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.

Drew Drew Womens Cascade Denim Blue Blue Nubuck Nubuck c88bd64




Cuero
Importado
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El Cascada deportiva proporciona la comodidad y se necesita estabilidad para alcanzar nuevas alturas!
Premium parte superior de cuero.
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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


Drew Drew Womens Cascade Denim Blue Blue Nubuck Nubuck c88bd64 - 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

Drew Drew Womens Cascade Denim Blue Blue Nubuck Nubuck c88bd64

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.


Drew Drew Womens Cascade Denim Blue Blue Nubuck Nubuck c88bd64

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

Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente Sandalia deportiva Tirra para mujer Teva Simplemente Taupe , Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Lago Azul / Gris Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Negro , Zapato de vestir uniforme al aire libre Vulc N Vent Slip para mujer de Columbia Zing / Iceberg , Propet Women s TravelActiv Ss Sandal Negro plata OluKai Ohana Punch / Ice Sandalias KEEN Mujer Whisper-w Rododendro / Marsala Skechers sandalia ancha de deporte 600-brilliancy para mujer Natural Sandalia Terra-Float Livia de Teva para mujer Gris azul Wolky Rio Sandal Red Patent Metal Chaco Mujer ZX1 Classic Sport Sandalia Trenza azul Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Negro Chaco Women s Z2 Classic Athletic Sandal Brocade Pine , Corkys Rock Mujer Slip On Amber Multi Sandalias Newport H2 de KEEN para mujer Rama de Malaquita / Plata , Sandalia KEEN Women s Venice H2 Midnight Navy / Coral caliente Sandalias KEEN Venice II H2-W para mujer Negro / gris acero Sandalia Drew Angela para mujer Dusty Multi Metallic Leather Skechers Performance On The Go 400 Tropical Flip Flop Negro / Blanco Tropical Chaco Mujer ZX1 Classic Sport Sandalia Paloma mandarina CLARKS Womens Wave Grip Negro sintético Propet Women s TravelActiv Ss Sandal Negro OluKai Kulapa Kai Fucsia / Negro , OluKai Kulapa Kai Dark Java / Dark Wood Sandalia para mujer NIKS Benassi Solarsoft Slide Blanco negro Sandalia Uneek O2-w de mujer KEEN Blanco / oro de la cosecha , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave , Sandalia NIKE Kawa Slide para mujer Iris oscuro / blanco / aluminio ,
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 Wolky Comfort Jewel Dos tonos blancos , Zapatillas de running New Balance Coast V3 para mujer Phantom / Overcast Sandalia deportiva Tirra para mujer Teva Trufa de ciruela Drew Womens Ariana Dusty Gold Zapatillas de running adidas Women s Rockadia W Trail Trace Púrpura / Blanco / Core Black Sandalia mujer rosa superfeet Negro Zapatillas para correr Mizuno Women s Wave Rider 21 Trébol / blanco , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris , Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul , Zapatillas para caminar Cinch de mujeres Devotion Plus de Ryka Vino / Negro , J.Renee Womens Mcwayfalls Negro Sam Edelman Women s Gigi Brillante multi retro serpiente de impresión floral Skechers Sport Relajación de mujer Breathe Easy Moneybags Sneaker Gris pardo Linea Paolo Felicia | Cuero perforado negro , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Vino tinto / blanco , Sandalia de cuña de tacón alto para mujer, B.O.C., NATURAL Chancleta Yoga Sling 2 Sanuk para mujer Vichy negro / blanco Sandalia Pikolinos Mujer Puerto Vallarta brandy , Final Fantasy XIII FF13 Serah Farron Zapatos de Cosplay Botas por encargo 2 Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Gris Sandalia de la diapositiva de Sunvue de las mujeres alrededor de la ciudad de Merrell Textura negra , Calzado de mujer Calvin Klein Patreece Slide Coñac Sandalia de viaje Via Spiga Women s Porter Cuero negro Bota de trabajo Darlin Darlin para mujer, color marrón oscuro, 10 B US Azúcar quemado Mocasines casuales Alpargata Deconstructed de TOMS para mujer Raya roja tejida , Trotters para mujer Dea Slingback Pump Blanco / negro Soft Kid / patente , Botines Joan Rain Wedge Chelsea Gloss Rain de mujer SOREL Colegiata Marina / Atmósfera , Sneaker de moda Kickstart para mujer Gris claro Zapatillas de deporte lavables Walker Propet para mujer Sr White , 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.

Drew Drew Womens Cascade Denim Blue Blue Nubuck Nubuck c88bd64 - 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 Drew Drew Womens Cascade Denim Blue Blue Nubuck Nubuck c88bd64 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 1603379
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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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