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.

CLARKS Zapatillas 3769 Tri Tri Etch para Cuero mujer Cuero negro 829f453




Cuero
Importado
Suela de goma
Parte superior de cuero
Forro de piel
Parte superior perforada
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


CLARKS Zapatillas 3769 Tri Tri Etch para Cuero mujer Cuero negro 829f453 - 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

CLARKS Zapatillas 3769 Tri Tri Etch para Cuero mujer Cuero negro 829f453

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.


CLARKS Zapatillas 3769 Tri Tri Etch para Cuero mujer Cuero negro 829f453

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 deporte de mujer Alegria Essence Blanco invierno Hush Puppies Cypress Knit de mujer con cordones Royal Navy Textile , Propet Walkers Evolution Oxford lavable de las mujeres Mora azul Holgazán Anna Slip-On de Eastland Women Negro , Chelsea Crew MADISON Bomba de terciopelo de terciopelo medio Oxford, mostaza Sadie Oxford de las mujeres de Eastland Arena , U-lite mujer perforada con cordones Wingtip cuero plano Oxford zapatos brogues vintage Azul , Propet Travelactive Slip-On Oxford para mujeres Heather negro / blanco , CLARKS Mujer Phenia Strand Lace Up Bronce / ante marrón Drew Shoe Women s Parade II Oxford Ternero negro , Indigo Rd. Gris oscuro Easy Spirit ON CUE Oxford para mujeres Gris pardo , Zapatillas de moda Native Unisex Apollo Moc. Snapper Red / Cabo Blue / Shell White , Nueve siete cuero de gamuza de las mujeres del dedo del pie redondo tacón grueso Comfort trabajo hecho a mano Botines Negro Mephisto Mens Cooper Gamuza de roble oscuro , Capezio Cadence para mujer Oxford Caramelo Cuero de Windup para mujer Caterpillar Lace Up Fashion Oxford Negro / Tostado , Klogs USA Moro Sneakers Frost Grey , Zapatillas de deporte lavables Walker Propet para mujer Gris Zapatillas de deporte Wisteria para mujer Spring Step Professional Blanco Easy Spirit Cacia Mary Jane para mujer Negro Drew Shoe Heather Synthetic Oxfords para mujer Cuero negro , Mephisto Women s Yliane Oxford Gris Claro Nubuck ZXD Round Toe High Top Sports Sneakers Lace Up Fashion Skateboard Pisos tachonado Zapatillas Zapatos Azul Sadie Oxford de las mujeres de Eastland Azul claro , Mephisto Women s Hawai Oxford Silver Monaco / Magic Cole Haan Women s 2. Zerogrand Stitchlite Ox Oxford Ambrosia Knit-optic White Shellys London Women s Hankie Oxford Flat Tela escocesa gris Mordenmiss Women s Laciness Madre Loafer Lace Up Mocasines de cuero Zapatos planos Negro
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 ASICS Gel-Nimbus 19 para mujer Negro / Onyx / Plata , Skechers Women s Go Run Forza 3 Rosado , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-corredor Rosa-gamma Azul , BZees Sandalia con cuña de beso para mujer Negro , Zapatillas de running NIKE Unisex Flyknit Racer Legion Blue / Black-gamma Blue NIKE W Zoom All Out Low 2 Womens Aj0036-200 Dusty Peach / Mtlc Red Bronze Zapatillas náuticas para mujer Saucony Originals Broncearse Zapatillas de running adidas Outdoor Tracerocker Trail para mujer Ice Purple / Ch Solid Gray / Easy Green , Zapatillas de running Kinvara 8 Saucony para mujer Azul marino Púrpura , Zapato de agua Tetrex Rapid Crest para mujer Merrell Carbón , La sandalia Beglad de mujer FLEXX Rose Gold Crackle , Zapatillas para caminar Lite-Agile Go Step de Skechers Performance para mujer Malla de carbón / verde azulado Zapatillas de running adidas Alphabounce Em W para mujer Colegial azul marino / Utility Black / Mystery Blue , Sandalias de tacón Gretchen de mujer de Jack Rogers Blanco , Chaco Zcloud 2 Sport Sandal para hombre Varsity Sun , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Berenjena / Sal marina Zapatillas de paseo Skechers Go Walk Lite-Luna para mujer Armada , Trotters Mujer Simmy Mary Jane Flat Grafito , Reebok Crossfit Nano 7.0 Track Shoe para mujer Azul / negro / plata / blanco Salvatore Ferragamo Mujer Varina Oxford Blue Vans Unisex Old Skool Zapatillas clásicas de skate Blue Ashes / Parisian Night , Zapatilla de running Titan II para hombre Diadora Khaki / Estate Blue Calzado deportivo DAWGS para mujer Golf Spirit Tan / blanco , Sandalia de la diapositiva Floella de la mujer de la marca de Lucky Picante BareTraps Women s Rebecca Slide Sandal Castaño Zapatillas de voleibol adidas Crazyflight X 2 de mujer Blanco / Plateado Metálico / Gris , Slipper Hailey de UGG para mujer castaña JOE N JOYCE Slippers Zuecos Zapatos Cuero sintético estrecho - Para hombre y mujer Azul oscuro , Mephisto Michaela Bootie para mujer Hazelnut Texas / Carla / Tobacco Bucksoft Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Amarillo 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.

CLARKS Zapatillas 3769 Tri Tri Etch para Cuero mujer Cuero negro 829f453 - 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 CLARKS Zapatillas 3769 Tri Tri Etch para Cuero mujer Cuero negro 829f453 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.

Oxfords
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 1472550
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2143822
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4979540
7613875
11597586
13543683
15138355
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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