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 de cuña Jubilee 10953 Propet para mujer cuña Sandalia Denim Multi aa70d32




Cuero
Importado
Sintético único
Eje mide aproximadamente Tobillo \ mide aproximadamente 0.75 pulgadas de pulgadas
Características una plantilla que se puede quitar para acomodar ortesis personalizadas
Ajustable cintas de gancho y lazo cierre y talón correa con elástico para un ajuste flexible
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 de cuña Jubilee 10953 Propet para mujer cuña Sandalia Denim Multi aa70d32 - 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 de cuña Jubilee 10953 Propet para mujer cuña Sandalia Denim Multi aa70d32

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 de cuña Jubilee 10953 Propet para mujer cuña Sandalia Denim Multi aa70d32

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 Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro Chaco Women s Zx2 Classic Athletic Sandal Trullo angular Sanda Verra Teva para mujer Surf Negro / Multi Sandalia Archie Fisherman del Dr. Scholl Nubuck azul marino Sandalias Columbia Sorrento Flip Athletic para mujer Salmón salvaje / ballena Chaco Mujer Z Eddy X2 Sport Sandalia Columnas Eclipse Sandalia Shaye Sport mujer Donald J Pliner Platino Chaco Women s Zx2 Classic Athletic Sandal Apex Gray Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Muy uva , Chaco Women s Z2 Classic Athletic Sandal Diamante Lavanda , Sandalia Drew Element para mujer Nubuck marrón , Sandalia Uneek O2-w de mujer KEEN Etherea / Whitecap , Sandalia Drew Angela para mujer Bone Smooth Leather , Chaco Mujer Zcloud Sport Sandal Agrupado azul , NIKE Women s WMNS Benassi Duo Ultra Slide, negro / negro Negro Negro Sandalia Drew Angela para mujer Beige Multi Print , Sandalias Newport H2 de KEEN para mujer Rama de Malaquita / Plata , Sandalias Swiftwater Mesh de Crocs para mujer Nuez , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 melocotón Sandalia de mujer Drew Alana Pewter-bronce , KEEN Sandalias Newport Eco-W para mujer Victoria / Dress Blues Wolky Comfort Jewel Choque de Multi Picasso Sandalia de susurro de mujer Keen Beet Red / Honeysuckle , Chaco Mujer ZX1 Classic Sport Sandalia Pep Pine , Chaco Mujer Zvolv Athletic Sandal Pane Angora KEEN Mujer, Sandalias de gladiador Maya Gris neutro / vapor OluKai Kulapa Kai Piña / Negro , Zapatillas de deporte lisas Terrex Climacool Voyager de Adidas Outdoor para mujer Negro / Negro / Blanco Tiza Sandalia Uneek O2-w de mujer KEEN Acero Gris / Vapor
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 NIKE Air Huarache Run Ultra SE para mujer LT OREWOOD BRN / LT OREWOOD Sandalias Bali KEEN para mujer Dapple Gray / Vapor Zapatillas de running ISO para mujeres de la guía Saucony Negro / Denim Zapatillas de running Kinvara 6 Saucony para mujer Blanco / azul / coral , Sandalia Drew Angela para mujer Bone Smooth Leather , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Hot Punch / Volt-total Orange , New Balance Women s w990v4 Running Shoe Negro / Plata , Zapatillas de running adidas Women s Edge Rc W Gris Dos / Plata Metálica / Energy Aqua Zapatilla de deporte Trail running wt690 para mujer de New Balance Teal / rosa , Sandalias mujer La Plume Jen Rhinestone dorado , Zapatillas de running TOPO FLI-Lyte 2 para mujer Azul rosado , NIKE Air Max Jewell Lea Womens Piedra pómez clara / Gris metálico Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Gunmetal / Thunder Sandalia de vestir Rubie de mujer chino de lavandería Silver Metallic , Zapatillas para caminar WW669V1 New Balance para mujer marrón , Converse Chuck Taylor All Star Core Ox Monocromo negro , Walking Cradles Womens Michaela Cuero negro , Kenneth Cole REACTION Jel-OUS 2 Woven Slip Slip Sandal para mujer Coral Multi , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2702- Silver Stardust & Silver Leathe , Sandalia de cuero Nirveli de Nine West para mujer Cuero natural oscuro TOMS - Slip-On de lienzo clásico para hombre Caoba roja TOMS Clásicos de mujer Rojo clásico , TOMS Womens Desert Wedge Bootie Gris pardo Tamaris Women s EDA 28206 Oxford Flat Coñac , Aravon Revsmart Oxford para mujeres Negro Timberland Womens Kenniston 6 \ Rubí , Botín Eli para mujer Bella Vita Gamuza negra para niños Botas de plataforma Mordenmiss Daisy Front para mujer Flor Mid Wedge Winter Wear Negro TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Blue Batik Textile Easy Spirit Women s Traveltime Mule 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.

Sandalia de cuña Jubilee 10953 Propet para mujer cuña Sandalia Denim Multi aa70d32 - 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 de cuña Jubilee 10953 Propet para mujer cuña Sandalia Denim Multi aa70d32 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 2620646
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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.