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.

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Cuero / Textil
Sintético único
Cemento la construcción proporciona estabilidad ligera
Nubuck cuero y nylon cincha superior
Transpirable revestimiento de malla tratado con Aegis
Microfibra plantilla envuelta eso es suave el toque y el trato Aegis antimicrobial solución
Moldeado mango de arco de nylon
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


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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

Sandalia Sandalia Jacardia Merrell Merrell para mujer mujer Negro 4fcadde

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.


Sandalia Sandalia Jacardia Merrell Merrell para mujer mujer Negro 4fcadde

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

Drew Womens Cascade Denim Blue Nubuck , Sandalias Newport H2 de KEEN para mujer Poseidón / Capri , Sandalia Terrell Post II Femenina de Merrell Tierra oscura , BZees Sandalia Deportiva Jive para Mujer Negro Chaco Women s ZX2 Classic Athletic Sandal Negro Chaco Women s Z2 Classic Athletic Sandal Paz azul , Sandalia de Sarasota KEEN para mujer Bisonte Sandalias Under Armour UA Ignite VII SL para mujer Blanco , Sandalia de Hollyleaf de las mujeres de Merrell Fucsia , Sandalia atlética al aire libre del cordón Vulc N Vent de Columbia para mujer Blanco, ostra Skechers Performance On The Go 400 Tropical Flip Flop Marina tropical , Sandalia de vestir Tillman de SoftWalk para mujer Lavado de oro Chaco Women s ZX2 Classic Athletic Sandal Mosaico del desierto Sandalia atlética del Caprizee de la mujer de Columbia Blanco negro Sandalia de espalda de mujer Merrell Women s Town Poseidón mobils by Mephisto - Sandalias de tacón Getha para mujer Black Queen / Magic Leather , Sandalia KEEN Clearwater CNX para mujer Anémona / acacia Chaco Mujer Zcloud X2 Sandalia Deportiva Lattice Cobalt Sandalia de mujer Drew Alana Naranja amarillo Sandalia Drew Angela para mujer Dusty Multi Metallic Leather , OluKai Ohana Paraíso / negro , Sandalias sintéticas Teva para mujer Terra Fi Lite Samba Coral Multi , OluKai Kulapa Kai Jalea de guayaba / negro , Zapatilla de deporte Sina 11 de Josef Seibel para mujer Nuss , Sandalias Swiftwater Mesh de Crocs para mujer Jean azul Sandalia Uneek O2-w de mujer KEEN Estate Blue / Harvest Gold Sandalias de deporte Nike Kawa Sport Shower Slide, negro / blanco Blanco negro Drew Womens Cascade Denim Blue Nubuck Sanda Verra Teva para mujer Atlántico Norte
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 Kids Air Max Zero Essential GS Negro / Negro / Negro Sandalia de mujer Drew Alana Verde / Fucsia , NIKE Mujeres Air Zoom Pegasus 33 Negro / gris frío / gris lobo / blanco NIKE Women s Tanjun Lava Glow / Blanco / Carmesí total , Sandalia para mujer Terrell Ari Wrap de Merrell Helecho , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Zapatillas adidas Outdoor Terrex Agravic Speed ​​Shoe de mujer Gris crudo, Gris crudo, Gris ceniza , New Balance Vazee Summit V2 Running Shoe Trail Runner para mujeres Deep Porcelain Blue / Reflection Zapatillas para caminar Skechers Performance Womens Go Walk 3 Crazed Azul marino / blanco Sandalias casual Debra 19 2 Strap de Josef Seibel para mujer Castagne , Earth Origins Tova Hibiscus Red , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro Sandalia de cuña FLY London Yand709fly para mujer Red / Off White Burrito Sandalia de tobillo con correa Gomera Finn Comfort para mujer Cognac Plisseelight , Zapato para caminar Sky Bolt para mujer Ryka Azul amarillo , Zapatillas Vibram VI-s Cross-Trainer para mujer Nightshade / Violet , Skechers Women s Go Walk Revolution Ultra Sneaker Negro / Multi ASICS para mujer Gel-Fit Yui 2 SE Brote Verde / Flash Coral / Blanco Ahnu Women s Yoga Flex Orquídea oscura Zapatos de entrenamiento Nike Free TR 6 para mujer Purple Shade blanqueado Lila 502 Vans Old Skool Unisex Adults Low-Top Trainers Negro Negro , Under Armour Ua ClutchFit Drive Ii para hombre Negro / oro blanco Sandalia Charlotte Slide de Naturalizer para mujer Cuero negro , OluKai Ohana Burbujeante / negro , Giro GF23110 Mujer Sica Vr70 Dirt Bike Shoes Negro mate Joules Women s Rockingham Botines Goma botánica negra Zapatillas de deporte lavables Walker Propet para mujer Sr Bone / White , Nine West Leather Lilianne Leather para mujer Vino , Trotters Liz Loafer para mujer Rosado TOMS Clásicos de mujer (Negro)

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.

Sandalia Sandalia Jacardia Merrell Merrell para mujer mujer Negro 4fcadde - 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 Sandalia Sandalia Jacardia Merrell Merrell para mujer mujer Negro 4fcadde 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 3164463
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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.


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