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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Malla de punto
Importada
Caucho único
Eje mide aproximadamente parte baja del arco
Tejer malla proporciona protección con un ponible diseño
Suela Las orejetas proporcionan grandes tracción y bajo protección
Talón El sistema GRID proporciona amortiguación de larga duración
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,

Brooks Womens Adrenaline GTS 18 Ébano / plata / rosa Calzado de running New Balance para mujer Vazee 2090 Rosa / amarillo , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Multicolor / Multicolor Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Clear Aqua / Glow Orange / White Zapatillas de running New Balance 990V3 para mujer Gris , Zapatillas de running adidas Pureboost X Tr Zip para mujer Lino / Traza Caqui / Oro táctil , Zapatillas de entrenamiento NIKE Women s Metcon 4 Blanco / Plata Metálica , Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres gris , Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Blanco / Menta-negro Zapato de running GEL-Excite 3 para mujer ASICS Carbón de leña / Grape / Aqua Splash , Zapatillas para correr NIKE Womens Sock Dart Glacier Blue / White Zapatillas de running adidas Adizero Boston 7 para mujer Gris / blanco / gris Zapatillas de running Mizuno Women s Wave Horizon 2 Monumento / Acuario Calzado de running New Zealand Vazee Pacev2 para mujer Dark Denim / Lime Glow NIKE Airs para mujer Zoom Pegasus 34 Low Top para correr con cordones Sneaker negro Zapato deportivo ASICS Gel-Kayano 24 para mujer Concha Rosa / Blanco / Begonia Rosa , Zapato de running 840v4 New Balance para mujer Gris claro Zapatillas de running adidas Mana Bounce de adidas Performance para mujer Claro Gris S12 / Blanco / Negro , Zapatillas de running NIKE Air Huarache para hombre Azul Gris / Blanco Cumbre Zapatillas de running Saucony PowerGrid para mujeres Gris / Té / Rosa , NIKE Men s Flex 2017 RN Gimnasio Azul / Blanco-foto Azul Zapatillas de running Saucony para mujer Echelon 6 Gris rosado , Zapatillas de running NIKE Air Max Flair para hombre negro NIKE Men s Flex 2017 RN Antracita / Volt / Cool Gray / Black , Zapatillas de running Adidas Women s Energy Cloud WTC W Tela Super Purple / Ray Pink / Unity Purple , Zapatillas de running adidas Ultraboost W para mujer Blanco / Blanco-2 / Blanco New Balance Women s w990v4 Running Shoe Komen Pink Zapatillas de running ASICS Gel-Kayano 23 para mujer Tiburón / Aruba Azul / Acuario Nike Air Max Zero QS femenino
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 para correr W630V5 para mujer New Balance Rosado Converse Chuck Taylor All Star Seasonal Color Hola Negro / pergamino / Dahlia rosa , Zapato deportivo ASICS Gel-Kayano 24 para mujer Negro / Phantom / Blanco Sandalia de cuero Maya New Balance para mujer Negro Sandalias KEEN Venice II H2-W para mujer Grape Kiss / Red Violet , Lanzamiento de Brooks Womens 5 Negro / Ebony / Primer gris Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco , Sandalia Barroll T Correa de Natural Soul para mujer Saddle Tan Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Plata / gris / rosa Zapato para caminar WW411v2 New Balance para mujer Negro Sandalia de vestir Bayleen II de mujer de Cole Haan Azul marino / azul marino SOREL Womens Ella T-Strap Sáhara , Taryn Rose Darla Loafer Flat para mujer Negro Zapatos de caminata SAS Low Me de cuero para mujer Negro , Zapatillas de deporte con cordones FitFlop para mujer Neoflex Azul indio / turquesa Zapatillas Skechers Performance Go Mini Flex-14006 para mujer Armada Zapatillas de running Golden Spike de Altra para hombre Negro / Diva Pink , Zapato Dantera, para hombre, L Low Waxed Court Blanco / gris violeta / blanco , Zapatillas de tenis Wilson Rush Pro 2.5 para mujer púrpura , NIKE WMNS Hyperdunk 2017 TB baja mujer 897812-601 rojo Sandalia de la diapositiva Indi Kate Spade New York para mujer Denim azul claro Sandalia Quasida Slide de Flexus by Spring Step para mujer Negro / Multi , Sandalia Mason para mujer Naturalizer Slide Cuero metalizado dorado claro , Vionic Women s Tide Ii Negro , Zapatillas clásicas de piel de oveja gemelas Sheep Touch para mujer - , Mordenmiss Nuevo verano de primavera y verano de flores hechas a mano zapatos planos de cuero Mocasines Estilo 2 Verde TOMS Kids 10009918 Alpargata-K Brillo iridiscente plateado , Nine West Leather Lilianne Leather para mujer Vino Zapatillas de deporte Slace Sport Sport D Lites Slip-On Mule Blanco , Trotters Liz Loafer para mujer marrón ,

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 B072Y2NN7R para Tr11 correr Saucony Women s Excursion Citron Tr11 Púrpura Citron 5d520f6 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 2895246
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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.