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 de/ running 19934 Saucony Zealot ISO 2 marino para mujer Azul marino/ azul/ sil b73cdf5




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 de/ running 19934 Saucony Zealot ISO 2 marino para mujer Azul marino/ azul/ sil b73cdf5 - 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 de/ running 19934 Saucony Zealot ISO 2 marino para mujer Azul marino/ azul/ sil b73cdf5

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 de/ running 19934 Saucony Zealot ISO 2 marino para mujer Azul marino/ azul/ sil b73cdf5

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 New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul , Zapatillas de running Mizuno Wave Rider 20 para mujer Negro / Blanco / Rosa Glo Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Wolf Gray / Metallic Rose Gold Zapatillas de running NIKE Free Run para mujer - SU16-10 - Blanco Mango / Sunset Glow , Zapatillas de running Reebok Women s Run Supreme 2.0 Mystic Maroon / Poison Pink / White Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral NIKE Men s Flex 2017 RN Negro / Hematita Metálica / Antracita / Gris Oscuro , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Negro-corredor Rosa-blanco Calzado deportivo New Balance para mujer Vazee Rush V3 Blanco / Vintage Indigo Zapato de running ASICS GT-2000 4 para mujer Plata / turquesa / sacador de lima NIKE Mujeres Air Max Thea Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout Under Armour Threadborne Fortis para mujer Negro (001) / Stealth Gray , Brooks Womens Hyperion Peacoat / azul marino / cacatúa , Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral 361 Zapatillas de running para mujer 361-Enjector Negro / Blanco / 0900 , Zapatillas de running adidas Ultra Boost para mujer Azul Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo Zapato de running ASICS GEL-Cumulus 20 para mujer Azure / Blue Print , Zapatillas de entrenamiento Nike Womens Metcon 3 Cool Grey / Solar Red-black-pure Platinum Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-blanco-negro , New Balance Women s w940v3 Running Shoe Plata , Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Azul nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Hematita Metálica / Negro-gris oscuro Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Negro , Zapatillas de running adidas Women s Edge Lux W Gris / gris / blanco cristal Zapatillas para correr Mizuno Women s Wave Enigma 6 Púrpura / blanco ,
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 para correr Mizuno Women s Wave Creation 15 Blanco / Eléctrico , NIKE Zapatillas de deporte / running Air Max para mujer Azul / verde / negro Sandalia Yucatan ECCO Mujer Paloma medieval / salvaje Sandalia NIKE Kawa Slide para mujer Cloro Azul / Azul Glaciar / Foto Azul adidas Adizero Lj Núcleo Negro, Ftwr Blanco, Naranja , NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco Zapatillas de running FuzeX PR para mujer de ASICS Pink Glow / White / Pink Ribbon Salomon Women s X-Mission 3 W-w Menta / Deep Dahlia / Athletic Green X Sandalias EVA Gizeh de mujer Birkenstock Rosado , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco Blanco Me Too OLYMPIA1 Zapato plano de cuero para mujer Nude Nappa , Zapatillas de running Alphabounce Em W para mujer de adidas Originals Rosa hielo / Trace rosa / gris uno , Sandalia Relax de Steve Madden para mujer Ante negro , Sandalia Huarache Easy Street Allure para mujer Negro nativo AP Mercury Liteknit Leche Rosa / Shell Blanco / Caucho Natural , Vans Auténtico (Neón) Orange Pop , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Punto negro / blanco , Zapatillas de deporte Diadora V7000 Premium para hombre Paja dorada , Clásicos de base Slip-on (Tm) para hombres de Vans True White , Vans Unisex Old Skool Zapatillas clásicas de skate Blanco negro Kate Spade New York Women s Isla Slide Sandal Blanco Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno turquesa Zapatillas de deporte Trinity Sneaker Dance de Bloch Blanco , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Lona negra lavada Naturalizer Michelle s Dress Dress Pump Gris pardo Sandalia Birkenstock de cuero Florida para mujer Cereza , Sandalia Birkenstock de cuero Florida para mujer Sfb Blue Birko-flor Pantalón con aros delanteros con abertura en la rodilla y parche Aria Heritage Heritage para mujer Índigo Tobillera BT Garin de BareTraps para mujer Gris pardo , Finn Comfort Women s Soho - 82743 Mar / Jazmín Nubuck / Okapi ,

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.

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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 de/ running 19934 Saucony Zealot ISO 2 marino para mujer Azul marino/ azul/ sil b73cdf5 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 3010351
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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.