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.

Zapato algodón B01HYUHJNO de running Saucony Zealot púrpura ISO 2 para mujer Coral/ púrpura/ algodón 3f2bfc5




100% sintético
Importado
Caucho único
Ligero Zapato de correr con amortiguación de apoyo diseñado para promover una marcha natural
ISOFIT funda interior suave transforma a pie para sentir como un calcetín
FLEXFILM superior para peso ligero asistencia
Everun plantilla proporciona amortiguación sensible
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


Zapato algodón B01HYUHJNO de running Saucony Zealot púrpura ISO 2 para mujer Coral/ púrpura/ algodón 3f2bfc5 - 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

Zapato algodón B01HYUHJNO de running Saucony Zealot púrpura ISO 2 para mujer Coral/ púrpura/ algodón 3f2bfc5

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.


Zapato algodón B01HYUHJNO de running Saucony Zealot púrpura ISO 2 para mujer Coral/ púrpura/ algodón 3f2bfc5

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 NIKE RN Flyknit 2018 Nylon para mujer Pulso carmesí / Vela / Híper carmesí / Negro Zapatillas de running Saucony Women s Freedom ISO Negro , Zapatillas de running NIKE Air Max Thea para mujer Fucshia Flash / Fuchsia Glow , En la ejecución de calzado de mujer en la nube menta , NIKE Men s Roshe Two Running Shoe Wolf Gray / Dark Gray / Volt Nike Men s Air Presto Essential Velvet Brown / Light Bone Zapatillas de running NIKE para mujer Air Huarache Run Ultra Elemental Rose / Wolf Gray / White Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Gris oscuro / Barely Volt / Wolf Gray Skechers GOrun 5 Zapatos de mujer Negro Negro Calzado de running NIKE RN 2017 2017 para mujer Solar Red / Black / University Red , Supra Martine Run Skate Shoe para hombre Rosa / Rosa-blanco adidas Women s Edge Lux 2 Sol / Plata / Naranja , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de running Saucony Women s Breakthru 2 Rosa / azul marino , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Espuma de mar / cielo cubierto , Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Zapatillas de running mujer HOKA ONE ONE Clifton 4 Cerezas Jubileo / Purple Passion , Zapatillas de running adidas Women s Edge Rc W Gris Dos / Plata Metálica / Energy Aqua Zapatillas de running NIKE RN 2017 para mujer gratis Vintage Verde / Provenza Púrpura-secoya Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple , Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Pasas oscuras / Pasas oscuras-blanco-pálido gris , Zapatillas de running Altra AFW1837F para mujer Torin 3.5 Rosado , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris oscuro , Calzado de running NIKE Kids Roshe One (GS) Racer Pink / Blanco-negro-blanco , Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / antracita , Zapatillas de running PUMA Women s Speed ​​500 Ignite Tobillo-Alta rojo Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa , Zapatillas de running Nike Kaishi para hombre Negro Zapatillas de running TOPO FLI-Lyte 2 para mujer Coral / 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 Sandalia Savannah de Ryka para mujer Gris oscuro , Zapatillas de running New Balance Coast V3 para mujer Ciclón ligero / plata Zapatillas de running ASICS Gel-Contend 4 para mujer Podar / Plata / Negro , Sandalia de mujer Drew Alana Estaño / Bronce , Chaco Women s ZX2 Classic Athletic Sandal Daga Zapatilla de deporte Sina 11 de Josef Seibel para mujer rojo Wolky Comfort Jewel Fantasía de serpiente blanca / negra posters OluKai Kulapa Kai Fucsia / Negro Chaco Z / Canyon 2 Cottage Poppy Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Negro / Estaño / Plata Metálica , Merrell Siren Edge Hiker para mujer Monumento Sandalias de peonía Easy Spirit para mujer Azul Artemisa Ante negro Bota Meri Alegria para mujer Melaza troceada , Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos castaña , Tecnología Vionic Men s Orthaheel White / Navy Walker Azul oscuro / negro SoftWalk Women s Topeka Flat Sabio 361 y grados; Blanco negro , JSport by Jambu Women s Wendy Flat Gris claro , Lakai Sheffield Skate Shoe Gamuza negra / gris Vans Auténtico (Leopardo metálico) negro / negro , Sandalia de diapositivas Henlee de Dolce Vita para mujer Tela roja Carlos by Carlos Santana Sandalia mujer Slide Liliana Estaño , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 2703- Gold Stardust & Gold , Botín de tobillo Mazin Cayto para mujer Hush Puppies Marron oscuro , Sorel - Botas sin cáscara para mujeres después de hora Secoya , Onitsuka Tiger Mexico 66 Gris vaporoso / blanco , Skechers Bikers para mujer Relaxed Fit en el exterior Sneaker Ante negro , MBT Shoe 700300-701U Elea 2 Beige Beige , Zapatillas de deporte de mujer Easy Spirit de Easy Spirit Gris ,

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.

Zapato algodón B01HYUHJNO de running Saucony Zealot púrpura ISO 2 para mujer Coral/ púrpura/ algodón 3f2bfc5 - 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 Zapato algodón B01HYUHJNO de running Saucony Zealot púrpura ISO 2 para mujer Coral/ púrpura/ algodón 3f2bfc5 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.