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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mezcla de poliéster
El estilo y / o el color del calcetín pueden variar según la disponibilidad.
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.


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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,

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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 RailFit 1 Clay Green (104) / Antracita Skechers Sport Skech Air Run High Fashion Sneaker Negro / rosa caliente , Sandalia de mujer Drew Alana Pewter-bronce Zapatillas deportivas 574S Sport New Balance para hombre Coco tostado / blanco adidas Adizero Defiant Bounce para mujeres Blanco / Blanco / Gris Uno , Sandalia de susurro de mujer Keen Shiitake / Malaquita , Sandalias Under Armour UA Ignite VII SL para mujer Negro / Neo Pulse NIKE Jamaza Sneaker Scarpe Running Nero Unisex ultraleggere Negro / Blanco-antracita Mizuno Women s Wave Daichi 2 Trail Runner Remolacha / Plata Sandalia NIKE Kawa Slide para mujer Cool Grey / Hyper Turquoise / Cool Grey , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Utiblk, Ftwwht, Mgsogr Reebok Women s Ridgerider 2.0 Trail Runner Lila Shadow / Coll. Skechers Women s Go Walk Revolution Ultra Sneaker Negro / Multi , Sandalias de cuña con espalda descubierta Stuart Weitzman para mujer Multi Noir Calzado de senderismo Go Joy 15602 de Skechers para mujer Blanco negro , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines C-blanco / rojo Skechers Women s Burst 2.0 New Avenues Sneaker Gris negro Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Blanco óptico FootJoy Colección Casual para Mujer Zapatos de Lino de Golf Tan / Tan Houndstooth Sandalia Carlos s Sanmie Slide de Carlos Santana para mujer Tan Multi , Sandalias planas de piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural Carlos by Carlos Santana Sandalia de mujer Zoey Slide Negro Calzado de ballet Freeform FF01 Femenino de Capezio Rosa claro , Under Armour Fusible para mujer FST Negro (002) / Glacier Grey Bota de mujer FRYE Dara Chelsea Carbón , BareTraps BT Ricarda tobillera para mujer Marron oscuro Botas de vaquero de mujer Black Star DUVAL (negro) Zapatillas de lona Sperry Top-Sider Firefish Canvas Gris , Mocasín de gamuza Kilty de Minnetonka para mujer Gris pardo

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 Zapatos descalzos CVT-Hemp de mujer CVT-Hemp Vibram FiveFingers y paquete de descalzos Premium Toesock Caqui e6bab74 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.

Corriendo
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

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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.