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 de running Altra running Women Zapatillas s Intuition Intuition 4 Negro e5b92b8




Sintético
Importado
Caucho único
Todo Nuevo Diseño superior
completo Suela de goma
EVA con A-Bound Top Layer e InnerFlex Entresuela
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 de running Altra running Women Zapatillas s Intuition Intuition 4 Negro e5b92b8 - 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 de running Altra running Women Zapatillas s Intuition Intuition 4 Negro e5b92b8

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 de running Altra running Women Zapatillas s Intuition Intuition 4 Negro e5b92b8

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

Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Blanco / Gris Oscuro Zapatillas de running NIKE Zooms Winflo 4 para mujer Antracita / Gris Oscuro-negro , Zapatillas de running Nike Zoom para mujer Hasta / Mtlc Red Bronze / Green Glow Zapatillas de montaña de montaña Akyra La Sportiva para mujer Esmeralda / menta , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / oscuro, pasas de rosa, mortal Zapatillas de deporte Nike Roshe Run para mujer Deep Granet / Black-gym Red-vry Berry NIKE Women s Tanjun Negro / Negro / Blanco , New Balance Women s w990v4 Running Shoe Komen Pink , NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / White , Zapatillas de running Mizuno Wave para mujer Wave Horizon Dark Shadow / Clownfish , Zapatillas para correr Saucony Freedom Iso Unisex borgoña Zapatillas de running New Balance Urgev2 para mujer Silver Mink / Alpha Pink , Zapatillas de running NIKE Air Max Torch 4 para mujer Negro / Plata Metálico / Rosa Flash , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Menta / Plata / Azul Zapatillas para correr Under Armour UA Speedform Fortis 2.1 para mujer Negro / Gris Glaciar / Gris Glaciar , Zapatillas de deporte con cordones Sneakers de hombre Carson Runner de PUMA Blue Atoll-white , NIKE Mujeres Downshifter 7 Negro / Mtlc Hematita Antracita Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Nube de lluvia / Artic Fox , Zapato de running ASICS GT-1000 6 para mujer Porcelana Azul / Humo Azul / Blanco , Zapatillas de running adidas Springblade Razor para mujer Power Teal / Frost Mint / Sol Gold Zapatillas de running NIKE Lunartempo 2 para mujer Rosado Zapatillas de running NIKE Airstream Sequent 3 para mujer Sequoia / Light Bone / Metallic Silver , Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Negro / Fantasma , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Deep Porcelain Blue / Overcast , Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / Campanula / Carbono , Zapatillas de running Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla del océano / Cirrus Blue / College Navy Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / Hola Lite
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 Nike Women s Free Rn Flyknit 2017 Multi-Color / Multi-Color Sandalia Merrell para mujer All Out Blaze Sieve Sport Estuco NIKE Women s Revolution 4 Running Shoe Platino Blanco / Blanco-puro Earth Origins Sophie Frost Grey Zapatillas de running Reebok para mujer Negro / Rosa ardiente / Blanco , OluKai Ohana Hibisco / Negro Zapato de correr con indicación de gel para mujer ASICS Relámpago / rosa fuerte / nectarina Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell PUMA Men s Roma Basic Sneaker Blanco / Rio Red / Steel Grey , Zapatillas SCARPA Neutron 2 Trail Running para mujer Resplandor gris / azul Mocasines y zapatillas Slip-ONS Monica de TOMS para mujer Cáñamo / cuero natural Chancleta Yoga Sling 2 Sanuk para mujer Borgoña Multi Radio Love , Supra Martine Run Skate Shoe para hombre Gris blanco , Enfermera Mates Womens Kris Blanco , Zapatilla de deporte Dansko para mujeres Gamuza de carbón , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Negro Bella Vita Mae-Italia Mujeres WW Sandalia de tobillo marrón claro con tacón abierto Cuero negro , CLARKS Womens Wynnmere Avah Peach Nubuck Zapatilla Reebok para mujer ROS Workout TR 2-0 Cross-Trainer Teal Dust / Forest Grey / Lemon Zest / Black , MOZO Zapatillas para mujer The Maven Sneaker Verde , Zapatillas de deporte PUMA para mujer con cremallera x Fenty Puma Blanco / Puma Negro Vans Old Skool Unisex Adults Low-Top Trainers (Adecuado) Frost Grey / True White , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Blanco negro , Zapato de bailarina Swing Tic-Tac-Toes para mujer rojo CLARKS Mujer Poppy Pump Flores Cuero crema Tommy Hilfiger Frank Sneaker para mujer Floral Zapatillas de moda Native Unisex Apollo Moc. Gris Bota del Oeste de Stockman Heritage de Ariat, Native Nutmeg, 7 D US Caído marrón / beige Bomba Dylan para mujeres Bella Vita Berenjena Berenjena La emperatriz de las mujeres de Skechers se ve bien Rosa topo

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 de running Altra running Women Zapatillas s Intuition Intuition 4 Negro e5b92b8 - 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 de running Altra running Women Zapatillas s Intuition Intuition 4 Negro e5b92b8 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 350325
2727897
3306406
7510354
8617295
13051857
14038485
17515799
18509371
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.