Fucsia adidas Yeezy Boost 700 Gris sólido / Tiza Blanco con núcleo negro Zapatillas de running Endurant para mujer ASICS Carbon / Flash Coral / Silver , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas de running Mizuno Women s Wave Catalyst 2 Turquesa / Coral Fiery , Zapatillas de running Saucony PowerGrid para mujeres Gris / Té / Rosa Zapatillas deportivas NIKE Lunar Apparent para hombre Negro / blanco-cool gris Zapatillas de running ASICS Gel-Nimbus 20 para mujer Limelight / Limelight / Safety Yellow , Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas Blanco adidas outdoor para mujer Terrex Swift R2 GTX Negro / Negro / Negro , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Melocotón espolvoreado / Vortex , Zapato de running Mach de HOKA ONE ONE para mujer Blanco / Hibisco , Zapatillas de running adidas Aerobounce W para mujer Core Black / Core Black / Gray Four Zapatillas de running Saucony para mujer Omni 16 Gris azul , Zapatillas de running New Balance 520v3 para mujer Artic Fox / Black Zapatillas de running NIKE Air Max 1 Ultra 2.0 SI para mujer Voltio platino blanco / puro , New Balance Women s w990v4 Running Shoe Negro / Plata , NIKE Women s Revolution 4 Running Shoe Negro Crimson Antracita Gris , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Blanco Blanco Zapatillas para correr ASICS Gel-Venture 6 para mujer Carbono / Gris Medio / Rosa Concha Zapatillas de running Reebok Women s Run Supreme 2.0 Aleación / Gris Estaño / Orquídea Celestial / Blanco , Zapatillas para correr Saucony Freedom Iso Unisex Gris Pantalón largo clásico NIKE Golf para mujer Tech Essentials Obsidian Black 410 , NIKE W Air Vapormax Flyknit C para mujer Ah8448-001 Atmósfera Gris / Blanco-blanco-caliente Punch Zapatillas de running adidas Alphabounce Beyond para mujer Blanco / Gris / Gris , Zapatillas deportivas NIKE Lunar Apparent para hombre Negro / blanco-cool gris Zapatillas de running Classic-M Vibram para hombre Negro Negro
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 ASICS Gel-Contend 4 para mujer Plata / aguamarina / verde fuerte Sandalia de vestir Tillman de SoftWalk para mujer Blanco , CLARKS Womens Wave Grip Negro sintético OLUKAI Haiku Elua Sandal - Mujer Cinder / Natural , Converse Chuck Taylor All Star Seasonal Color Hola Carbón Zapatillas de running adidas Alphabounce Rc W para mujer Núcleo Negro / Plata Metálico / Gris Cinco , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Negro , Sandalia Sunkiss Cara de Sperry Top-Sider para mujer Broncearse Zapato para caminar Ryka Women s Sky Colony Blue / Soft Blue / Chrome Silver Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink / Black - Mango brillante , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco , Zapato Cross-Trainer Grafik 2 para mujer Ryka Azul marino , Yo también mujer Avalon Dark Alpaca Tumbled Metallic Zapatillas para caminar Go Flex 2-14992 de Skechers Performance para mujer Gris pardo Zapatilla deportiva con cordones Slip-On Onitsuka Tiger Mexico 66 Azul marino / apagado blanco Zapato de agua nativo de Lennox para hombres Bone White / Shell White / Torch Red / Dublin Block , Converse Chuck Taylor All Star Core Ox Armada Converse Chuck Taylor All Star Core Ox Mono negro NIKE Men s in-Season 7 Cross Trainer Gris oscuro / Barely Volt / Aurora Green , Supra mujer Skytop Sneaker Blanco apagado / blanco hueso , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante negro Converse Chuck Taylor All Star Core Ox Casi negro Zapatillas de tenis New Court 896v2 Hard Court para mujer Negro / Seafoam Zapatillas de skate auténticas Lo Pro para mujer Skyway / Blanc De Blanc , Supra Vaider LC Sneaker Gris - Blanco / Gris , Clásicos de base Slip-on (Tm) para hombres de Vans Clásico Negro / Blanco , Bota de mujer Eddie Bauer K-6 Pimenton Xelement 2505 Righteous Womens Black Zipper Botas para moto - 7 - Bota de 7 ojos para punta de acero de Arce de Dr. Martens Work para mujer Gris TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Espiga de carbón

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 10 Zapatillas 10 para Zapatillas correr Plata correr/ Berry fe0e956 - 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 10 Zapatillas 10 para Zapatillas correr Plata correr/ Berry fe0e956 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.