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.

NIKE Boys Gris Star Runner (GS) Boys Runner Gris 32c2f09




Ideal para niños
Durable
Ligero
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


NIKE Boys Gris Star Runner (GS) Boys Runner Gris 32c2f09 - 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

NIKE Boys Gris Star Runner (GS) Boys Runner Gris 32c2f09

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.


NIKE Boys Gris Star Runner (GS) Boys Runner Gris 32c2f09

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 running NIKE Juvenate para mujer Orquídea / Orquídea / Midnight Navy White , NIKE Women s Tanjun Negro / Metálico Rojo Bronce Zapatillas Nike Womens Air Max 2015 Negro / Blanco / Cumbre Blanco / Refleja Plata Converse Chuck Taylor All Star Seasonal Color Hola Blanco / Vapor Rosa / Blanco Converse Chuck Taylor All Star Seasonal Color Hola Carbón , Zapatillas de baloncesto Jordan Kids Jumpman Team II Negro / infrarrojo 23-cool gris-infrarrojo 23 , Brooks Womens PureGrit 6 Negro / Ebony / Diva Pink , Converse Chuck Taylor All Star Seasonal Color Hola Ratón NIKE W Zoom All Out Low 2 Womens Aj0036-200 Dusty Peach / Mtlc Red Bronze , Zapatillas de running Spira Phoenix para mujer con resortes Royal / Black / White , Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado , Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Mono negro , Converse Chuck Taylor All Star Seasonal Color Hola Azul / blanco / rojo NIKE Mujeres Air Zoom Pegasus 33 Platino / Negro / Gris / Rosa explosión , Zapatillas de running NIKE Unisex Flyknit Racer Negro / gris oscuro / blanco Calzado de running rápido para mujer Reebok, tejido flexible Cool Shadow / Porcelain / Blue Lagoon / White / Blue Zapatillas de running Adizero Prime Sp de adidas Originals Núcleo Negro, Ftwr Blanco, Naranja Calzado de running rápido para mujer Reebok, tejido flexible Blanco / gris ceniza / negro / polvo del desierto Skechers Women s Go Ride 7 Running Shoe Blanco negro , Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco negro , Zapato de running ASICS GEL-Kayano 22 para mujer Onyx / Plata / Carbón de leña , Sandalia deportiva NIKE Kawa Slide para hombre Lobo gris / antracita / gris lobo , Zapatillas Nike Women s Flex Trainer 5 Antracita / Blanco / Gris Oscuro / Hipervioleta Zapatillas de running NIKE Roshe Two para mujer Blanco / Blanco-Puro Platino , Converse Chuck Taylor All Star Seasonal Color Hola Brillo de sol , Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico Zapatillas NIKE Women s Revolution 3 para correr Negro / Metálico Gris Oscuro / Antracita , Zapato de senderismo KEEN para mujer Versago Dahlia roja / coral profundo , Zapatillas de running adidas Alphabounce Cr W, mujer Core Black / Core Black / Core Black
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 W1540V2 New Balance para mujer Blanco azul Zapatillas NIKE Air Max Thea Print Blanco / Blanco / Morado Oscuro para correr 6.5 Mujeres EE. UU. - Sandalia para mujer NIKS Benassi Solarsoft Slide Rosa Fuego / Prisma Rosa Chaco Mujer Zvolv Sport Sandal Lluvia , Zapato de running GEL-Excite 3 para mujer ASICS Rosa fuerte / Blanco / Amarillo intermitente Calzado deportivo New Balance Vazee Urge v1 para mujer Púrpura / amarillo , Earth Origins Selby Azul marino Nike Hombres Air Zoom Pegasus 33 Rage Green White Black 301 OLUKAI Mujer Kia I Wahine Negro Negro , Sandalia plana Nanci de Naturalizer para mujer Negro , Sandalia de cuña Karmen de mujer Alegria Pewter Way Drew Shoe La alegría de las mujeres Azul cielo ante , Sandalias de tacón de soñador Easy Street para mujer Pewter Metallic , Zapato Cross-Trainer X Conviction Women de ASICS Aqua Splash / Diva Blue / White Skechers Women s Walk Walk Joy-Invite Sneaker Azul marino / Multi , Zapato de entrenamiento cruzado RYKA Women s Influence Acero Gris / Cromo Plata / Azul Buzo / Zuma Rosa Zapatillas de skate DC SE Pure High-Top TX SE de mujer Negro , Charles by Charles David Sandalia para mujer Nicki Slide Negro Sandalia de diapositivas de ocio para mujer de Seychelles Blanco , Sandalia FitFlop de cuero con diapositivas en piel de mujer Delta Blanco Urbano La Plume Diapositiva de estiramiento de la mujer Blanco , Bota de moda Emma Tall BEARPAW para mujer Esmeralda , Bota de mujer Shellys London Erin Hiker Floral negro Sandalia Birkenstock de cuero Florida para mujer Negro Bota de tierra para mujeres de la tierra Corteza de cuero de grano completo , Botín Simba para mujer Freebird Negro , EMU Australia Emu Australia Emu Australia Emu Australia Negro Botín de tobillo Blake de Sam Edelman para mujer Woodland Brown Terciopelo en piel de ante Zapato femenino Keli de Alegria Romance azul Arcopedico Womens Lolita Slip-On Azul

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.

NIKE Boys Gris Star Runner (GS) Boys Runner Gris 32c2f09 - 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 NIKE Boys Gris Star Runner (GS) Boys Runner Gris 32c2f09 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.

Corriendo
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.