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.

Sandalia Savannah gris de Savannah Ryka para mujer Negro mujer/ gris helada 8484dd7




Textil
Importado
Sintético suela
Plataforma mide aproximadamente 0.75 pulgadas
4.9 oz
sintético y tejido textil superior con tres puntos de ajuste.
Anatómicamente diseñado Q-contorno Plantilla con profundidad talón para extra estabilidad para reducir exceso de pronación y arco aumentado contacto.
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


Sandalia Savannah gris de Savannah Ryka para mujer Negro mujer/ gris helada 8484dd7 - 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

Sandalia Savannah gris de Savannah Ryka para mujer Negro mujer/ gris helada 8484dd7

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.


Sandalia Savannah gris de Savannah Ryka para mujer Negro mujer/ gris helada 8484dd7

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

KEEN Hombres Newport H2 Sandal Negro Chaco Mujer ZX1 Classic Sport Sandalia Origami Black , Sandalia deportiva Bondi de Donald J Pliner para mujer Azul , Sandalias KEEN Venice II-W para mujer Bleacher / Raven Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7643 Negro Chaco Zcloud 2 Sport Sandal para hombre Pirámide Verde , Sandalias KEEN Mujer Whisper-w Vapor / acero gris , Chaco Women s Z2 Classic Athletic Sandal Eclipse estático Wolky Comfort Jewel 30800 Cuero azul oscuro , Sandalia de mujer Drew Alana Verde / Fucsia Sandalia Savannah de Ryka para mujer Coral / gris , OluKai Ohana Tropical / Hielo Sandalia para mujer Merrell, alrededor de la ciudad Negro Chaco Women s Zx3 Classic Athletic Sandal Lluvia de frambuesa OluKai Ohana Piña / Negro , Earth Origins Sophie Azul marino , Pale Jade Sandalia Teva Mujer Terra-Float Livia Lux Negro , Sandalia KEEN Women s Venice H2 Negro / Gris neutro , MBT Lila 6 Sport Sandalia W Azul índigo Sandalia Uneek O2-w de mujer KEEN Blanco / oro de la cosecha , Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose , Sandalia Shannon del Dr. Comfort Women rojo Sandalias KEEN Womenport Newport Hydro-W Higo de uva / higo de verano Sandalia deportiva Kana para mujer ECCO Negro Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco Sandalia Drew Element para mujer Nubuck negro , Teva Terra Sandal para hombre Madang Blue Sandalia deportiva Nolita para mujer Anne Klein Tela 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 Sandalias OLKAKAI Ohana - Hombre Poi / Poi Converse Chuck Taylor All Star Seasonal Color Hola Azul Sandalia de deslizamiento ultra Comfort NIKE para mujer Pure Platinum / Wolf Gray , Zapatillas de running adidas Women s Rockadia W Trail Gris / negro / rosa , Zapatos OLUKAI Pehuea - Mujer Pale Gray / Carboncillo , Hush Puppies Mujeres Power Walker Ii Loafer cuero azul marino Tierra mujeres Tayberry Oro Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul Mediano / Negro-aluminio-electro Verde Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Phantom / Black / White Munsell Sandalias Harper Huarache de SoftWalk para mujer Negro / peltre , Sandalia de vestir Desi Easy Street para mujer Nude Patent / Nude Zapatillas para caminar ProFit TravelFit Slide para mujer Negro rojo , Zapatillas deportivas Superga Womens 2750 Cotu Canvas Blanco negro , Zapatillas para mujer Del Rey TOMS Black.chambray Zapatillas de entrenamiento cruzado Froma FM de PUMA Tazon 6 para hombre Quiet Shade-alto riesgo rojo , Zapatillas de deporte Fenty x Trainer Mid de PUMA para mujer Puma Black / Puma White , ALMA NATURAL Sandalia Adalia Slide para mujer Caqui Sandalia Charlotte Slide de Naturalizer para mujer Alabastro Sandalias planas de mujer de Easy Street Blanco , Sandalia de raso de satén Feeltheluv de mujer de las nueve oeste Satén negro , CAPEZIO Mujer cambre (1128W) -Petal Pink Pétalo rosa Sandalia de vestir Taimi de Naturalizer para mujer Negro , Zapatillas de ciclismo X-Alp Launch II Pearl iZUMi para mujer Negro / gris oscuro , TOMS Womens Desert Wedge Bootie Gamuza de arpillera negra , Trotters Lucy Ballet para mujer Blanquecino , Botín de mujer Dominicaa ALDO de ALDO Caqui Bota de sarga Rockel Total Motion para mujer Rockport Nueva Taupe Shiny Python , Bernie Mev Women s Brighten Fabric Casual Estaño , Clásicos de base Slip-on (Tm) para hombres de Vans Carbón , CLARKS Mujeres Sillian Bella Mary Jane Flat Nubuck sintético de arena

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.

Sandalia Savannah gris de Savannah Ryka para mujer Negro mujer/ gris helada 8484dd7 - 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 Sandalia Savannah gris de Savannah Ryka para mujer Negro mujer/ gris helada 8484dd7 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.

Sandalias y toboganes deportivos
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 640520
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11310431
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17335065
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.