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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Parte superior sintética
Importada
Phylon suela de espuma
de una sola pieza cuero sintético correa tiene jersey forro para un suave sentir
inyectado Entresuela de Phylon se dobla como una suela para peso ligero amortiguación
Flex surcos ayudan al deslizar el movimiento con tu pie
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


NIKE Men blanco s Benassi Just Do de It Athletic Sandal s Niebla de medianoche/ blanco 87df8e9 - 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

NIKE Men blanco s Benassi Just Do de It Athletic Sandal s Niebla de medianoche/ blanco 87df8e9

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.


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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 adidas Alphabounce Rc W para mujer Gris oscuro Heather / Dark Solid Gray / Dark Gray , Zapatillas de running Nike Air Zoom Elite 8 para mujer Negro / Gris lobo / Gris oscuro / Blanco Zapatillas de running NIKE Lunarsolo para mujer Pulso de Carbono / Púrpura claro , Zapato de running GEL-Excite 3 para mujer ASICS Rosa fuerte / Blanco / Amarillo intermitente , Zapatillas de running Saucony Women s Triumph ISO 2 Verde azulado / negro / blanco Inov-8 Mujer F-Lite 260 (W) Cross Trainer Gris negro Zapatillas de running ASICS Gel-Contend 4 para mujer Negro / Negro / Carbono Zapatillas de running ASICS Gel-Kayano 23 para mujer Negro / Plateado / Flash Coral , Zapatillas de running NIKE para mujer Lunarstelos Bright Crimson Metallic Silver 602 , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris , Zapatillas de calle YKTM Performance Women Boost 4.0 Street Running Shoe Road Negro , Under Armour Women s Thrill 3 Rhino gris / blanco / azul infinito , Zapatillas de running ASICS Gel-Kayano 23 para mujer Tiburón / Aruba Azul / Acuario Zapatillas de running W1540V2 New Balance para mujer Blanco azul , NIKE Men s Flex 2017 RN Dark Sky Blue / Gym Blue-glacier Grey , NIKE Mujeres gratis TR 7 blanco , Mizuno Women s Wave Prophecy 5-w Gunmetal / Fuchsia Purple Under Armour Women s Drift 2 True Ink (918) / Blanco Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Blanco Zapatillas de running ASICS GT-2000 6 para mujer Brillante rosa / negro / blanco Zapato de correr ideal para rejilla de mujer Saucony Negro / Gris / Pr Calzado de running Nike Flyknit Lunar2 para mujer Pure Platinium / Atomic Purple / Corte Púrpura / Blanco Zapatillas de calle YKTM Performance Women Boost 4.0 Street Running Shoe Road Blanco Zapato de running ASICS Gel-Quantum 360 cm para mujer Gecko negro / blanco / verde Zapato de running Saucony Zealot ISO 2 para mujer Té / Algodón , Zapatillas de running Zoot para mujer Del Mar Deep Purple / Purple Haze / Honey Dew Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Plata / rosa Camiseta adidas Swift Run W para mujer Claro Marrón / Blanco / Núcleo Negro Zapatillas de running Saucony PowerGrid para mujeres Gris / Té / Rosa
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 para mujer Air Max Motion LW Green Abyss / Neo Turq Calzado de running Fila Memory Faction 2 de Fila Monumento / Sugarplum / Blanco KEEN Mujer, Sandalias de gladiador Maya gris , Earth Origins Sophie Women s Sandal Azul marino , Sandalias KEEN Mujer Ana Cortez Flip-W Negro , Brooks Womens Ariel 18 Gris / gris / rosa Amortiguación 635v2 New Balance para mujer Negro , Sandalia NIKE Kawa Slide para mujer Blanco negro , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue , Zapatillas de trail running Inov-8 Women s Race Ultra 290 Gris / Berry / Azul , Sandalias de diapositivas casuales Easy Street Womens Conca Fabric Open Toe Negro Zapatillas de running NIKE Air Max Torch 4 para mujer Blanco / menta Zapatillas para caminar adidas Outdoor Terrex Voyager DLX W para mujer Negro / Negro / Gris Cinco Zapatos de tierra Masio Cuero azul de primera calidad , Bomba de cuña Dreamstride Women s Dreams Negro , Zapatos de mujer Spira Classic Walker 2 con resortes Negro , Katy Perry Women s The Alanna Sandal Arena rosa David Tate para mujer Nelly Nubuck negro , Zapatillas para mujer Babolat Propulse BPM rosa , nativo AP Mercury Liteknit Shell White / Shell White / Natural Rubber Reebok Zapatillas de deporte Royal Aspire 2 Fashion para mujer Negro / Plata Metálico Lakai Riley Hawk Skate Shoe Textil de medianoche , Sandalias Poppy de mujer de Eastland Broncearse , Capezio DS24 Rockit Dance Sneaker para mujer Negro , Zapatilla Klogger Klogger de Mujeres Tempur-Pedic Hashbrown , Glerups Unisex G-01 - Botas de fieltro Gris , Bota de mujer Serra de TOMS Oxford Tan Suede , Encaje Wander de mujer Palladium Sue Tuxedo Oxford Negro , Zapatillas de deporte Aika de Cloud Footwear para mujer Beige Fantasy , CLARKS Mujer Ashland Bubble Slip-On Cuero marrón

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.

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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 NIKE Men blanco s Benassi Just Do de It Athletic Sandal s Niebla de medianoche/ blanco 87df8e9 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

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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.