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 2 running Saucony Zealot ISO 2 para Zealot mujer Té Té/ Algodón 443672f




100% sintético
Importado
Caucho único
Ligero Zapato de correr con amortiguación de apoyo diseñado para promover una marcha natural
ISOFIT funda interior suave transforma a pie para sentir como un calcetín
FLEXFILM superior para peso ligero asistencia
Everun plantilla proporciona amortiguación sensible
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

Zapato de 2 running Saucony Zealot ISO 2 para Zealot mujer Té Té/ Algodón 443672f

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 2 running Saucony Zealot ISO 2 para Zealot mujer Té Té/ Algodón 443672f

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

NIKE Women s Revolution 4 Running Shoe Negro antracita blanco NIKE Mujeres Downshifter 7 Blanco / puro platino Nike Men s Air Presto Essential Cargo Khaki / Cargo Khaki-rattan , NIKE WMNS Zoom Fly Women 897821-411 Vasto gris Zapatillas de running New Balance 1340v3 para mujer Gris oscuro Zapato deportivo ASICS Gel-Cumulus 18 para mujer Aqua Splash / White / Pink Glow , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Utilitario Negro / Calzado Blanco / Gris Medio Heather Solid Gray , NIKE Mens N98 Tribute Track Jacket Blanco / negro / negro Zapatillas de running W1540V2 New Balance para mujer Blanco azul Zapatillas de running NIKE Air Max Thea para mujer Laguna Blanca / Azul / Verde Fantasma Saucony Triumph ISO 3 Running Sneaker para mujer Rosa / Berry / Plata NIKE Women s Tanjun Blanco Blanco Negro Zapatillas de running Endurant para mujer ASICS Aluminio / Plata / Acuario NIKE Men s Benassi Just Do It Athletic Sandal Midnight Navy / Blanco / Midnight Navy Zapatillas de running New Balance Prism V2 para mujer Sea Spray / Water Vapor , Zapatillas de running New Balance para mujer Fresh Foam Zante V3 Performance Añada Indigo / Sal Marina Calzado de running NIKE RN 2017 2017 para mujer Negro / Blanco / Industrial Azul / Racer Pink ASICS La sacudida de las mujeres Glacier Grey / Carbon / Bright Rose , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro / Plata / Tinta Peacoat , Zapatillas para correr NIKE Air Huarache Run Ultra SE para mujer LT OREWOOD BRN / LT OREWOOD BRN , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Cerúleo / Glacier Blue , Zapatillas de running Mizuno Wave para mujer Wave Horizon Plata / rosa , Zapatillas para correr Saucony Freedom Iso Unisex Aceituna Zapatillas de running NIKE RN 2017 para mujer gratis Vintage Verde / Provenza Púrpura-secoya Zapatillas de deporte Nike Air Max Mercurial 98 para hombre Negro , NIKE Tanjun ENG Sneaker para mujer Pink Pearl Pink Sunset Glow , Zapatillas de calle YKTM Performance Women Boost 4.0 Street Running Shoe Road rojo , Under Armour Speedform Velociti para mujeres Lime Fizz / Fresco verde / blanco , Zapatillas de running NIKE LunarSolo LT Atomic Pink / Crimson Tint para mujer NIKE ,
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 Under Armour Women s HOVR Phantom NC Azulejo azul (302) / grafito Sandalia de mujer W Verra Sport de mujer Teva Atlántico Norte Sandalias Drew de mujer de Drew Piel azul pizarra Earth Origins Sophie Platino , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Negro / Carbono / Rojo , Zapato casual bajo NIKE Lupinek Flyknit para hombre Vela / Negro / Antracita , Inov-8 Roclite 280 Trail Runner para mujeres Verde oscuro / verde azulado / rosa , Zapatillas de running Mizuno Wave para mujer Inspire 10 Blanco Zapatillas de running Under Armour Speedform Intake 2 para mujer Merlot / Tropic Pink / Merlot , adidas Womens Response LT W Plata blanca , Cobb Hill Rockport Mujeres Renee CH Flip Flop Bronce , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Porcelana Azul / Blanco / Asics Azul , Alegria Womens Keli Exclusive Ric Rack Rainbow CLARKS Sandalia de vestir Leisa Gianna para mujer Broncearse SOREL Sandalia casual Ella de mujer Natural , Zapatillas de running NIKE Air Huarache para hombre Negro / Negro-Negro-Blanco , UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Azul verde , JBU by Jambu Women s Gwen Rain Shoe Cazador , Entrenador campeón Gusto Cross Trainer Malla rosa clara Diadora N.9000 Arrowhead Campanula Blue , Sandalias planas de piscina Walking Cradles para mujer Abrigo blanco de cachemira / corcho , Circus de Sam Edelman Sandalia de mujer Raquel Slide Almendra Sam Edelman Women s Tai Sanda Cuero plateado / metalizado suave , Sam Edelman Sandalia Glenn para mujer Cuero negro Sandalia de la diapositiva de junio de las mujeres de Propet Negro , TOMS - Slip-On de lienzo clásico para hombre Universidad de la raya negra , Cape Robbin Jean-1 Silver Rhinestone Cristales adornados Cono talón Slouchy Scrunch Boot Durango Womens Lady Rebel 10 \ Briar Brown / Rusty Red , Nike Tanjun Sandal Womens Cool Grey / Pure Platinum / White , Easy Devitt Oxford Flat para mujer Oro

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 Zapato de 2 running Saucony Zealot ISO 2 para Zealot mujer Té Té/ Algodón 443672f 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 4564180
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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.