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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Sintético y malla
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


Salming Salming Miles Running Miles Shoe Mujeres B079DQYPWL Turquesa da71c9a - 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

Salming Salming Miles Running Miles Shoe Mujeres B079DQYPWL Turquesa da71c9a

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.


Salming Salming Miles Running Miles Shoe Mujeres B079DQYPWL Turquesa da71c9a

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 Casual de Cuero Cortez Clásico NIKE para Mujer Blanco / rojo universitario , Zapatillas de running NIKE Roshe Two para mujer Wolf Gray / Wolf Gray , La Sportiva Lycan Running Shoe - Mujer Stone Blue / Jade Green , Calzado de running NIKE Kids Flex 2017 RN (GS) Rabia Verde / Negro / Azul Cloro , Under Armour Threadborne Slingflex para mujer Cubierto Gris (943) / Apollo Gray , Zapatillas de running NIKE Juvenate para mujer Negro / Blanco / Negro / Blanco , Under Armour Charged Reactor Run Run Antracita (100) / gris oscuro , Converse Chuck Taylor All Star Seasonal Color Hola Papel rosa Zapato de running ASICS GEL-Kayano 22 para mujer Coral / Pink Glow / Flash Yellow NIKE Women s Tanjun Anthacite / Igloo White , Converse Chuck Taylor All Star Seasonal Color Hola Chili Paste / Atlantic / Egret , Zapato de running ASICS GEL-Kayano 22 para mujer Gris plata / Pistacho / Coral Fiery Zapatillas Merrell Women s Encore Nova Slip On Bracken Smooth Zapatillas de running Adidas Performance Ultra s Boost M para hombre Gris / plata , Zapatillas de entrenamiento cruzado Heather para mujer New Balance 711v3 Gris azul Zapatillas de deporte casual Gazelle de adidas para hombre Azul marino blanco Converse Chuck Taylor All Star Seasonal Color Hola Blau , Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / Navy NIKE Men s Roshe Run Vino de Oporto Negro , Zapatillas de running Saucony Women s Triumph ISO 4 Vizi Rojo / Blanco NIKE Kids gratis Rn (grande) Racer Pink / Black Lava Glow , Converse Chuck Taylor All Star Seasonal Color Hola Negro / garceta / cuero Cordura negro Zapatillas de running Reebok Zprint 3D para mujer Nube gris / Rebel Berry / Poison Pink / Pewter , Zapatillas NIKE Women s Revolution 3 para correr Ghost Green / Sunset Glow / White Converse Chuck Taylor All Star Seasonal Color Hola Blanco / Vapor Rosa / Blanco , Reebok Classic Sneaker Equipo Navy / Platino , Zapatillas náuticas para mujer Saucony Originals Negro / Bronce Converse Chuck Taylor All Star Seasonal Color Hola Inked / Egret , Zapatillas de running NIKE Unisex Flyknit Racer Citrus brillante / Laguna azul-negra
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 Lanzamiento de Brooks Womens 4 Negro / Fucsia Púrpura / Azul deslumbrante Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Azul Zapatillas de running Mizuno Wave para mujer con sombra Fucsia Púrpura / Plateado / Azulejo Azul , Zapato de running Fuzex Rush para mujer de ASICS Gris medio / negro / blanco Sandalia deportiva Teva W Hurricane 4 para mujer Negro , Drew Shoe Gemini cuero de mujer, malla, caucho, zapatillas de moda Ante negro , Zapatillas de entrenamiento WX624v2 para mujer New Balance Blanco , Adidas Sport Performance Mujeres Terrex CC Voyager Sleek Sneakers Verde ceniza / Aero Verde / Amarillo semi-congelado , Sandalias planas EVA-Nu de Eileen Fisher para mujer Negro , Sandalias Abbott Gladiator de Cobb Hill para mujer Nubuck negro Kenneth Cole REACTION Womens Cake Shop 2 Soft Gold Synthetic Sandalia Sharene de Calvin Klein para mujer Arena del desierto , Zapatillas para caminar Lite-Agile Go Step de Skechers Performance para mujer Azul marino / blanco Heather Drew Shoe La alegría de las mujeres Negro New Balance WW577 Hook and Loop Walking para mujer Blanco , New Balance 611v1 Cush + Cross Trainer para mujeres Negro Zapato Ryka Women s Vestige RZX Cross-Trainer Gris oscuro , New Balance WX811v2 High Top Cross Trainer para mujeres Gráfico de espacio negro / exterior Zapatilla de pies Cush N de Prop & eacute; t para mujeres Armada Vans Unisex Old Skool Zapatillas clásicas de skate (C & l) Vestido Blues / Material Mix Calzado de baloncesto NIKE Women s Court Borough Low Blanco , Zapatillas de deporte ligeras JiYe para hombre Zapatillas de deporte deportivas para correr para mujer Negro rojo Sandalia Rockdale Lanea Woven Slide para mujer Tejido negro 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 7039 - Escala de plata Zapatilla de hombre viejo amigo castaña Bella Vita Womens Mel-Italy Gamuza italiana gris topo , Dolce Vita Saxon Oxford para mujer Negro Zapatilla de deporte sin cordones sin mangas lavada Chillax de Keds para mujer Melocotón rosa Botas Harley-Davidson Womens Adrian, de cuero, negras, de corte medio Negro , Zapato casual Alt-tec para mujer Hi-Tec Luna azul ,

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.

Salming Salming Miles Running Miles Shoe Mujeres B079DQYPWL Turquesa da71c9a - 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 Salming Salming Miles Running Miles Shoe Mujeres B079DQYPWL Turquesa da71c9a 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.


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