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 Terra-Float de Livia de Sandalia Teva azul para mujer Gris azul 47ed35f




100% Textil
Importado
Caucho suela
Correa superior
Etileno plantilla de acetato de vinilo
Gancho y cierre de lazo
Float-lite suela
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 Terra-Float de Livia de Sandalia Teva azul para mujer Gris azul 47ed35f - 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 Terra-Float de Livia de Sandalia Teva azul para mujer Gris azul 47ed35f

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 Terra-Float de Livia de Sandalia Teva azul para mujer Gris azul 47ed35f

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 deportiva Marc Jacobs Women s Somewhere Amarillo / Multi , Sandalia Drew Element para mujer Nubuck negro , Sandalia Jacardia Merrell para mujer Gris pardo , Sandalias KEEN Mujer Whisper-w Higo de verano / Crabapple , KEEN Hombres Newport H2 Sandal Cuervo / Aluminio Zapato de agua H2 Go Skechers Performance para mujer Azul marino / gris Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro Sandalia Yucatan ECCO Mujer Frosty Green / Moon Rock / Frosty Green , Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Sodalita , Sandalia deportiva Alise Les Filles para mujer Dusty Lilac Nappa Chaco Mujer Zcloud X2 Athletic Sandalia Runa Teal , Wolky Comfort Jewel Black Smooth , Zapatos de Paloma de Oro GP5937 Cierre de Cierre Fácil Cierre Sandalias Deportivas / de Agua para Hombres y Mujeres 7668 negro azul , Sandalia deportiva Nolita para mujer Anne Klein Tela Coral BZees Sandalia Deportiva Jive para Mujer marrón Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Acuífero , Sandalia KEEN Women s Venice H2 Gasolina / Crepúsculo , Propet Women s TravelActiv Ss Sandal Negro plata , Wolky Comfort Jewel Fantasía multicolor negra , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth Calzado casual Slip-On de Bernie Mev para mujer Estaño Blast sandalias sexy y cómodo con tacones altos oro Europa y los Estados Unidos gran tacón alto sandalias con damas Blanco Chaco Women s Zx2 Classic Athletic Sandal Trullo angular Sandalia Deportiva Jordan Spires de Via Spiga Paja de gamuza , Sandalia KEEN Women s Uneek Slide-w Negro Negro , Sandalia de mujer KEEN Rose Negro / Gris neutro , Calzado casual Slip-On de Bernie Mev para mujer Bronce , Sandalia Drew Angela para mujer Beige-multi Sandalia Uneek O2-w de mujer KEEN Estate Blue / Harvest Gold ,
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 Earth Origins Sophie Frost Grey , KEEN Hombres Newport H2 Sandal Tierra oscura / Acacia , Chaco Women s Z1 Classic Athletic Sandal Negro Zapatillas de trail running Salomon Women s Speedcross 3 Negro / Negro / Plateado Metálico-x Zapatillas de trail running Merrell para mujer Bare Access Arc 4 Negro / Aventurina , Skechers Women s Go Walk Joy-Paradise Sneaker Azul marino / blanco , Zapatillas de running Saucony Women s Jazz Low Pro Azul blanco , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Negro Plano original de la mujer del Dr. Scholl s Collection Collection Kimberned Pointed Toe Vestido azul nubuck Apex Emmy Slip para mujer Broncearse Zapatos de entrenamiento Nike Free TR 6 para mujer Purple Shade Lilac 502 , FitFlop Womens Serene Python Print Slip-On Blanco Urbano , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Negro Zapato de entrenamiento cruzado Yui con tratamiento en gel femenino de ASICS Glaciar / Paradise Green / Diva Pink Clásicos de base Slip-on (Tm) para hombres de Vans Carbón , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Púrpura , Zapatillas de deporte CrazyMove Pro de adidas by Stella McCartney Core Blanco / Hi-res Blue S18 / Semi Solar Red , SCHUTZ Sandalia Yaslin Slide para mujer Escarlata , Sandalia Dolce Vita Halle Slide para mujer Tela natural / negra Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita De lino blanco Charles by Charles David Sandalia mujer Kadia Slide Arena del desierto , Zapatillas de volante Mizuno Wave Lightning Z4 para mujer Negro / real , Bomba de vestido Daisie de mujer Steve Madden Peltre negro Bomba de vestido de pasión Easy Street para mujer Patente rojo brillante , Zapatilla Diana de mujer verde de Patricia Negro Trotters Lucy Ballet para mujer Cerezo negro Zapatillas de deporte Apollo Chukka de hombre nativas Barracuda Blue / Barracuda Blue Clásicos de base Slip-on (Tm) para hombres de Vans Carbón , Zapato femenino Keli de Alegria Lustre castaño Mocasín de gamuza Kilty de Minnetonka para mujer Cereza

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 Terra-Float de Livia de Sandalia Teva azul para mujer Gris azul 47ed35f - 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 Terra-Float de Livia de Sandalia Teva azul para mujer Gris azul 47ed35f 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

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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.