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 Ryana Marilyn 2962 Slide para para mujer Sandalia Blanco negro a93dcc0




Hecho por el hombre
Importado
Sintético suela
sintético doble ajustable correa superior con impresión divertida y elegante.
Almohada EVA cepillado suave con forma de corazón inserción de poliuretano.
Deporte Media suela de EVA Lite
Sport Suela de Lite EVA con flocado.
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 Ryana Marilyn 2962 Slide para para mujer Sandalia Blanco negro a93dcc0 - 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 Ryana Marilyn 2962 Slide para para mujer Sandalia Blanco negro a93dcc0

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 Ryana Marilyn 2962 Slide para para mujer Sandalia Blanco negro a93dcc0

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

Sandalia de mujer Kelley de Eastland Aceituna , Sandalia para mujer FRYE Ally Criss Cross Slide Secoya , Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero negro Loeffler Randall - Sandalia de nudo con tachuelas Lucia, de mujer Índigo Sandalias planas de piscina Walking Cradles para mujer Negro , BareTraps Women s Beverly Slide Sandal Gris , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Stuart Weitzman Sandalia Rosita para mujer Ballet Taupe , Sandalia Slide de Elina de Franco Sarto para mujeres Azul mediterráneo , BareTraps Women s Beverly Slide Sandal Negro Sandalia Beau Slide Kate Spade New York para mujer Oro , Sandalia Trotters Ellie Wedge Slide para mujer Goldwash , Sandalia Mason para mujer Naturalizer Slide Cuero metalizado lila Sandalia Trotters Ellie Wedge Slide para mujer Goldwash , Sandalia Aerosoles Midday Women Slide Combo verde claro , Sandalia de la diapositiva Indi Kate Spade New York para mujer Neopreno azul Chanclas Deportivas Cooper Slide de Marc Jacobs para mujer Negro Sandalia de la diapositiva de Stella de las mujeres de Rachel Zoe Plata Sandalia Rockdale Lanea Woven Slide para mujer Latte tejido , Sandalias planas para piscina Walking Cradles para mujer Abrigo de cuero / corcho estampado serpiente natural Sandalia del vestido de la inversión de LifeStride de las mujeres Plata Sandalias planas de mujer de Easy Street Coñac Sam Edelman Sandalia Bernice Slide para mujer Cuero negro , Sandalia Shanny Slide de Jessica Simpson para mujer Rojo / Multi Sandalia Aerosoles Midday Women Slide Tela blanca , FitFlop Glitterball Slide sandalias de mujer zapatos Negro , Sandalias Jerusalem Sandalia Arad Slide marrón , Sandalia Gilda de Onex para mujer Beige Eldorado , Sandalia para mujer KAANAS con plataforma tejida Akumal Azul
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 para correr Mizuno Women s Wave Enigma 6 Quarry-diva rosa-negro Sandalias Bali KEEN para mujer Negro / Imán , Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad , Sandalia deportiva ECCO Cruise para mujer Negro Negro , Sandalia deportiva Tirra para mujer Teva Gris oscuro Zapatillas de running Adidas Performance Ultra s Boost M para hombre Colegial azul marino / colegiado azul marino / azul marino nocturno Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres Negro Zapatillas de running New Balance 990V3 para mujer Negro Zapatillas de running Kineta Relay para mujer Saucony Carbón , Trotters Mujer Simmy Mary Jane Flat Negro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Negro Negro Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul humo / azul oscuro / Cantalupo Calzado para caminar WW812 para mujer New Balance Negro Zapatillas de running Golden Spike de Altra para hombre Negro / Diva Pink Vans Unisex Old Skool Zapatillas clásicas de skate (1966) azul / blanco / rojo Earth Womens Allegro Ballet Flat Pale Grey Metallic Printed Suede Zapatillas de tenis ASICS Gel-Solution Speed ​​3 para mujer Negro / Plata / Rosa Diva , Converse Chuck Taylor All Star Core Hola rojo Sandalia Dolce Vita Delana Slide para mujer Cuero negro Sandalia Raelyn Wedge Slide de Easy Street para mujer Tan Pebble Zapatillas Ivana de Spring Step para mujer Beige , Bota Deia para mujer de TOMS Drizzle Grey Leather , Zapatillas Sidi Drako SRS Push - Mujer Negro / azul claro , Zapatillas de mujer de Overland Emma Classic Sheepskin Arrendajo azul , Bota Deia para mujer de TOMS Gamuza / lana negra , Trotters para mujer Dea Slingback Pump Negro Negro , CLARKS Womens Un Haven Lace Nubuck azul marino TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Negro Trotters Liz Loafer para mujer Light Tan Tonal ENMAYER Mujeres Moda retro negra Mocasines con punta redonda Resbalón en piso al aire libre Zapatillas con peludo y hebilla Negro # 4

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 Ryana Marilyn 2962 Slide para para mujer Sandalia Blanco negro a93dcc0 - 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 Ryana Marilyn 2962 Slide para para mujer Sandalia Blanco negro a93dcc0 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2741367
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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.