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.

Zapatillas Neutral de 10 running Brooks Womens Ghost 10 Zapatillas Neutral acolchadas Pavo real negro/ rosa/ Living Coral 7acf78b




tela
Importado
Caucho único
Piérdete en tus pensamientos con un suave y bien amortiguado montar en el Brooks & reg; Fantasma 10.
Predecesor: Ghost 9.
Soporte Tipo: Neutro.
Amortiguación: Alta energización Amortiguación.
Superficie: La carretera. 12 mm. a alto soporte de arco. Malla de ingeniería superior proporciona transpirabilidad y estiramiento necesario Ligero sintético superposiciones. cierre con cordones. Lengua de peluche y collar. forro para un gran sensación en el calzado. plantilla de espuma. El ADN es una mezcla de amigable con el planeta BioMoGo y receptivo Brooks DNA, y tiene un enfoque de género diseño de amortiguación, haciendo una dinámica paseo. Choque segmentado Pad acomoda cualquier pie de aterrizaje
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


Zapatillas Neutral de 10 running Brooks Womens Ghost 10 Zapatillas Neutral acolchadas Pavo real negro/ rosa/ Living Coral 7acf78b - 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

Zapatillas Neutral de 10 running Brooks Womens Ghost 10 Zapatillas Neutral acolchadas Pavo real negro/ rosa/ Living Coral 7acf78b

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.


Zapatillas Neutral de 10 running Brooks Womens Ghost 10 Zapatillas Neutral acolchadas Pavo real negro/ rosa/ Living Coral 7acf78b

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 Saucony Women s Ride 10 Running Shoe Gunmetal / rosa Sanda Verra Teva para mujer Negro , Zapato de tenis ASICS Gel-Dedicate 5 para mujer Blanco / Porcelana azul / azul índigo , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Colegio Navy / Burdeos , Converse Chuck Taylor All Star Seasonal Color Hola Naranja solar , Zapatillas NIKE Women s Revolution 3 para correr Pure Platinum / Concord / Violeta persa / Gamma Blue Zapatillas de running Pure Cloudfoam de adidas para mujer Carbono / carbono , Zapatillas de running Saucony Women s Excursion Tr11 GTX Gris morado , Zapatillas de running NIKE Zoom All Out Low para hombre Azul , Zapatos ligeros de agua de deslizamiento Quick-Adult Mesh Aqua para mujer Pooluly Gris 306 Sandalia de vestir Townsend de Eastland para mujer Negro Pantalón largo clásico NIKE Golf para mujer Tech Essentials Rojo / Escuadrón Azul OLUKAI Womens Paniolo Slide Natural / Natural Stuart Weitzman Womens Carmina Noir Gloss , Sam Edelman Sandalia Dariel para mujer Oro fundido / Cuero metalizado , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Hueso , Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Blanco invierno , Zapato Cross-Trainer X Conviction Women de ASICS Aqua Splash / Diva Blue / White , Zapatillas de tenis con bota forrada de invierno Nepal de Vivobarefoot para mujer Marron oscuro , Vans Herren Authentic Core Classic Sneakers (Bum de goma) True White / True White Sandalia Rockdale Lanea Woven Slide para mujer Latte tejido , Sandalia de cuña Ollie Wedge Chinese Laundry para mujer Rich Brown Smooth Jack Rogers Women s Lauren Sandal Negro , Sandalias Dior J Pliner Dacia para mujer Negro Sandalia de mujer Frisco Slide de Franco Sarto Mezclilla Sandalia para mujer Merrell, 1SIX8 Linna AC + Slide Monumento Sam Edelman Women s Slide Sandal de mujer Azul / Multi Zapatillas Cross-Trainer de Enzo WN para mujer de PUMA Sombra silenciosa , Bella Vita Womens Tex-Italia Grey Suede , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Navy Classic ,

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.

Zapatillas Neutral de 10 running Brooks Womens Ghost 10 Zapatillas Neutral acolchadas Pavo real negro/ rosa/ Living Coral 7acf78b - 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 Zapatillas Neutral de 10 running Brooks Womens Ghost 10 Zapatillas Neutral acolchadas Pavo real negro/ rosa/ Living Coral 7acf78b 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

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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.