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.

Skechers B0719KM3RN Women s Go MEB s Go Razor 2 Carbón 99d6373




NA
Importado
Prácticamente FITKNIT inconsútil la parte superior es de apoyo, cómodo y transpirable
Ligero, durable y flexible web paramétrica suela proporciona tracción multi-superficie - Golpe de pie medio zona promueve la eficiencia en cada zancada
4mm compensación de gota de talón - Tire de la pestaña del talón función para accesible fácil de encendido y apagado
Ultra ligero, VUELO receptivo Amortiguación GEN entresuela - moldeada contador de talón para un estable y seguro ajuste
Hot melt construcción de collar para mayor comodidad - Plantilla extraíble para un ajuste personalizado
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


Skechers B0719KM3RN Women s Go MEB s Go Razor 2 Carbón 99d6373 - 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

Skechers B0719KM3RN Women s Go MEB s Go Razor 2 Carbón 99d6373

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.


Skechers B0719KM3RN Women s Go MEB s Go Razor 2 Carbón 99d6373

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

Brooks Women s Ravenna 8 Negro / Diva Pink / Plum Caspia Zapatillas para correr ASICS Gel-Venture 6 para mujer Indigo Azul / Negro / Coral Zapatillas para correr Mizuno Wave Inspire 12 para mujer Plata blanca Zapato de running ASICS GEL-Cumulus 20 para mujer Negro / Flash Coral , New Balance Women s w990v4 Running Shoe Komen Pink , Calzado de running casual para mujer New Balance WL555, para mujeres Negro , Zapatillas de running NIKE Mens Air Zoom Structure 19 Hypr Jd / Cdr-spc Blue-hypr Crimson , Zapatillas de running New Balance Boracay V3 para mujer Supercell / Ozone Blue / Bleached Sunrise , Under Armour Women s Thrill 3 Rhino gris / blanco / azul infinito Zapatillas de running NIKE Lunarglide 9 para mujer Negro / Blanco / Antracita Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Pure Platinum Black White Volt 002 Zapatillas de running ASICS Gel-Kayano 23 para mujer Poseidón / Plata / Cacatúa Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Azul marino / rosa / verde azulado Zapatillas para correr Saucony Women s Cohesion 10 Gris / menta , Calzado de running New Balance para mujer Vazee 2090 Azul marino / amarillo , Zapatillas de running Mizuno Wave para mujer Inspire 14 Folkstone Gray / Pink Glo , Zapatillas de running adidas Pureboost Xpose Clima para mujer Easy Coral / White / Light Maroon Under Armour Women s HOVR Sonic NC Grafito (105) / Plata Metálica Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Clear Aqua / Glow Orange / White , Zapatillas para correr Nike Downshifter 6 para mujer Blanco negro Zapatillas de running Gel-Moya para mujer de ASICS Gris / plata Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Negro-corredor Rosa-blanco Zapatillas para correr Mizuno Women s Wave Enigma 6 Darkshadow-capri , Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Blanco negro , Zapatillas de running ASICS GT-2000 6 para mujer Brillante rosa / negro / blanco , Brooks Women s Adrenaline GTS 17 Plata / púrpura Cactus flor / Bluebird Zapatillas de running Classic-M Vibram para hombre Negro Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell Zapatillas ONEMIX Air Running para mujer Zapatillas deportivas atléticas ocasionales Azul claro ,
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 adidas Adizero Lj Núcleo Negro, Ftwr Blanco, Naranja Zapatillas de senderismo acuáticas para mujer Astral TR1 Trek Turquesa turquesa Skechers Women s Go Flex Max Loafer Gris pardo , Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React Azul , Zapatillas Erika Propet para mujer Hueso , Skechers Sport Skech Air Run High Fashion Sneaker Negro / rosa caliente , Zapatillas de running ASICS GT-2000 6 para mujer Brillante rosa / negro / blanco , Zapatillas para caminar al aire libre Go Skechers Performance para mujer Chocolate , Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Blanqueado Lime Glo / Vintage Indigo , Zapatillas para caminar para caminar al aire libre Go Skechers Performance para mujer Negro Drew Shoe Women s Tulip Comfort Croc negro , Skechers Performance Women s On The Go City 3 Optimize Negro , YDN Mujeres Open Toe High Heels Mules Slip on Bombas Diapositiva Transparente Vestido Sandalias Zapatos Azul Tecnología Vionic Men s Orthaheel White / Navy Walker Blanco negro Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Moca , Zapatillas Skechers Go Walk Lite-15360 para mujer Blanco Lakai Ellis Skate Shoe Azul / naranja , Vans Auténtico Vestido azul / azul / rojo Birkenstock Mujer, Madrid Sandalia deslizante Magic Galaxy Silver Birko-flor La Plume Jen de las mujeres Beige Metálico , Sandalia del vestido de la inversión de LifeStride de las mujeres Plata , Sandalia para mujer FRYE Ally Criss Cross Slide Secoya OluKai Ohana Blanco negro , Zapatillas de ciclismo SPI X-Alp Launch SPD Pearl iZUMi para mujer Perla ahumada / Highrise Sandalia Birkenstock de cuero Florida para mujer Blanco Pantalón largo clásico NIKE Golf para mujer Tech Essentials Cool Gray / White-Max Orange , Blundstone 510 Unisex Slip-On Boot Negro , Easy Devitt Oxford Flat para mujer Negro , Blundstone Super 550 Series Boot Negro , Propet Walkers Evolution Oxford lavable de las mujeres Mora azul ,

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.

Skechers B0719KM3RN Women s Go MEB s Go Razor 2 Carbón 99d6373 - 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 Skechers B0719KM3RN Women s Go MEB s Go Razor 2 Carbón 99d6373 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 702252
3014467
4112640
4842949
9296374
10974501
11470792
14723278
19043995
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.