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 34 Blue de running NIKE Zapatillas Air Zoom Pegasus 34 para mujer Hot Punch/ Thunder Blue 2f1498e




Número de modelo: 880560602
Género: para mujer
Color: HOT PUNCH / THUNDER AZUL
Hecho en: China
Marca Nuevo con Original Caja
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 34 Blue de running NIKE Zapatillas Air Zoom Pegasus 34 para mujer Hot Punch/ Thunder Blue 2f1498e - 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 34 Blue de running NIKE Zapatillas Air Zoom Pegasus 34 para mujer Hot Punch/ Thunder Blue 2f1498e

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 34 Blue de running NIKE Zapatillas Air Zoom Pegasus 34 para mujer Hot Punch/ Thunder Blue 2f1498e

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

Zapato de running ASICS para mujer Torrance Mysterioso / Negro / Aluminio Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa , Zapatillas de running Reebok para mujer Denim Glow / Faux Indigo / Coral / Blanco , NIKE Women s Tanjun Blanco negro Supra Martine Run Skate Shoe para hombre Rosa / Rosa-blanco Zapatillas de running ISO para mujeres de la guía Saucony Niebla / Púrpura Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Pale Gray / Pale Gray-dark Gray , Under Armour Men s Speedform Intake Negro / Tropic Pink / Msv , Camiseta adidas Swift Run W para mujer Gray Two / Ice Pink / White , Zapatilla para mujer Saucony tipo A8 Blanco | , Zapatillas de running adidas Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco Skechers Sport Skech Air Run High Fashion Sneaker Black Knit / Hot Pink Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Blanco negro Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris oscuro Zapatillas de running adidas Women s Response Tr W para mujer Super Purple / White / Energy Aqua , Zapatillas para correr W630V5 para mujer New Balance Rosado , Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / Campanula / Carbono , Amortiguación 635v2 New Balance para mujer Baya Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Volt / Blue Lagoon / Blanco Zapatillas de running Reebok Women s Run Supreme 2.0 Negro / Rosa solar Zapatillas de montaña de montaña Akyra La Sportiva para mujer Slate / Jadegreen Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Berenjena / Sal marina , Zapatillas de running NIKE Lunartempo 2 para mujer Blanco / Negro Pure Platinum Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul Zapatillas de running NIKE Air Huarache para hombre Cool Grey Pure Platinum 012 , Brooks Women s Ravenna 8 Very Berry / Aqua Splash / Lime Punch Zapatillas de running Saucony para mujer Omni 16 Teal / Citron , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running NIKE Lunartempo 2 para mujer Hyper Violet / Black / Concord / Gamma Blue ,
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 Omnium de mujer Teva Pizarra , Nike Men s Air Presto Essential Gym Red / Team Red-team Red New Balance Ws574v1 para mujeres negro Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco / Semi Solar Pink NIKE WMNS Zoom Fly Women 897821-411 Hielo Azul / Zorro Azul , adidas Performance Ultraboost X para mujer Claro / Gris / Gris Medio / Gris Oscuro Heather Bomba natural Nora para mujeres champán , Brooks Womens Adrenaline GTS 18 Negro Negro , NIKE Men s Benassi Just Do It Athletic Sandal Negro / Rojo de desafío , Zapatillas de running Saucony Women s Breakthru 2 Rosa / azul marino , Sandalia Drew Cascade para mujer Nubuck negro , Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas negro Sandalia Wavey para mujer OTBT Cacao Sandalia Gora Yoga Sanuk para mujer Natural Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Gris oscuro , Zapatos ocasionales diabéticos Maggy X Black de Dr. Comfort para mujer Negro , Zapatillas de tenis Hard Court WC696V3 de New Balance para mujeres Plata blanca , Sandalia Buxom Slide de Charles David para mujer Plata , Sandalia para mujer Cait Donald J Pliner Negro , Jig Foo Latin Salsa Rumba Chacha Practice Ballroom Zapatos de baile para mujeres Negro ~ 50 tonos de bronce Zapatos de vestir de baile Colección, Comodidad Boda de la tarde Bombas: Salón de baile para mujeres Zapatos para América, Tango, salsa, swing, Theather Art por Fiesta Party (2.5 \ & 3 \ 2702 Copper Sparkle & Ultra Copper , Noble Outfitters Mens todo alrededor del dedo del pie cuadrado resistente Negro , Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Negro Zapatilla Glamour de mujer verde Daniel Black Kidskin Trotters para mujer Dea Slingback Pump Black Micro / Satin Franco Sarto, artista, colección, Freda, mujer, bota Whisky Cobb Hill Rockport - Aubrey Dress Pump para mujer Estaño Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 14 , MaxMuxun Mujer Zapatos Perforated Lace UP Brogues Wingtip Comfort Oxfords Negro U-lite mujer perforada con cordones Wingtip cuero plano Oxford zapatos brogues vintage 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 34 Blue de running NIKE Zapatillas Air Zoom Pegasus 34 para mujer Hot Punch/ Thunder Blue 2f1498e - 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 34 Blue de running NIKE Zapatillas Air Zoom Pegasus 34 para mujer Hot Punch/ Thunder Blue 2f1498e 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 647820
678321
1068277
1300529
5180296
9124782
9387212
15361756
17557812
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.