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 mujer de/ senderismo B01B72H4VM Ultra Cardiac II para mujer The North Face Shady Blue/ Nasturtium Orange 147892f




Sintético y malla
Caucho suela
Transpirable parte superior de malla jacquard
Cuarto construcción de cable geo para un mejor ajuste
FlashDry tecnología superior absorbe la humedad lejos de la piel para mantener la sensación de los pies más fresco y menos propenso a formar ampollas
Protección puntera de caucho
CRADLE estabilidad del talón la tecnología promueve colocación adecuada del pie y devuelve energía
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 mujer de/ senderismo B01B72H4VM Ultra Cardiac II para mujer The North Face Shady Blue/ Nasturtium Orange 147892f - 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 mujer de/ senderismo B01B72H4VM Ultra Cardiac II para mujer The North Face Shady Blue/ Nasturtium Orange 147892f

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 mujer de/ senderismo B01B72H4VM Ultra Cardiac II para mujer The North Face Shady Blue/ Nasturtium Orange 147892f

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 trail running Xa Pro 3D GTX W para mujer Salomon Negro , Zapatillas de running adidas Galaxy Trail W para mujer Gris oscuro / gris Tech / naranja brillante , Zapatilla de running Vasco para mujer Pendulum II Neutral Gray / Horizon Blue Vionic Womens Cypress Trail Walker Gris / verde azulado Zapatillas de senderismo The North Face Ultra Fastpack II GTX para mujer Tnf Black / Bright Navy , Reebok Women s Ridgerider 2.0 Trail Runner Negro / Rosa Solar / Plata / Estaño , Zapato de entrenamiento Under Armour Charged Core para mujer Stealth Gray / Pink Chroma / Graphite Botas de mujer New Balance 510 Boots Grey Suede / Textile Zapatillas de running New Balance para mujer 590v3 Negro / Castlerock , Zapatillas de running adidas Women s Rockadia W Trail Carbono / Acero sin procesar / Trace Scarlet Zapatillas de trail running New Balance WT410V4 para mujer Azul marino / azul claro Salomon Women s Speedcross 4 W Trail Runner Azul pizarra / Azul balneario / Verde fresco , Zapatillas para correr Under Armour UA Fat Tire 2 para mujer Premier Purple / Rhino Gray / Black , Salomon Women s Speedcross 4 CS W Trail Runner Mallard azul Zapatillas de trail running New Foam Gobi Neutral New Balance para mujer Gris / rosa , Zapatillas de running Trail Trail Norvan VT de Arc teryx - Mujer Caraibes / Blue Nights , Zapatillas de trail running WT10v1 Minimus para mujer New Balance Azul negro Salewa Literas Literas para mujeres Negro / Pinky , Zapatillas de running Salomon Women s Speedcross Pro 2 W Trail Negro Altra Women s Lone Peak 3 Trail Runner Azul Calzado de trail running Vasco transistor FS para mujer Glaciar gris / azul , El corredor de Badwater Trail para mujeres Hi-Tec Azulejo azul / azul Curacao / Blarney Zapatillas de running MT-onemix Unisex Modern City Lightweight Running Jogging Rosado azul , Zapatillas de running Skechers GOtrail para mujer Taupe / rosa Columbia Montrail Women s Variant X.S.R. Gray Ash, Phantom Purple Zapatillas de running New Balance para mujer 590v3 Reflection / Vintage Indio Zapatillas de running Trail para acolchado 510v4 de New Balance para mujer Trueno , MT-onemix Women Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Abeja amarilla I Zapatillas New Balance para mujer Fresh Foam Gobi Trail Running Blanco / oro rosa ,
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 deportiva ECCO Terra 2S para mujer Camello Chaco Mujer Diana Sport Sandalia Pulse Eclipse , Sandalia Drew Angela para mujer Bone Smooth Leather , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Turq / Black-volt-rio Teal , Zapatillas Reebok para mujer Trilux Run PNT Sneaker Prnt - Negro / Blanco , Zapatillas de running ASICS Gel-Kahana 8 para mujer Negro / Island Blue / Pink Glow Birkenstock Kid s Arizona Sandal Negro , Vestido de dama de honor único de Abby L019 para mujer vestido de fiesta de la boda vestido de cono Sandalias de microfibra Plata , Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Azul / Denim Sandalias Joanie II de mujer SOREL CAMELLO , Sandalia Savannah de Ryka para mujer Coral / gris , Inov-8 Mujer F-Lite 275 (W) Cross Trainer Coral rojo Entrenador cruzado Minimus 20v7 de New Balance para mujeres Negro Zapatillas de deporte Unisex Skate Trainers de Vans Authentic True White Vans Unisex Old Skool Zapatillas clásicas de skate (Gamuza / Lona) Frost Grey / True White Adidas W Adipower Boost Boa Golf Shoe para mujer Gris / Misterio Ruby , Sandalia Cato Dolce Vita para mujer Cuero negro , Sam Edelman Women s Bay Slide Sandal Cuero de silla Sandalia de tacón Alto Posey de Tahari para mujer Multi Sparkle , Sandalias planas Nori de Easy Street para mujer Negro , Sandalia Dolce Vita Halle Slide para mujer Denim de Ceniza , FRYE Punta Redonda de Victoria Femenina Gamuza Victoriana, Punta Redonda - 79117-Ept Elephant Soft Oiled Suede , Bomba Stillson para mujer de Sam Edelman Dusty Rose Suede Bomba Rieker Women s Sina 78 Blanco gris / polvo blanco , Bota de caza de 8 pulgadas 800G Silencer para mujeres Lacrosse Mossy Oak Break Up Infinity Zapatillas de tela Raissa de Nine West para mujer Luz natural , Zapatos de mujer de Babe Oxford de zapatos deseados Tan / Natural Corral Boots Womens A3228 marrón , Hush Puppies Dasher Mardie Oxford para mujeres Marfil Arcopedico Womens Lolita Slip-On Aceituna ,

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 mujer de/ senderismo B01B72H4VM Ultra Cardiac II para mujer The North Face Shady Blue/ Nasturtium Orange 147892f - 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 mujer de/ senderismo B01B72H4VM Ultra Cardiac II para mujer The North Face Shady Blue/ Nasturtium Orange 147892f 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.

Trail Running
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1820720
4770477
8116553
12075734
13762763
14455986
15069194
15099251
17908097
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.