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.

Zapatos de B071HM7YTQ césped del Negro escuadrón Zapatos femenino Boombah - 14 opciones de color - Varios tamaños Negro/ púrpura 2cfa0c8




El elegante y aerodinámico diseño del escuadrón Turf juega tan rápido como se ve elección óptima en césped artificial así como natural hierba.
Se adapta fiel al tamaño. 10.5 oz
Malla Windows: Promueve ventilación y permite que el calor escapar, guardar tus pies frescos
sin costura Parte superior de Microweld: Reduce la creación de calor fricción y ofertas el más respirable parte superior de apoyo posible
De Boombah - el líder en rendimiento deportes de ingeniería vestimenta y equipo.
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


Zapatos de B071HM7YTQ césped del Negro escuadrón Zapatos femenino Boombah - 14 opciones de color - Varios tamaños Negro/ púrpura 2cfa0c8 - 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

Zapatos de B071HM7YTQ césped del Negro escuadrón Zapatos femenino Boombah - 14 opciones de color - Varios tamaños Negro/ púrpura 2cfa0c8

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.


Zapatos de B071HM7YTQ césped del Negro escuadrón Zapatos femenino Boombah - 14 opciones de color - Varios tamaños Negro/ púrpura 2cfa0c8

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

Pantalones cortos para correr Nike AeroSwift 2 \ Universidad Rojo / Negro , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / Rojo , Zephz Tumble Cheerleading Shoe Ladies , Zapatillas de deporte NIKE Youth Presto Fly Grade School Mesh Negro / Negro-Negro Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / real Zapatillas de running New Balance KJ880V5 para jóvenes (Little Kid / Big Kid) Gris / rosa Nfinity Adult Evolution Cheer Shoes Blanco , Chass y eacute; Zapatillas de deporte NIKE Youth Presto Fly Grade School Mesh Light Bone / Velvet Brown , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Blanco negro Mariposa Lezoline Tenis de mesa TB Zapatos y ndash; Zapatos para Ping Pong y ndash; 6-11 y ndash; Azul / verde o azul / rojo Zapatos y ndash; o zapatillas de deporte de mujer , Nintendo Titan Cheer Shoe , Mariposa Lezoline Tenis de mesa TB Zapatos y ndash; Zapatos para Ping Pong y ndash; 6-11 y ndash; Azul / verde o azul / rojo Zapatos y ndash; o zapatillas de deporte de mujer Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Blanco / rosa , Kaepa Youth Stellarlyte Cheer Shoe (par) Blanco , Nfinity Adult Evolution Cheer Shoes Blanco Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / púrpura Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Azul marino / Columbia , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Blanco negro Zapatillas de deporte NIKE Youth Presto Fly Grade School Mesh Gimnasio Rojo / Blanco Gris Oscuro / Blanco , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / Rojo Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Royal / Red / White Zapatillas de deporte NIKE Youth Presto Fly Grade School Mesh Negro / antracita-lobo gris , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Azul marino / Columbia Nintendo Titan Cheer Shoe , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / gris , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / gris Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / púrpura , Nfinity Adult Evolution Cheer Shoes Blanco ,
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 Chaco Mujer Zcloud X2 Sandalia Deportiva Plata metálica Sandalia atlética Salomon Women s Crossamphibian Swift W Mallard azul , Skechers Performance Women s Go Run 400 Obstruct Rosa caliente Zapatillas de running Saucony Women s Triumph ISO 4 Blanco azul , Zapato de running ASICS GEL-Kayano 22 para mujer Gris plata / Pistacho / Coral Fiery Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / verde lima , Chaco Mujer Diana Sport Sandalia Biselado negro Zapatillas de running Salomon Women s Sense Mantra 3 Lucite Verde / Llamativo / Negro , Tierra Hera Champagne Impreso Gamuza , Zapatillas de running NIKE Roshe Two Flyknit V2 para mujer Blanco / Lobo Gris-blanco-blanco Zapatillas Reebok Women s Runner 2.0 MT Track Cráneo gris / Verde azulado / Blanco / Negro / Plateado Metálico Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Berenjena / Sal marina , CLARKS Sandalia Saylie Moon para mujer Peltre Metálico Cuero Easy Spirit Women s Traveltime Mule Azul marino / ante blanco Zapatillas de tenis Grandpro Stitchlite de Cole Haan para mujer Tango Rojo , Vans Old Skool Unisex Adults Low-Top Trainers (35u) (Tablero de ajedrez) Negro / Rojo Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre True White Sandalia de la diapositiva de Stella de las mujeres de Rachel Zoe Plata Sandalia plana Marsela de Spring Step para mujer Negro , Sam Edelman Sandalia Glenn para mujer Oro fundido oscuro CORRAL bordado de mujer con tachuelas bordado floral vaquera punta cuadrada - R1373 marrón , Bota de montar Eresa para mujer de ALDO Marrón medio Zapatilla de vals femenina Foamtreads Visón de terciopelo , Haflinger para mujer como Sasha Slip On Slipper Rosa , SOREL Womens Nadia Bootie Sal marina Botas Gaerne SG-12 Blanco verde , Botín de mujer Dominicaa ALDO de ALDO Oro Nine West de mujer Garroy Leather Oxford Flat Cuero negro , Zapatillas de deporte lavables Walker Propet para mujer Bronce Easy Spirit Women s Traveltime Mule Azul marino / ante blanco

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.

Zapatos de B071HM7YTQ césped del Negro escuadrón Zapatos femenino Boombah - 14 opciones de color - Varios tamaños Negro/ púrpura 2cfa0c8 - 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 Zapatos de B071HM7YTQ césped del Negro escuadrón Zapatos femenino Boombah - 14 opciones de color - Varios tamaños Negro/ púrpura 2cfa0c8 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.

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

Geometry Construction Reference

Thirteen elementary straightedge 2968935
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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.