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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Sintético
Importado
Sintético suela
Peso es de 4.2 oz
Impreso correas con sintético correas de soporte del talón. Talón ajustable cierre de correa y elástico oculto gore en el empeine
Anatómicamente diseñado q-contorno Plantilla con profundidad talón para extra estabilidad para reducir exceso de pronación y arco aumentado asistencia
suave eva entrega todo confort diario
Ligero suela eva con una muy flexible dibujo de la banda
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 Belmar Athletic de Athletic Ryka para de mujer mujer Azul marino 0cdb1cc

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 Belmar Athletic de Athletic Ryka para de mujer mujer Azul marino 0cdb1cc

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

Sandalia Terrell Post II Femenina de Merrell Atlantis , Drew Womens Cascade Nubuck azul marino , ALMA NATURAL Sandalia de pescador Beatrice para mujer Lagarto negro adidas Sandalia atlética Anyanda Flex Y W para mujer Easy Mint / Silver Metallic / White , Sandalias KEEN Venice II-W para mujer Bleacher / Raven , OluKai Kulapa Kai Dragon Fruit / Black , Sandalia KEEN Women s Venice H2 Gruta Celeste / Azul , Sandalia Helen de Propet para mujer Gunsmoke , Chaco Women s Z2 Classic Athletic Sandal Trine en blanco y negro Sandalias KEEN Venice II H2-W para mujer Negro / gris acero , Tacones Pleaser para mujer Infinity 902 Negro claro FLEXUS Zapatos Mujer Flexus Fabrizia Sandal | Sandalia Kea Vent mujer Columbia Haute Pink / Deep Blush Propet Women s TravelActiv Ss Sandal Oro / negro Sandalia KEEN Women s Exo-W Uneek Vapor / Paloma Sandalias KEEN Mujer Whisper-w Coral azul / Báltico Zapatilla de deporte Sina 11 de Josef Seibel para mujer Azafrán , Sirena de mujer Merrell Sandalia atlética Q2 Negro , KEEN Hombres Newport H2 Sandal Negro Chaco Women s Z1 Classic Athletic Sandal Negro , Sandalia para exteriores ECCO Women s Offroad Lite Rainer Sandal Outdoor Blanco / Sombra Blanco , Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Blanco Earth Origins Sophie Negro Chaco Mujer Zvolv Athletic Sandal Camino Orange , Sandalia deportiva W Sirra de mujer Teva para mujer Sabio del desierto , Drew Womens Cascade Nubuck marrón , Skechers Performance On The Go 400 Tropical Flip Flop Negro / gris Sandalia KEEN Clearwater CNX para mujer Anémona / acacia , Chaco Mujer ZX1 Classic Sport Sandalia Burst Blue ,
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 Revs New Classic Masaje de reflexología Sandalias para hombres & Mujer. Mira. Diseño. Terapia. Cojín Comfort Con plantilla de masaje & Absorción de impactos Único. Negro Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial Borgoña / Cardenal / Misterio Rojo , Zapatillas de running Topo Athletic Ultrafly 2 - Mujer Rosa / amarillo ASICS GelGlyde Shoe para mujer Negro / Blanco / Coral , Salomon Sonic Aero W L39349700 Trail Runner Tibetano Rojo / Sangría / Lima Punch. Zapatillas para correr New Balance Rushv3 para mujer Piscis / Pigmento Zapatillas de running Mizuno Wave Wave Evo Ferus para mujer Azul Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Gris Zapatos OLUKAI Pehuea - Mujer Stormy Blue / Stormy Blue , FitFlop Womens Loosh Luxe & iquest; Cuero desnudo Zapatillas de running NIKE Air Max 90 para mujer Vela / radiante esmeralda negro Dr. Martens Myles Black Brando Sandal Negro , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , CLARKS Womens Un Reisel Myra Slide Sandal Cuero negro , Sandalia de cuero Camilia de TOMS para mujer Cuero marrón Bomba Nikole para mujer Via Spiga Cuero negro , Sandalias de mujer Wolky Caviar beige , Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Blanco negro Drew Shoe Women s Hope cuero terapéutico moda Oxfords Negro , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Beige Sandalia de la diapositiva de Stella de las mujeres de Rachel Zoe Plata Sam Edelman Sandalia Glenn para mujer Cuero negro Sandalia Cato Dolce Vita para mujer Ante amarillo 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 1606- Pearl Gold Boombah Women s A-Game - Juego de calas intermedias moldeadas - 8 opciones de color - Varios tamaños Negro / real Bomba de vestimenta Cambredge para mujer Jessica Simpson Black 03 , V Italia 1111 marrón, azul Botas altas de cuero para mujer con forro de lana azul + marrón , Mocasín de gamuza Kilty de Minnetonka para mujer Gamuza turquesa Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Azul Sandalias con cordones Joanie de SOREL para mujer Rojo brillante

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 Sandalia Belmar Athletic de Athletic Ryka para de mujer mujer Azul marino 0cdb1cc 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 637613
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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.