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 de 11796 mujer B074KMSX44 Drew Alana Pewter-bronce 59adf82




Cuero
El hermosa flor diseño de esto sandalia única en su especie lo hace un verdadero cabeceo
Cuero superior con dos Velcro® ajustable correas para una costumbre ajuste
Cuero completo revestimientos y un natural, dedo oblicuo
Grueso Poliuretano ULTRON® plantilla con gamuza cubierta superior
extraíble Con cierre de Velcro®, plantilla de doble profundidad acomoda la prescripción ortesis
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 de 11796 mujer B074KMSX44 Drew Alana Pewter-bronce 59adf82 - 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 de 11796 mujer B074KMSX44 Drew Alana Pewter-bronce 59adf82

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 de 11796 mujer B074KMSX44 Drew Alana Pewter-bronce 59adf82

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 Terrell Ari Convert Sport de Merrell para mujer Negro , Sandalia Drew Element para mujer Nubuck azul marino OluKai Kulapa Kai Piña / Negro , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth , Sandalias KEEN Venice II H2-W para mujer Cornstalk / Safari , Propet Women s TravelActiv Ss Sandal Negro plata , PUMA - Arco para mujer WNS, Marshmallow Silver Marshmallow-puma Silver , Sandalia Tanga Loea de mujer OLUKAI Negro Negro , Northside Womens Savannah Sport Closed Toe Sandal Negro / gris , Sandalias Newport H2 de KEEN para mujer Plaza Taupe / Provincial Blue Sandalias Columbia Sorrento Flip Athletic para mujer Salmón salvaje / ballena Sandalia Uneek O2-w de mujer KEEN Opal azul / azul provincial Flip-flop Teva Women s Mush II Rombo Red , Chaco Women s Zx2 Classic Athletic Sandal Trullo angular , Chaco Mujer ZX1 Classic Sport Sandalia Pep Pine , Wolky Women s Rio Sandalia Red Patent Metal Sandalias Newport H2 de KEEN para mujer Rama de Malaquita / Plata Sandalias Newport H2 de KEEN para mujer Raya Black Drew Womens Cascade Nubuck azul marino Sandalia Drew Angela para mujer Dusty Multi Metallic Leather Drew Womens Cascade Nubuck azul marino Sandalia de mujer Drew Alana Pewter-bronce , Tacones Pleaser para mujer Infinity 902 Negro Negro , Sandalias Newport H2 de KEEN para mujer Cuervo / Capri Sandalias Swiftwater Mesh de Crocs para mujer Negro Sandalias sintéticas Teva para mujer Terra Fi Lite Samba Coral Multi , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso Sandalia Drew Angela para mujer Bone Smooth Leather , Wolky Comfort Jewel 20570 Cuero rojo
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 NIKE Women s Tanjun Blanco Blanco Negro , Sandalia deportiva Marc Jacobs Women s Somewhere Negro / Multi , Under Armour Women s RailFit 1 Antracita (100) / Acero Under Armour Women s Charged Rebel Black (004) / Purple Rave Sandalia Yucatan ECCO Mujer Pastel multicolor OluKai Kulapa Kai Aqua / Negro , CLARKS Sandalia Saylie Jade para mujer Armada Sandalia de mujer W Verra Sport de mujer Teva Atlántico Norte , Zapatillas de trail running Nitrel v1 FuelCore para mujer New Balance Gris negro , Zapatillas de running Cavu para mujer HOKA ONE ONE Vintage Indigo / Hibiscus , Earth Womens Allegro Ballet Flat Rose Gold Pearlized Soft Leather Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Champagne / Metálico , Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Azul marino / ondulado / estiramiento Zapatillas Skechers Go Walk Lite-15360 para mujer Armada , Zapatillas de deporte Camper Peu Cami 46104 para mujer Negro 1 , Zapato para caminar WW411v2 New Balance para mujer Gris BareTraps Keely Sandal para mujer Caramelo Entrenador cruzado New Foam Crush V1 de New Balance para mujer Negro , Zapatillas de deporte Fenty X High Top Trainer de mujer PUMA Whisper White / Whisper White PUMA Mujer Ignite Flash Evoknit Satin En Pointe Wn Sneaker Puma negro-periscopio-metálico Beige Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Blanco negro Supra Vaider LC Sneaker Negro / Cobre / Blanco , Sandalia para mujer Qtee Slide de Anne Klein Negro sintético Sandalia para mujer Merrell, 1SIX8 Linna AC + Slide Monumento , Zapatos rusos de Pointe Rubin Pointe, caña media flexible de U-Cut Mocasines casuales Alpargata Deconstructed de TOMS para mujer Peltre Metálico Cuero Sandalia Birkenstock de cuero Florida para mujer Negro , VANELi Womens Reanne Hebilla de Gamuza / Gamuza Negra , Skechers Womens Bikers - Yacht Spot Natural , Propell M3910 Vista Walker Oxford para hombres 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 de de 11796 mujer B074KMSX44 Drew Alana Pewter-bronce 59adf82 - 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 de 11796 mujer B074KMSX44 Drew Alana Pewter-bronce 59adf82 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.


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