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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Textil y sintético
Importado
Caucho único
Eje mide aproximadamente parte baja del arco
Fresco Entresuela de espuma
TPU Silla de montar
moldeada Foam Comfort Heel
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

Zapatillas de Balance running 19347 New Balance Foam Cruz para V1 para mujer New Balance Blanco Blanco 2933518

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.


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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 New Balance 520v3 para mujer Artic Fox / Black , Zapatillas de running Reebok Zpump Fusion 2.0 para mujer Carbón / Negro / Icono Rosa / Blanco ASICS Metrolyte II Shoe Women s Walking Hielo medio gris / gris medio / albaricoque , Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa , Zapatillas de running Gobiv2 para mujer New Balance Semillas con Grove Zapato de running Saucony Zealot ISO 2 para mujer Coral / púrpura / algodón Zapatillas de running de malla cruzada Nova Vam Cruz V1 para mujer New Balance Reflexión / Blanco , NIKE Men s Benassi Just Do It Athletic Sandal Negro / Juego Rojo / Blanco Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink Black Bright Mango 600 Zapatillas de running Comfort Ride 575v2 New Balance para mujer Azul gris / Coral Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Multi-Color / Multi-Color Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul Mediano / Negro-aluminio-electro Verde , Zapatillas de running Under Armour Speedform Intake 2 para mujer Ghost Gray (101) / Estaño , Zapatillas Reebok Women s Runner 2.0 MT Track Negro / carbón / ácido rosa / blanco Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , NIKE Men s Flex 2017 RN Negro / blanco / antracita / gris fresco Zapato deportivo ASICS Gel-Excite 4 para mujer Uva / Plata / Begonia Rosa Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Negro / Fantasma , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Negro / Onyx / Plata , Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Zapatillas de running adidas Alphabounce Beyond para mujer Blanco / Gris / Gris Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino Zapatillas de running NIKE Air Zoom Vomero 13 para mujer Ocean Bliss / Noise Aqua Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Shadow Brown / Metallic Pewter / Rush / Maroon , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Rojo de alto riesgo / rojo de alto riesgo (hombres) Zapatillas para correr W630V5 para mujer New Balance Gris oscuro Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Gris oscuro / Barely Volt / Wolf Gray , Zapatillas para correr Go Run Forza Boston 2016 de Skechers Performance para mujer Azul marino / verde azulado Zapatillas para correr NIKE Womens Sock Dart Midnight Navy / Golden Beige ,
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 de running adidas Ultraboost W para mujer Táctil Azul / Táctil Azul / Negro Zapatillas de running AMPLICA para mujer ASICS Azul índigo / Azul índigo / Coral destello Zapatillas de running Kineta Relay para mujer Saucony Té de menta , Zapatillas de running NIKE Unisex Flyknit Racer Blanco / White-sail-pure Platinum , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Core Black / Footwear White Sandalia Drew Element para mujer Nubuck negro Sandalia de mujer Drew Alana Blanco negro , Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva , Zapatillas de running NIKE Juvenate para mujer Glacier Blue / Mica Blue / White Zapatillas de trail running Salomon Women s Sense Pro Trail Mystic Purple / Black / Pink Salomon Women s XA Enduro W Trail Runner Cielo nocturno Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño , Sandalia de gamuza Poppy de TOMS para mujer Desierto de color topo / cáñamo Bota Deia para mujer de TOMS Gamuza / Lana gris hierro forjado , Calzado de senderismo Skechers Go s 15601 para mujer Azul marino / blanco , KEEN Kids Seacamp Ii CNX Water Shoe Danubio negro / azul , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Seafoam / Blanco , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Carbón , Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on Rosa / Lima Zapatillas clásicas Vionic para mujer Walker Gris pardo , Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Taupe Cabana , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker azul Sandalia de cuero Lucili de Nine West para mujer Negro / Cuero Multi Sandalia Ryana Marilyn Slide para mujer Frapé , Sandalia Taryn Rose de mujer Iris Knit Slide Sonrojo , Adtec - Bota de trabajo marrón de 9 \ Broncearse , Botas de campo acolchadas TuffRider Ladies Alpine en piel sintética - , Pantuflas de piel de oveja australiana natural de LAMB Slippers Azul oscuro Trotters para mujer Dea Slingback Pump Armada Abilene de cuero sin curtir con punta de bota para mujer - 5070 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 Zapatillas de Balance running 19347 New Balance Foam Cruz para V1 para mujer New Balance Blanco Blanco 2933518 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 2230659
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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.