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.

New Balance Women s w990v4 Gray Running Shoe Women Gray 19293/ Castlerock 8b45f51




75% piel de cerdo de gamuza / 25% Malla
Hecho en Estados Unidos
Caucho único
ENCAP tecnología de suela intermedia proporciona soporte y durabilidad máxima
Encap anillo de poliuretano con etileno vinilo núcleo de acetato en talón
Doble densidad espuma de cuello
12 mm de caída: debido a varianzas creadas durante el desarrollo y fabricación procesos, todo referencias a 12 mm de caída son aproximados
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


New Balance Women s w990v4 Gray Running Shoe Women Gray 19293/ Castlerock 8b45f51 - 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

New Balance Women s w990v4 Gray Running Shoe Women Gray 19293/ Castlerock 8b45f51

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.


New Balance Women s w990v4 Gray Running Shoe Women Gray 19293/ Castlerock 8b45f51

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

ASICS Womens Gel-Fortitude 8 Gris medio / blanco / azul de porcelana Zapato deportivo para acolchado 635v2 New Balance para mujer Silver Mink , Zapatillas de running NIKE Air Max 2017 para mujer Té Berry Puerto Vino Lobo Gris 601 Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Blanco / Fresco Grey-metallic Silver , Mochila para correr Newton Women s Motion 7 Fuschia / Teal 9 - , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Phantom / imán , Under Armour Women s HOVR Sonic NC Negro (003) / Marfil , Zapatillas de running Under Armour Speedform Intake 2 para mujer Gris Zapatillas de running Saucony Women s Breakthru 3 Azul / menta Zapatillas de running adidas Ultraboost W para mujer Gray Five / Carbon / Ash Pearl Calzado de running para mujer New Balance Cypher Castlerock / White , Zapatillas de running NIKE Lunar Skyelux para mujer Golpe caliente / Marrón nocturno Saucony Women s Ride 10 Running Shoe Niebla / Rojo Under Armour Threadborne Fortis para mujer Gris Nublado (103) / Blanco Zapatillas de entrenamiento NIKE Metcon Dsx Flyknit para mujer Negro / Cool Gray-hyper Punch , Zapatillas para caminar Skechers Go Run-Mojo para mujer Negro , Zapatillas para correr Pacev2 para mujer New Balance Silver Mink / Black Brooks Women s Dyad 9 Primer gris / Capri Breeze / Plata , Zapatillas de running adidas Cloudfoam Super Flex W para mujer Misterio Azul / Blanco / Naranja Brillante Zapatillas para correr Nike Zoom All Out Flyknit para mujer Blanco / Negro - Pure Platinum Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / Summit White-lagoon Pulse Zapato deportivo ASICS Gel-Kayano 24 para mujer Gris medio / negro / Atlantis , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Cool Grey / Court Purple / Dark Grey , Zapatillas de running Pure Cloudfoam de adidas para mujer Blanco / Blanco / Negro , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Lobo Gris Puro / Negro-puro Platino-caliente , Zapatillas de running NIKE Mens Air Zoom Structure 19 Luz retro / blanco-verde azulado , Zapatillas de running Reebok Women s Supreme 3.0 MT Nube gris / Caribe trullo / polvo de asteroides , NIKE Air Max 90 Womens Style: 325213 Mujeres 325213-048 Lobo gris / blanco-gris oscuro ,
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 Sandalia deportiva Alise Les Filles para mujer Dusty Lilac Nappa , Sandalia de susurro de mujer Keen Poseidon / Blue Danube New Balance Women s Wlazrhe Púrpura Zapatillas náuticas para mujer Saucony Originals Negro / Plata Zapatillas de running NIKE Lunar Skyelux para mujer Blanco / puro Platinum-off blanco Skechers Women s Go 4-14918 Calzado para caminar Blanco negro , Sandalias mujer La Plume Jen Armada , Zapato con cordones CLARKS Wave Kick para mujer Malla negra , Taipán de mujer de David Tate Verona Champagne Lamb Bernie Mev Women s Gummies Victoria Calzado para caminar Pavo real , Sandalia Alegria Airie Slide para mujer Goto Black , Sam Edelman Womens Barry Mulberry Pink Vaquero Saddle Leather Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Plata , Zapatilla para caminar Slip-On Go Flex de Skechers Performance para mujer Púrpura , Sandalias de mujer Wolky Azul acero , Zapatillas de deporte cruzadas WX40V1 de New Balance para mujer Blanco negro Calzado de trabajo resistente a los resbalones de Fila, Memory Comfort Trainer Blanco / Blanco / Plata Metálica Zapatillas de entrenamiento adidas Performance Sonic Allegra para mujer Solar Rojo / Plateado / Blanco , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco / Lt Blue Mesh Clásicos de base Slip-on (Tm) para hombres de Vans Negro Negro , Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sam Edelman Women s Bay Slide Sandal Marfil negro / negro Sandalia Evelia Slide de Bandolino para mujer Tela dorada , Therafit Eva Women s cuero ajustable correa Slip On Sandal para Fascitis plantar / Dolor en el pie Broncearse , Sandalias clásicas para mujer del Dr. Scholl s Shoes Black Global Stripe Ringor Dynasty Turf Shoe- Lanzamiento Azul marino / blanco , Zapatilla Thong LUKS de mujer Darlene Micro Chenille Thong Blanco , LifeStride Women s Lively 2 Pump Soft Taupe , UGG Mujer Alena mocasín castaña PIC / PAY Benji - Cordones de las mujeres Cremallera vintage Arranque - Media altura Cuero de gamuza envuelto Bloquear el talón del tobillo Bootie Ante 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.

New Balance Women s w990v4 Gray Running Shoe Women Gray 19293/ Castlerock 8b45f51 - 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 New Balance Women s w990v4 Gray Running Shoe Women Gray 19293/ Castlerock 8b45f51 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 259651
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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.