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.

Newton Running El destino El de los hombres Running Amarillo Amarillo naranja 39c5fd4




Malla
Eje mide aproximadamente Low-Top del arco
Engineered malla superior para estructura liviana y ventilación
ETC plantilla proporciona antibacteriano comodidad en los pies
Acción / reacción tecnología para absorción de impactos añadida y lanzamiento de energía
EVA entresuela para duradero amortiguación ligera
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


Newton Running El destino El de los hombres Running Amarillo Amarillo naranja 39c5fd4 - 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

Newton Running El destino El de los hombres Running Amarillo Amarillo naranja 39c5fd4

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.


Newton Running El destino El de los hombres Running Amarillo Amarillo naranja 39c5fd4

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 New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Deep Porcelain Blue / Overcast , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul / Coral vivo , Zapatillas de running adidas Alphabounce Rc W para mujer Core Black / White / Utility Black , Camiseta adidas Swift Run W para mujer Trace rosa / gris tres / cristal blanco , Zapatillas de running Adidas Performance para mujer Ultra Boost Blanco / Blanco / Plata , Zapato de running COMUTORA para mujer ASICS Mar tempestuoso / Mar tempestuoso / Blanco , Zapatillas de running ASICS Gel-Contend 4 para mujer Gris piedra / plata / rosa caliente , Zapatillas de running NIKE Air Max Flair para hombre Blanco / puro platino , Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Niebla / Blanco / Negro Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Gris Zapatillas de running ASICS Gel-Nimbus 20 para mujer Gris Medio / Gris Medio / Rosa Concha , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Pure Platinum / Black / Cool Gray , Zapatillas de running NIKE Air Huarache para hombre Blanco / Blanco-negro , Zapatillas para correr Mizuno Women s Wave Paradox 3 Capri-rosa , Zapatillas de running adidas Alphabounce Em W para mujer Trace Olive / Trace Cargo / Gray One , Newton Running Womens Kismet 4 Primavera Verde / Amarillo , Zapatillas de running New Balance Boracay V3 para mujer Blanco negro , Zapatillas de running NIKE para mujer Air Max Thea Prm - Zapatillas de running Kinvara 6 Saucony para mujer Azul / Citron , Zapatillas de running Reebok Print Smooth Ultk para mujer Negro / polvo de asteroide / blanco Zapatillas de running New Balance 990V3 para mujer Gris / rosa Zapatillas de running NIKE RN 2017 para mujer gratis Vintage Verde / Provenza Púrpura-secoya Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow Short NIKE Tempo para mujer clave de cal / luz aqua / carbón clásico / plata mate Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Saúco , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Blanco / Gris Oscuro Zapatillas de running Nike Kids Air Max Zero Essential GS White Wolf Gray 100 Zapatillas NIKE Kids Revolution 3 (GS) para correr Juego Royal Black Wolf Gray Wht
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 NIKE Femenil Flex 2016 Rn Running Azul Brillo / Blanco / Azul Racer / Azul marino medianoche , Calzado de running NIKE Boy s Lunarstelos (GS) Midnight Navy / Black-antracita , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Porcelana Azul / Blanco / Asics Azul , Sanda Verra Teva para mujer Cuerda elástica , Zapatillas para correr NIKE Lunarcharge Essential para mujer Blanco / Blanco-Blak Zapatillas deportivas para mujer ASICS Ayami Valor Verde azulado / gris , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Blanco negro! Sandalias EVA Gizeh de mujer Birkenstock Rosa Eva , Bomba Nikole para mujer Via Spiga Patente Rouge OOFOS Unisex Ooriginal Sport Thong Flip Flop Negro Negro , CLARKS Sandalia plana Brizo Sammie para mujer Microfibra perforada azul marino Tierra Hera Floral azul , Zapato de entrenamiento WX824 para mujer New Balance Blanco , Calzado de senderismo Go Joy 15603 de Skechers para mujer Blanco negro , Cole Haan Studiogrand Knit Cross Strap Sneaker Black Knit Zapatillas clásicas Tretorn Nylite Bold Platform Classic para mujer Blanco vintage , Vans Auténtico (Digi Hula) Blanco clásico / Blanco verdadero , Vans Herren Authentic Core Classic Sneakers Armada , Zapatillas NIKE WMNS Lunar Force 1 Duckboot Mujer Casual Lifestyle Negro / Negro-blanco Zapatillas de golf Go Golf Blade de Skechers Performance para mujer Gris azulado , Sandalia Beau Slide Kate Spade New York para mujer Blanco , Calzado Aria Pointe 121 para mujer Capezio Pétalo rosa , JOE N JOYCE Paris Sandalias de suela suave SynSoft Cristal Burdeos Bota de moda de lona negra Winsted Dr. Martens Negro Sandalia de vestir Taimi de Naturalizer para mujer Cuero negro , Ovation Ladies Synergy Front Zip Paddock Boots Negro , Timberland PRO Hightower 8 \ Cuero de grano completo Kaffe NIKE Kids Jr. Tiempo Rio IV (FG) Calzado de fútbol para suelo firme Negro / Blanco / Láser Naranja / Volt , Mujer Alegria Belle Mary Jane Flat Catedral , Bota de lluvia Tinsley para mujer Sam Edelman Negro mate ,

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.

Newton Running El destino El de los hombres Running Amarillo Amarillo naranja 39c5fd4 - 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 Newton Running El destino El de los hombres Running Amarillo Amarillo naranja 39c5fd4 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 2106433
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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.