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 de running 2017 NIKE RN 2017 para mujer gratis mujer 19762 Wolf Gray/ Off White-pure Platinum-black cb0f93c




malla
goma único
LIGERO TRANSPIRABILIDAD de capa única el material de punto es suave y transpirable para ayudarte a permanecer genial de tu primera milla a su último.
ADAPTADOR Ajuste dinámico FIT envolturas de tecnología tu pie por un seguro, bloqueado sensación. \\ u2013 fuerte, cables livianos \\ u2013 integrar con tus cordones para adicional soporte de arco.
FLEXIBLE CONFORT El peso ligero, Nike de perfil bajo Suela libre flexible con tu pie para un cómodo, paseo natural.
Mínimo diseño de lengua mejora transpirabilidad en parte superior del pie
Plantilla se ajusta a la forma de tu pie para una completa comodidad
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 de running 2017 NIKE RN 2017 para mujer gratis mujer 19762 Wolf Gray/ Off White-pure Platinum-black cb0f93c - 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 de running 2017 NIKE RN 2017 para mujer gratis mujer 19762 Wolf Gray/ Off White-pure Platinum-black cb0f93c

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 de running 2017 NIKE RN 2017 para mujer gratis mujer 19762 Wolf Gray/ Off White-pure Platinum-black cb0f93c

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 Laguna Zoot para mujer Aguamarina / maracuyá / gris Sandalia de vestir Tillman de SoftWalk para mujer rojo Zapatillas adidas Originals Adizero Defiant Bounce Tennis, de adidas Originals Leyenda Ink / Shock Pink / White Zapatillas de running La Sportiva para mujer Trailido Bushido Fiordo , Calzado de entrenamiento cruzado TR 6 NIKE para mujeres en temporada Gris oscuro / Plateado metalizado / Rosa forzado / Gris claro , UB-ONEMIX Calzado deportivo de malla ligera transpirable al aire libre para caminar Menta / amarillo Zapatos Salomon Sense Pro Max Running Trail para mujer Virtual Pink, Black, Enamel Blue Sandalias con tachuelas de cuero para mujer de Royou Yiuoer Sandalias con tira en T Rosado Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Glicina / Rosa Violeta , Sandalia LifeStride para mujer Eden Blanco , NIKE W Air Max 270 Mujeres Ah6789-103 Blanco / Corte Morado-menta-negro Taipán de mujer de David Tate Verona Champagne Lamb , Zapatilla para caminar nebulosa D de Geox para mujer Mar de agua Cobb Hill Rockport Women s Angelina Dress Pump Negro / Multi , Skechers Women s Breathe Easy Sweet Jam Sneaker Natural , Zapato de entrenamiento cruzado RYKA Women s Influence Negro / rosa atómico / azul real / gris forja , Reebok Crossfit Nano 7.0 Track Shoe para mujer Negro / Ceniza / Goma , Zapatillas de entrenamiento cruzado Vibram para mujer V-Train Gold s Gym Negro / Gris / Azul Reflejo Zapatillas de deporte WF616 para mujer New Balance Negro Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker rojo , Vince Camuto Sandalias Ellanna Slide para mujer Brillante plata Sandalia Anika de Lucky Brand para mujer Picante DC Youth Rebound Skate Shoes Negro / Fucsia KAANAS Sandalias planas de alpargatas tejidas de Martinica para mujer Natural-1 , Zapatillas de salón de baile muy finas para mujer EK1613LED Edición limitada con 3 \ Dark Tan Satin Twisted X Ladies Brown / Purple Zapatos casuales marrón , BEARPAW Mujeres Ashlyn Azul cobalto CLARKS Mujer Flores Dahlia Tobillo Botín Grey Suede / Leather Combi , Drew Shoe Heather Synthetic Oxfords para mujer brandy , TOMS Clásicos de mujer Negro sobre negro

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 de running 2017 NIKE RN 2017 para mujer gratis mujer 19762 Wolf Gray/ Off White-pure Platinum-black cb0f93c - 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 de running 2017 NIKE RN 2017 para mujer gratis mujer 19762 Wolf Gray/ Off White-pure Platinum-black cb0f93c 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 2953026
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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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