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 Low NIKE para para mujer Lunarepic 18511 Low Flyknit 2 Azul e6dd49f




Sintético
Goma único
Modelo Número: 863780405
Género: para mujer
Color: GLACIAR AZUL / METÁLICO PLATA
Hecho En: Vietnam
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 de running Low NIKE para para mujer Lunarepic 18511 Low Flyknit 2 Azul e6dd49f - 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 Low NIKE para para mujer Lunarepic 18511 Low Flyknit 2 Azul e6dd49f

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 Low NIKE para para mujer Lunarepic 18511 Low Flyknit 2 Azul e6dd49f

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 running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Negro-Blanco-Racer Azul , Zapatillas de running NIKE Air Max Thea para mujer Laguna Blanca / Azul / Verde Fantasma , Calzado de running Pearl iZUMi para mujer Em Road H3 v2 Shadow Gray / Meadow Mauve , Calzado de running Fila Memory Faction 2 de Fila Monumento / Sugarplum / Blanco Zapatillas de running adidas Duramo 8 para mujer Turquesa , Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink Skechers Performance Women s Go Run 400 Obstruct Frambuesa Zapatillas de running ASICS Gel-Cumulus 19 para mujer Begonia Pink / Begonia Pink / Baton Rouge NIKE Women s WMNS Praktisk, Negro / Negro-Antracita Negro / Negro-antracita , PUMA Men s Roma Basic Sneaker Cuero blanco / negro , NIKE Women s Revolution 4 Running Shoe Ocean Bliss / Blue Force / Glacier Blue PUMA Men s Roma Basic Sneaker Blanco / Alto Riesgo Rojo / Blanco , Zapatillas de running a campo traviesa Fastwitch 8 para mujer Saucony Verde azulado / blanco NIKE Men s Roshe Two Running Shoe Paramount Blue Wolf Gray R Zapatillas de running NIKE Air Huarache para hombre Industrailblue / Midnightnavy , ASICS Kanmei Shoe Casual para mujer Gris medio / gris medio / carbono , Zapato de running retro con diseño clásico de ASICS para mujer Coral / Coral , Zapato de running ASICS GT-1000 6 para mujer Concha Rosa / Begonia Rosa / Blanco , Supra Martine Run Skate Shoe para hombre Andora-blanco , Zapatillas de running Reebok Women s Run Supreme 2.0 Negro / Rosa solar NIKE Men s Flex 2017 RN Antracita / Volt / Cool Gray / Black Zapatillas para correr ASICS Gel-Venture 6 para mujer Castlerock / Silver / Honeydew Zapatillas running para mujer Saucony Liteform para mujer borgoña Zapatillas de running adidas Women s Edge Lux W Negro / Energía / Blanco , Zapatillas de running NIKE Air Max Flair para hombre Blanco negro , NIKE WMNS Zoom Fly Women 897821-411 Apenas gris / Sequoia Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco Zapatillas de running NIKE Air Max Flair para hombre Wolf Gray Cool Gray Black 003 New Balance W880gb6 para mujeres Gris azul
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 Terrell Post II Femenina de Merrell Pizarra , Calzado de running rápido para mujer Reebok, tejido flexible Colegial azul marino / Laguna azul / Raincloud / Blanco Zapatillas de running Mizuno Wave para mujer Inspire 13 Dark Slate / Raspberry Earth Origins Sophie Sedona Brown Zapatos OLUKAI Pehuea - Mujer Dark Plum / Dark Plum , Zapatillas para caminar Skechers Women s Go 4-Brisk Blanco negro MBT Zapatillas para caminar Sirima 6s MJ para mujer Negro , Pikolinos Womens Java W0K-0964 brandy Vionic Rest Farra - Sandalias de apoyo para mujeres Patente roja Botín \ Noir / Antico Sandalia Colette Alegria para mujer Noche del salvaje oeste , Easy Spirit Women s Traveltime Mule rojo PUMA Women s Ignite Dual Netfit WN Cross Trainer Paradise Pink-puma White , Zapatillas de entrenamiento cruzado Vibram para mujer V-Train Gold s Gym Negro / Gris / Azul Reflejo , Prop & eacute; t Olivia Oxford para mujeres Gris pardo Sandalia de la diapositiva de Florene de las mujeres de Lucky Brand Anacardo asado Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1682- Tan satinado , Bota Ecuestre Madeln-WC Tommy Hilfiger para Mujer Marrón / Carbón de leña , Corral Women s Patch Western Snip Toe Boots Marrón, coñac Bomba de vestido de pasión Easy Street para mujer Patente de arruga de color topo Bastien Industries Mocasines de puesta a tierra de ballet de piel Buffalo Hide de mujer Saddle Tan Blondo Maddox Waterproof Boot - Mujeres Nubuck negro , UGG Aira Knit Slip on Slipper Crema , TOMS Kids 10009918 Alpargata-K Plumas de hoja de taupe desierto ED Ellen DeGeneres Mujeres Lilliane2 Bisque Lather / Patent Pu Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Rojo clásico Sanuk Women s Vee K Shawn Chukka Boot Lavado negro , CLARKS Womens Un Haven Lace Cuero negro TOMS - Slip-On de lienzo clásico para hombre Caoba roja , CLARKS Mujeres Sillian Bella Mary Jane Flat Nubuck sintético marrón oscuro

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 Low NIKE para para mujer Lunarepic 18511 Low Flyknit 2 Azul e6dd49f - 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 Low NIKE para para mujer Lunarepic 18511 Low Flyknit 2 Azul e6dd49f 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 2155262
2801114
3127230
6265919
7417400
10249455
14719807
17700976
18373732
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.