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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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


PUMA 19996 Leadcat Botanical Women Slide PUMA Slide - 42d056a - movie-store.online

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.


PUMA 19996 Leadcat Botanical Women Slide PUMA Slide - 42d056a

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 Yucatan ECCO Mujer Negro , Sandalias KEEN Mujer Whisper-w Negro / Imán , Wolky Comfort Jewel 12800 Blue Nubuckleather , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Pimenton Sandalia Terrell Post II Femenina de Merrell Dusty Olive BZees Sandalia con cuña de beso para mujer Negro KEEN Hombres Newport H2 Sandal Quemado de oliva / amarillo dorado , Sandalia Belmar Athletic de Ryka para mujer Gris / Coral Sandalia Atlética Merrell para Mujeres, Alrededor de la Ciudad Vertical Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Sandalia Uneek O2-w de mujer KEEN Opal azul / azul provincial , Skechers Performance On The Go 400 Vivacity Flip Flop Gris pardo Sandalia atlética Salomon Women s Crossamphibian Swift W Espectro Azul / Aura Astral / Cerámica , Sandalia NIKE Kawa Slide para mujer Negro / Vivid Pink Sandalia atlética Salomon Women s Crossamphibian Swift W Espectro Azul / Aura Astral / Cerámica , Wolky Comfort Jewel Gray Picasso Crash , Sandalia Yucatan ECCO Mujer Negro , Chaco Mujer Zcloud X2 Sandalia Deportiva Runa Teal , Nike Benassi Duo Ultra Slide Blanco Chaco Women s ZX2 Classic Athletic Sandal Edredón Melocotón Sandalia para mujer Teva Terra-Float Travel Knit Legion Blue Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Laguna Azul Sandalia de mujer KEEN Rose Blue Opal / Provincial Blue , Wolky Comfort Jewel Rojo Multi Multi Fantasy de colores Sandalia de susurro de mujer Keen Imán / Sangría KEEN Hombres Newport H2 Sandal Cuervo / Aluminio , Teva Terra Sandal para hombre Cross Terra Black , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Zorro ártico Wolky Comfort Jewel 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 EM Road N1 v2 Pearl iZUMi para mujer Grey / Deep Wisteria , Wolky Comfort Jewel Gray Croco , Under Armour Speedform Slingwrap para mujer Negro (001) / Glacier Grey , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Azul marino / rosa / verde azulado , Zapatillas para correr Mizuno Wave Inspire 12 para mujer Blanco negro , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Negro 1786 , Sandalia de vestir de tela Leisa de nueve mujeres del oeste Gray Multi Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Armada Sandalia Nora Star de mujer Mephisto Patente amarilla Bota Meri Alegria para mujer Melaza troceada New Balance 711v3 Heather Cross Trainer para mujeres Acero / Negro Sandalia de cuero Genie Taos para mujer Oro , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Estaño Zapato de entrenamiento WX824 para mujer New Balance Blanco , Zapatillas de golf adidas Climacool Knit Cblack / D, mujer, de adidas Energy Blue 50 sombras de blanco 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- Satén de marfil , Bota con cordones Twisted X Boots WAL0007 All Around para mujer Sillín engrasado / rosa , Bomba de vestido LifeStride Sevyn para mujer Piedra Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 3 Bomba de Angel II para mujeres estilo suave Elegancia de hueso Aquatalia de Marvin K. Botines de gamuza para mujer Tobillera Ceniza , Botín Colibri Nf63 para mujer El Naturalista Negro / Zanahoria , Cama Stu Maggie Oxford para mujeres Cuero de teca Mason , Zapatillas sin cordones Slip-On Sport Sport Roper para mujer, punta redonda, marrón marrón , Bota Kikka para mujer Dromedaris Sillín Drew Shoe Women s Drew Slip Resistant Leather, caucho, espuma, oxfords Negro ED Ellen DeGeneres Womens Stella Open Toe sandalias de cuff del tobillo Tela de Bisque Zapato femenino Keli de Alegria Chrome Cube , CLARKS Mujer Ashland Bubble Slip-On Negro , Skechers You de Women s One Slip-on Shoe Blanco 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 PUMA 19996 Leadcat Botanical Women Slide PUMA Slide - 42d056a 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

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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.