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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Malla ^ Sintético
Importado
BioMoGo Amortiguación de la media suela de ADN se adapta dinámicamente a tu paso, manteniéndote en control de crucero, milla tras milla.
El Progresivo extendido Diagonal Rollbar (PDRB) guía su cuerpo de nuevo en su movimiento natural camino con agregado control.
A diseño espacioso abastece a plano o voluminoso pies.
A Heel-Segmented Crash Pad proporciona transiciones suaves.
El plantilla proporciona amortiguación adicional para un zapato de felpa en el zapato sensación.
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

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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.


Adicción Adicción a las/ mujeres 19144 de Brooks 13 Evening Blue/ Teal Victory/ Purple Cactus Flower 3cd9a14

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 NIKE Lunartempo 2 para mujer Pink Blast / Clr Jd / Black / Brly Grn , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris oscuro , Zapatillas para mujer Reebok Zprint Her MTM para mujer Negro / Gris Ceniza / Blanco , Zapatillas de running Under Armour Charged Bandit 3 para mujer Purple Rave (959) / Penta Pink , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul humo / azul oscuro / Cantalupo Zapatillas de running New Balance para mujer Fresh Foam Zante V3 Breathe Pack Vivid Ozone Blue / White Zapatillas de running NIKE Lunartempo 2 para mujer Hyper Violet / Black / Concord / Gamma Blue Zapatillas de running ASICS Gel-Contend 4 para mujer Negro , Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover Zapatillas de running NIKE Air Max 2017 para mujer Blanco negro , Zapatillas de entrenamiento Nike Womens Metcon 3 Gris oscuro / verde fantasma , Zapato de running de punto fuzeX para mujer ASICS Rosa Glo / Flash Coral / Negro , Zapatillas de running adidas Originals Women s Tubular Dawn Originals Utiblk / Utiblk / Crywht Zapatillas de running adidas Adizero Boston 6 W para mujer Utilidad Negro / Platino / Negro , Zapatillas de running adidas Supernova Glide 8 para mujer Lgh Solid Gray / Lgh Solid Gray / Silver Metallic , Zapatillas de running ASICS Gel-Contend 4 para mujer Blanco / Aruba / Plata Zapatillas de volante NIKE Air Zoom Hyperace para mujer Wolf Gray / Volt-black Zapato deportivo ASICS Gel-Cumulus 18 para mujer Silver / Pink Glow / Lapis Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Onyx / Plata , Zapatillas de running Kinvara 6 Saucony para mujer Rojo / naranja / menta , Zapatillas de running adidas Alphabounce Beyond W, para mujer Núcleo negro, núcleo negro, tela gris cinco , Skechers Women s Go MEB Razor 2 Blanco negro Zapatillas de running W1540V2 New Balance para mujer Blanco azul Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Brt Mng / Wlf Gray-pch Crm-plst P , Zapatillas de running NIKE Air Max Thea para mujer Blanco / Blanco-blanco , Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue Zapato de running ASICS GT-2000 4 para mujer Turquesa / rosa fuerte / azul marino , PUMA Womens Fierce Rope Satin EP Melocotón Beige / Puma Blanco / Perla
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 para correr ASICS Gel-DS Trainer 22 para mujer Acuario / Aqua Splash / Flash Coral Zapatillas NIKE Kids Revolution 3 (GS) para correr Binario Azul / Electrolime / Azul Supremo , Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Lago Black / Gray NIKE Mujeres Downshifter 7 Golpe caliente / negro / blanco , Zapatillas de deporte casual Gazelle de adidas para hombre Marrón / Blanco / Metálico / Dorado NIKE Women s Tanjun Coral Stardust / Sail Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro Zapatillas de trail running New Balance WT00 Minimus Zero v2 para mujer New Balance Púrpura / rosa , Zapatillas de running Mizuno Wave para mujer Inspire 14 Folkstone Gray / Pink Glo Cobb Hill Rockport Women s Angelina Dress Pump Negro Trotters Mujer Simmy Mary Jane Flat Sabio , Zapatillas de deporte GEL-Fit Sana 2 de ASICS para mujer Flash Coral / Blanco / Azul de buceo Sandalia de tacón Edline de mujer Ivanka Trump Ante negro , Botas de tobillo Pallaphoenix K Palladium para mujer Rosado , Adidas Mujeres Aleki X Cross Tainer Amarillo / Amarillo / Aqua , LI-NING Zapatillas de baloncesto Samurai III Wade Culture para hombre Zapatillas de deporte Mujeres grises BareTraps Women s Cherilyn Slide Sandal Negro Viónico con Orthaheel Aubrey Estaño Sandalia con tacón Florence Block de Via Spiga para mujer Patente de arena Sandalia Merrell Terrano Ari Slide para mujer Negro , ALMA NATURAL Sandalia Kiki Slide para mujer Negro Trotters Lucy Ballet para mujer Cerezo negro , Bomba de cuña Yaralla J.Renee para mujer Taupe metálico , Botas Trixie Joe Rocket para mujer (negro, talla 5) - , XYD mujeres Slingback medias gatito talón bombas punta estrecha hebillas zapatos de vestir con elegantes arcos rojo , TOMS Clásicos de mujer Negro sobre negro , Trotters Lucy Ballet para mujer Coñac Mercurial Superfly V FG Rojo / Negro , Madden Men s M-Grassy Oxford Ante negro , Nurse Mates Womens Fleet 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.

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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 Adicción Adicción a las/ mujeres 19144 de Brooks 13 Evening Blue/ Teal Victory/ Purple Cactus Flower 3cd9a14 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 3919195
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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.