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
Sintético único
Eje mide aproximadamente parte baja del arco
REVlite espuma de la entresuela
Bootie Construcción
Sin costura aplicación material
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,

Zapatillas de running adidas Aerobounce Pr W para mujer Aero Blue / Aero Blue / Noble Indigo Zapatillas de running NIKE Air Max 2017 para mujer Concord / Blanco / Violeta Persa , Zapatillas de running New Balance Foam Zantev2 para mujer New Balance Rosa / negro Zapatillas de running Nike Zoom para mujer Hasta / Mtlc Red Bronze / Green Glow Zapatillas de running Newton Kismet II para mujer Rosa / Blanco , Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Silver Mink / Outerspace NIKE Men s Flex 2017 RN Negro Blanco Anth Cool Gray , Zapatillas para correr NIKE Womens Sock Dart Lino / Blanco-blanco-negro Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo Zapatillas de running Fresh Foam 1080v6 de New Balance para mujer Azul rosado , NIKE Men s Roshe Two Running Shoe Max naranja / negro / noche profunda , Zapatillas para caminar Reebok Women s Cloudride LS DMX Blanco negro , NIKE Men s Flex 2017 RN Hyper Orange / Black , Zapatillas de running AMPLICA para mujer ASICS Gris Medio / Plata / Aruba Azul , Zapatillas de running Fresh Foam 1080v6 de New Balance para mujer Gris azulado Zapatillas de running Nike Kids Air Max Zero Essential GS River Rock / Black-CobbleStone Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Solar Rojo / Negro / Mango Zapatillas de running Saucony Progrid Kinvara 3 para mujer Blanco / Citron / Rosa Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta Zapatillas de running NIKE Lunarglide 9 para mujer Negro / blanco / gris oscuro / gris lobo , Zapatillas de running Saucony Women s Ride 8 GTX Blanco / negro / rosa NIKE Men s Flex 2017 RN Antracita Gris Negro NIKE Men s Benassi Just Do It Athletic Sandal Oro negro / metálico , Calzado de running NIKE Kids Roshe One (GS) Blanco / puro platino / blanco , Calzado deportivo New Balance para mujer Vazee FuelCell V1 Saúco / Plata Zapatillas de running adidas Questar Ride para mujer Carbono / Negro / Gris Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul , adidas Performance Ultraboost X para mujer Carbono / Cristal Blanco / Núcleo Negro ,
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 deporte NIKE Womens Air Max 95 Running 307960 Sneakers Shoes Pure Platinum Hyper Violet 001 , Chaco Mujer Zcloud X Sandalia Deportiva Durazno estándar Chaco Mujer ZX1 Classic Sport Sandalia Dart Peach , NIKE Women s Tanjun Paramount Blue White Lava Glow , Zapato de running NIKE Kids Tanjun Print (GS) Antracita / Metallic Platinum , NIKE Womens Roshe Two Running Shoes Negro Negro , Sandalia deportiva Tirra para mujer Teva Negro / Blanco Multi , Sandalia Yucatan ECCO Mujer Frosty Green / Moon Rock / Frosty Green Zapato de vestir uniforme al aire libre Vulc N Vent Slip para mujer de Columbia Acero, Cornstalk , Zapato de running ASICS GEL-Venture 5 para mujer Gris plateado / turquesa / sacador de lima , Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird , Zapato acuático Supervent de mujeres de Columbia Grafito, Canyon Blue Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Rosa caliente , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Gris oscuro Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Azul marino / verde azulado , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Estaño New Balance 711v3 Disney Cross Trainer para mujeres Puntos negros / blancos / rojos New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Seafoam / Blanco Zapatillas de paseo Slip-on Go Step Lite de Skechers Performance para mujer Gris pardo Clásicos de base Slip-on (Tm) para hombres de Vans Negro / Marshmallow , Zapatillas de golf PUMA Summercat Sport para mujer Peacoat / Blanco , Sandalia Cato Dolce Vita para mujer Orchid Suede , Zapatillas de voleibol Wh-Pk Tornado 9 Woms Whist de Mizuno para mujer Blanco / rosa , Dr. Martens Women s Dupree Oxford Negro , Azul de Betsey Johnson Zapatillas SB-Haven para mujer Marfil Easy Spirit Women s Motion Lace up Oxford Cuero de peltre Zapatos de tierra serval Rojo brillante Zapatillas de deporte FRYE 79981 Alexis Low Lace Sneaker Blanco negro Cole Haan Stitchlite para mujer Oxford Negro , Betunia Loafer de mujer Aerosoles Negro Acolchado

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 Calzado B078RB5PVW deportivo/ New Balance para Balance mujer Vazee Rush V3 Blanco/ Vintage Indigo aaf0337 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 1765691
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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.