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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Ariaprene
Importado
Caucho único
Eje mide aproximadamente Parte superior del arco
Sin cordones cross-trainer alto-superior con transpirable inserciones de malla y Corriendo a través de la arriba para un seguro ajuste
acolchado lengua y cuello con lazos de tracción
Agarre suela de tracción
Pivot puntos en la suela , talón y dedo del pie para 360 grados de movimiento
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,

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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 running NIKE Air Zoom Pegasus 34 para mujer Negro / Tinta / Provenza Púrpura / Plata Metálica NIKE Men s Lunar Fingertrap Cross Trainer Negro Negro Zapatillas para correr Under Armour UA Flow Criss Cross para mujer Negro / Blanco / Blanco , Zapatillas de running Kinvara 8 Saucony para mujer Azul Sandalia deportiva Tirra para mujer Teva Atlántico Norte , Zapatillas de trail running WT10v1 Minimus para mujer New Balance Azul negro Zapatillas de running NIKE Air Max Zero SI para mujer Negro / Ligero Bone-cool Gris-total Crimson Zapatillas de running ONEMIX para hombre y mujer Zapatillas de deporte para correr negro rojo , Calzado vertical Dynafit felino - Mujer Fucsia Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Phantom / imán , OLUKAI Honoka a Sandals - Mujer Negro Negro Sandalia Patrese de estilo suave para mujer Patente blanca Adidas Mujeres Aleki X Cross Tainer Amarillo / Amarillo / Aqua Zapatillas Superga 2750 Cotu para mujer Grey Silver , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Baya , Entrenador Ryka Eva Nrg Cross para mujeres Azul blanco Vans Unisex Old Skool Zapatillas clásicas de skate Denim 2 Tone Blue True White Vans Auténtico (Neón) Orange Pop , Converse Chuck Taylor All Star Core Hola Azúcar en bruto , Sandalia de la diapositiva de la boya de las mujeres de Donald J Pliner Armada Sam Edelman Women s Bay Slide Sandal Cuero de silla Zapatilla de zapatilla de zapatilla 100% auténtica de lujo de las señoras de Snugrugs con suela suave castaña Calvin Klein Women s Gayle Pump Arena del desierto , Zapatillas Shimano SHRP3W Road Performance Shoe para mujer Negro Me Too Arena Sandalia para mujer New Luggage Goat Spore Revitalign Oceanside - Zapatillas de confort para mujer Negro Vionic Womens Hadley Estaño Calcetines de tacón alto para mujer, botines de media pierna con cordones, botines de tacón medio, botines de tacón de ante elástico para mujeres Terciopelo azul marino-6.5cm , Propet Women s Tour Walker Strap Sneaker Negro Propet Women s Tour Walker Strap Sneaker Blanco

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 B01HSH7MCG de 20000 entrenamiento cruzado Fierce PUMA para mujer Fierce Core Puma White-puma Black 03f9502 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.

Fitness y entrenamiento cruzado
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 4412576
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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.