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.

New Women Balance Women s w940v3 15509 w940v3 Running Shoe Azul 1054066




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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


New Women Balance Women s w940v3 15509 w940v3 Running Shoe Azul 1054066 - 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

New Women Balance Women s w940v3 15509 w940v3 Running Shoe Azul 1054066

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.


New Women Balance Women s w940v3 15509 w940v3 Running Shoe Azul 1054066

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

Zapato de running ASICS Gt-2000 5 para mujer Negro / Carbón / Flash Coral Zapatillas de running Saucony Women s Triumph ISO 2 Verde azulado / negro / blanco , NIKE Women s Tanjun Equipo Red White , Saucony Women s Ride 10 zapatillas para correr Blanco | , PUMA Men s Roma Basic Sneaker Blanco / Nueva Marina , Zapatillas de running NIKE Zooms Winflo 4 para mujer Negro / Blanco / Gris Oscuro , Zapatillas para correr Saucony Women s Cohesion 10 Gris / negro / azul Zapatillas de running NIKE Lunartempo 2 para mujer Blk / Verde , Zapatillas para correr Nike Flex Experience RN 5 Blanco / Azul Real Profundo , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Mega Blue / Obsidian-concord , Zapatillas de deporte Reebok Print Run Dist para hombre Negro / Aleación , Zapatillas de running NIKE Air Max Torch 4 para mujer Negro / Plata Metálico / Rosa Flash EN Zapatilla Cloudster para mujer Blanco negro , Amortiguación 635v2 New Balance para mujer Plata , Zapatillas de running adidas Alphabounce Beyond W, para mujer Gris dos grises tres grises uno , Zapatillas de running Kinvara 8 Saucony para mujer Berry / rosa , Zapato de running Saucony Zealot ISO 2 para mujer Azul marino / azul / sil New Balance Women s w990v4 Running Shoe Gray / Castlerock Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Blanco / gris fresco , Calzado de running NIKE RN 2017 2017 para mujer Barely Volt / White-wolf Gray , Zapatillas para correr NIKE Women s Air Zoom Structure 21 Armería Azul / Armería Azul marino-cirro Azul , Zapatillas para correr New Balance W840V3 para mujer Gotita , Zapatillas de running ZIGOR para mujer Zapatillas de mujer Flyknit Epic React gris NIKE Women s Tanjun Light Bone / White-black , Zapatillas de running New Balance 1340v3 para mujer Gris oscuro , Zapatillas de running adidas Ultraboost W para mujer Pizarra negra / negra / oscura , Zapato de correr ideal para rejilla de mujer Saucony Gris blanco NIKE Mujeres Downshifter 7 Cool Grey / Lava Glow / Dark Grey / White , Zapatillas de running NIKE Air Huarache para hombre Hyper Turquesa / Blanco
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 Mujer adidas Cf Qt Racer W Gris Uno / Blanco / Gris Dos Chaco Mega Z / Cloud Barrado negro + blanco Zapatillas para correr Nike Women s Flex Experience Rn 5 Negro / Bluecap / Hyper Cobalt / White , Sandalia Terrell Post II Femenina de Merrell Negro , Sandalias OLKAKAI Ohana - Hombre Dark Java / Ray , Mujer adidas Cloudfoam Qt Racer Negro / Blanco / Carbono Skechers sandalia ancha de deporte 600-brilliancy para mujer Negro Zapatillas de deporte Onitsuka Tiger Mexico 66 Abedul / Verde 1 , Zapatillas de running New Balance para mujer 590v3 Negro / rosa , Salomon Women s Speedcross 4 W Trail Runner Rojo tibetano / sangría / negro , Clorts Zapato de agua cerrado para mujer Sandalia de senderismo de secado rápido 3H025 Morado oscuro Zapatilla deportiva Skechers You-14958 para mujer Blanco JSport by Jambu Women s Wendy Flat Armada , Sandalia para mujer con correa de tobillo Naot para mujer Lustre negro, cuervo negro, serpiente negra Zapatillas de paseo Skechers Performance para mujer Go Walk 2 Slip-On Negro , Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Gris blanco , Zapatillas Fm Cross-Trainer para mujer Tazon 6 WN de PUMA Puma Black-puma Team Gold , DC Youth Rebound Skate Shoes Gris / Negro / Gris , Calzado de mujer Calvin Klein Patreece Slide Blanco suave , Plataforma de campo Leisa para mujeres de CLARKS Cuero Multi Marrón Lauren by Ralph Lauren Sandalia mujer Davan Slide Paja , Sandalias Fifi19 Slide de Donald J Pliner para mujer corcho 50 tonos de oro 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) 7004- Oro , Zapatilla Saraih9 de Easy Spirit para mujer Negro Fizik R5B Donna BOA Zapato con Fuschia Trim Antracita / gris oscuro , Jabot para mujer Vaneli Truffle Suede / Matching Elastic , Botín para mujer Pleaser Asp1018 / Bpu / M Black Faux Leather / Black Matte Wedge del desierto de las mujeres de TOMS Trigo-trigo , Botín de tobillo Carey Fergalicious para mujer Gris pardo Zapato femenino Keli de Alegria Sugar Skulls Dottie

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.

New Women Balance Women s w940v3 15509 w940v3 Running Shoe Azul 1054066 - movie-store.online


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 New Women Balance Women s w940v3 15509 w940v3 Running Shoe Azul 1054066 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

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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.


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