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 Merrell para Sport mujer Merrell Terran Ari Post Negro Sport Negro 5ed7f20




Cuero y Textil
Importado
Sintético suela
Plataforma mide aproximadamente 0.5 pulgadas pulgadas pulgadas
Cuero Y parte superior tejida
EVA Plantilla para cojín
Merrell Air Cushion en el talón absorbe choque y agrega estabilidad
Neopreno para comodidad acolchada
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 Merrell para Sport mujer Merrell Terran Ari Post Negro Sport Negro 5ed7f20 - 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 Merrell para Sport mujer Merrell Terran Ari Post Negro Sport Negro 5ed7f20

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 Merrell para Sport mujer Merrell Terran Ari Post Negro Sport Negro 5ed7f20

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

Sandalias Newport H2 de KEEN para mujer Cáñamo / azul provincial , OluKai Ohana Sour Tart / Ice Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Black Sport MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Amarillo , KEEN Sandalias Newport Eco-W para mujer Negro / Imán , Sandalia Yucatan ECCO Mujer Coral Blush PUMA Women s Leadcat AO Lthr WNS, Negro / Blanco Blanco negro , Sandalia atlética de Rka Women s Glance Azul / verde azulado , Calzado casual Slip-On de Bernie Mev para mujer Estaño Sandalia deportiva Nolita para mujer Anne Klein Tela Coral , Sandalias Newport H2 de KEEN para mujer Cáñamo / Gorra blanca Sanda Verra Teva para mujer Granadina Chaco Mujer Zcloud Sport Sandal Noche Emplumada , Chaco Mujer ZX1 Classic Sport Sandalia Paloma mandarina , Chaco Women s Z2 Classic Athletic Sandal Cañón , Chaco Women s ZX2 Classic Athletic Sandal Negro Teva Terra Sandal para hombre Cross Terra Black Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva , Sandalia Helen de Propet para mujer Gunsmoke Chaco Women s Z2 Classic Athletic Sandal Bungee maya , Sandalia de mujer Drew Alana Pewter-bronce , Sandalia Savannah de Ryka para mujer Coral / gris , Sandalia Yucatan ECCO Mujer Orquídea , Sandalia Yucatan ECCO Mujer Irritante , BZees Sandalia con cuña de beso para mujer Blanco Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Pimenton BZees por Naturalizer Dream Negro-floral Sandalia Savannah de Ryka para mujer Coral / gris CLARKS Sandalia Saylie Jade para mujer Armada ,
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 Calzado de running con punta de material compuesto, resistente al deslizamiento, Meiera 2, de Fila, para mujeres Monumento / Castlerock / Safety Yellow Sandalia universal W Sanborn de Teva para mujer lirio de tigre Zapato NIKE Roshe One femenino Deep Garnet / Bright Crimson-pure Platinum NIKE Air Huarache Run Ultra Blanco / Blanco-Blanco-Negro (Mujer) Zapatillas de running NIKE Air Max Thea Ultra PRM para mujer Negro / negro-cool gris Zapato de agua Evo de mujer Teva Negro Zapatillas para caminar ProFit TravelFit Slide para mujer Oro / negro , Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple , Skechers Performance Women s Go Run 400 Obstruct Gris / azul NIKE Women s Revolution 4 Running Shoe Gunsmoke / Ocean Bliss / Dark Grey , Nike Hombres Air Zoom Pegasus 33 Lobo gris / negro / gris oscuro / azul Vionic Women s de Orthaheel, Shore Slide Sandal Estaño Prop & eacute; t Olivia Oxford para mujeres Bronco Brown , Zapatillas para caminar Ryka Women s Devo Plus 2 Negro / púrpura Calzado de senderismo Skechers Go s 15601 para mujer Negro , Botín para mujer Pleaser Asp1018 / Bpu / M Black Faux Leather / Black Matte , Entrenador campeón Gusto Cross Trainer Malla azul claro K-Swiss Tubos de mujer Infinity CMF Cross Trainer Baby Blue / White Zapatillas de running Golden Spike de Altra para hombre Lima / rosa , ASICS para mujer Gel-Fit Yui 2 SE Negro / Limelight / Plata Vans Herren Authentic Core Classic Sneakers Negro Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre True White Vans Unisex Old Skool Zapatillas clásicas de skate (1966) azul / blanco / rojo , Sandalia de girasol Aerosoles para mujer Negro Sandalia Rockdale Lanea Woven Slide para mujer Tejido azul marino , Prop & eacute; t Womens Aurora Slide Rubí Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy Bota de mujer FRYE Melissa Button Short 2 Negro Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Verde , Steve Madden Women s Jazie Oxford Nubuck 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.

Sandalia Merrell para Sport mujer Merrell Terran Ari Post Negro Sport Negro 5ed7f20 - 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 Merrell para Sport mujer Merrell Terran Ari Post Negro Sport Negro 5ed7f20 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 383557
3234015
5107046
9703878
11570242
13610513
16391987
16788921
17633884
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.