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.

Zapato de running ASICS running Gel-Quantum 360 cm para para mujer/ Blanco/ Nieve/ Plata 509d4e7




Sintético
Importado
Talón Altura: 0
Ajuste: Fiel al tamaño
Superior: Características sintéticas
de este artículo incluyen: APMA, Atlético, Amortiguado, Ejercicio, Tacón plano, encaje Up, rendimiento Para ejecutar, correr, Inconsútil, sintético
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


Zapato de running ASICS running Gel-Quantum 360 cm para para mujer/ Blanco/ Nieve/ Plata 509d4e7 - 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

Zapato de running ASICS running Gel-Quantum 360 cm para para mujer/ Blanco/ Nieve/ Plata 509d4e7

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.


Zapato de running ASICS running Gel-Quantum 360 cm para para mujer/ Blanco/ Nieve/ Plata 509d4e7

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 Pure Cloudfoam de adidas para mujer Blanco / Blanco / Negro , Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Sigilo gris / blanco / agua , Zapatillas de running NIKE Air Max Thea Ultra FK para mujer Pure Platinum / Pure Platinum , Zapatillas de running New Balance 520v3 para mujer Artic Fox / Black Zapatillas de running NIKE Air Max Flair para hombre Negro / Neo Turq NIKE Women s Revolution 4 Running Shoe Ocean Bliss / Blue Force / Glacier Blue , Adicción a Brooks 12 Antracita / Alto riesgo Rojo / Plata Zapatillas de running Gel-Moya para mujer de ASICS Azul negro , Zapatillas de running New Balance 990V3 para mujer Gris NIKE Men s Roshe Two Running Shoe (Se) Blue Jay / Blue Jay , NIKE Mujeres Downshifter 7 Cool Grey / Lava Glow / Dark Grey / White , Zapatillas de running Gobiv2 para mujer New Balance Poisonberry / Black , Zapatillas para correr Topo Athletic Runventure 2 - Mujer Gris / amarillo Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer 007-black Wolf Gray Anthracite White Zapato de running Indigo Ryka para mujer Gris / menta Zapatillas de correr Nike Air Max 1 Essential para mujer Negro / Gimnasio Rojo / Vela / Gm Md Marrón , Saucony Women s Ride 10 Running Shoe Azul marino / mezclilla Zapatillas de running Pure Cloudfoam de adidas para mujer Cristal blanco / Crystal White / Talc NIKE Mens N98 Tribute Track Jacket Blanco negro Zapatillas de running adidas Women s Cf Racer Tr W, mujer Blanco / blanco / plata mate , Skechers Performance Women s Go Run 400 Obstruct Negro Saucony Women s Ride 10 Running Shoe Niebla / Rojo , Zapatillas de correr AS-Lite Nimbus 18 Lite-Show para mujer ASICS Negro / Plateado / Tiburón , Zapatillas de running acolchadas 420v4 para mujer New Balance Gris azul Zapatillas para correr Nike Flex Experience RN 5 Racer Blue / Metallic Silver / Black / White Zapatillas de running Kinvara 8 Saucony para mujer Berry White Under Armour Women s HOVR Sonic NC Blanco Blanco , Zapato de tenis ASICS Gel-Dedicate 5 para mujer Blanco / Porcelana azul / azul índigo Zapatillas de running NIKE Flex Run 2015 para mujer Blanco negro ,
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 de running NIKE Unisex Flyknit Racer Pure Pltainum Zapatillas deportivas 574S Sport New Balance para hombre Varsity naranja / blanco Sandalia Belmar Athletic de Ryka para mujer Gris oscuro Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Bosque ártico Converse Chuck Taylor All Star Seasonal Color Hola Negro / rojo / blanco NIKE Men s Roshe Two Running Shoe Gris oscuro / puro platino KEEN sandalia de Venecia para mujer Mineral azul / ámbar Zapatillas de trail running Inov-8 Trailtalon 235 (W) para mujer Teal , Zapatos OLUKAI Pehuea - Mujer Honu negro / negro , Atletismo femenino de Apex Blanco / Vinca , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Pacífico / Plata Splendid Women s Brooklyn Sandal Plata , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Baya Carlos by Carlos Santana Sandalia Adalyn para mujer Negro NIKE Women s Air Max 95 Zapatillas de running Mizuno Wave para mujer Inspire 13 Blue Radiance / Blueprint / Fiery Coral Zapatillas de running Mizuno Wave para mujer Inspire 13 Blue Radiance / Blueprint / Fiery Coral , Sandalia Dansko para mujer Minka Negro Full Grain , JBU by Jambu Women s Wildflower Sandal Gris claro , KEEN Women s Uneek 8MM Sandal Zinfandel / Vino morado , Chaco Mujer Fallon Sandal Java Vans Old Skool Unisex Adults Low-Top Trainers (Cuadros primario) Negro / blanco , Sandalia Juno Slide de Vionic Women s Rest Cereza , Sunny Equestrian Boot de Tommy Hilfiger para mujer Gris negro , PUMA zapatillas de deporte Fenty x Bow Trinomic para mujer Sweet Lavender / Sweet Lavender CLARKS Mujer McKella Morris Sneaker Nubuck gris sintético , Zapatilla de fumadores Classic Stripe Smoking de Soludos Natural / Negro , Bota de trabajo de cuero impermeable con aislamiento de viuda negra RefrigiWear para mujer, negra Zapatos OLUKAI Pehuea - Mujer Tapa / Tapa , Correa de crepetray para mujer ECCO de mujer Oxford Arcilla / Polvo mutados ,

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.

Zapato de running ASICS running Gel-Quantum 360 cm para para mujer/ Blanco/ Nieve/ Plata 509d4e7 - 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 Zapato de running ASICS running Gel-Quantum 360 cm para para mujer/ Blanco/ Nieve/ Plata 509d4e7 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 2052156
7570237
8656612
12594036
13523240
13586023
15246054
17902466
18619860
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.