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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Modelo de caucho
Número: 852461005
Sexo: mens
Color: NEGRO / BLANCO-OSCURO GRIS
Hecho en: Vietnam
Marca Nuevo con Original Caja
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

Pantalón Gris largo clásico NIKE 18337 Golf mujer para mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 2963509

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.


Pantalón Gris largo clásico NIKE 18337 Golf mujer para mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 2963509

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 Vibram para mujer, negras, 39 EU / 8 M ES - Zapatillas de running NIKE Mens Air Zoom Structure 19 Negro / Volt / Blue Lagoon / Blanco Zapatillas para correr Saucony Xodus Iso 2 para mujer Berry Coral Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / antracita Zapatillas Reebok Women s Plus Lite HB Track Heather Black - Black / Asteroid Dust / Rbk Rubber Gum / Chalk , Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Phantom / Black Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Gamma Blue / White / Rio Teal Zapatillas de running Saucony para mujer Omni 16 Gris azul , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Racer Pink / Vastgrey Camiseta adidas Swift Run W para mujer Core Black / Core Black / White , Zapatillas de running TOPO FLI-Lyte 2 para mujer Azul rosado , Nike Men s Air Presto Essential Gris Zapatillas de running FuzeX PR para mujer de ASICS Pink Glow / White / Pink Ribbon , Under Armour Threadborne Fortis para mujer Gris Nublado (103) / Blanco Zapatillas de running NIKE Huarache Run SE para mujer Blanco / negro-blanco / negro NIKE Women s Wmns Air Vapormax Flyknit 2, negro / blanco-gris oscuro Negro / Blanco-gris oscuro Zapatillas para correr Saucony Koa Tr para mujer Gris azul , Zapatillas de running NIKE para mujer Lunarepic Flyknit Shield Negro / Metálico Plata-oscuro Gris-sigiloso , Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey Saucony Women s Guide 10 Zapatillas para correr Azul claro / azul Zapatillas de running New Balance 1340v3 para mujer Gris oscuro Zapatillas de running Saucony para mujer Stretch N Go Breeze borgoña , Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Puerto / Aqua , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla oceánica / Blanco-azul Glow-hyper Jade Zapatillas de running Saucony para mujer Echelon 6 Gris rosado , Zapatillas de running NIKE Women s Air Zoom Structure 20 Racer Pink / White-pure Platinum Amortiguación 635v2 New Balance para mujer Plata , Zapatillas de running adidas Aerobounce Pr W para mujer Aero Blue / Aero Blue / Noble Indigo ,
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 Calzado casual Slip-On de Bernie Mev para mujer rojo , Sandalia de vestir Tillman de SoftWalk para mujer Negro Reebok Classic Sneaker Equipo Navy / Platino , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Lobo gris / blanco - negro , KEEN Hombres Newport H2 Sandal Tierra oscura / Acacia KEEN Hombres Newport H2 Sandal Brindle / Sunset Reebok Classic Sneaker Blanco negro , Zapatillas de trail running Salomon Speedcross 4 para hombre Lima ácida , Calzado de running Trail para mujer New Balance para amortiguar 620v2 Trueno / negro Zapatillas de running ASICS Gel-Noosa Tri 9 para mujer Blanco / Amarillo intermitente / Ciruela Sandalias de cuero de Viv Viv de TOMS Rose Gold Specchio , Zapatillas de running New Balance 990V3 para mujer Armada Earth Womens Allegro Ballet Flat Rose Gold Pearlized Soft Leather Aravon Women s Pc Three Strap Sandal Cuero negro Zapatillas de ciclismo Pearl Izumi X-Alp Seek VI para mujer Bloque Belga , Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on Azul marino / blanco , Sandalias Rainbow 301ALTS Single Layer Premier Leather Bosque verde Vans Auténtico Lavado / azul marino , Zapatillas de skate solar NIKE SB Check para hombre NIKE Gris oscuro / gris oscuro , Kenneth Cole New York Mujeres Osmond Fringe Pom Detail Slide Sandalia Blue Denim Sandalia para mujer Cava Donald J Pliner White Geo Jungle , Sandalia Dolce Vita Hazle Slide para mujer Cuero natural Bota Jemima para mujer Dr. Martens Negro Lamo Women s Open s Open Toe Wrap Flat castaña TOMS Clásicos de mujer Lienzo natural Salón muy fino Salsa de tango latino Zapatos de baile para Mujeres 1620 1.3 pulgadas Talón + Plegable Paquete de cepillo Satén negro , Hush Puppies Womens Chowchow WT Oxford Gamuza añil vintage , Kelsi Dagger Frontera de Mujeres de Brooklyn Oxford Trigo Calzado para caminar CLARKS Women s Wave Andes Sage Nubuck Slip clásico de mujer de SAS en Moca

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 Pantalón Gris largo clásico NIKE 18337 Golf mujer para mujer Tech Essentials Negro/ Blanco/ Gris Oscuro 2963509 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 7362907
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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.