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.

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Importado
Suela de goma
Plantilla extraíble
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


Prop & eacute; t Mujeres Lavable Lavable Walker Suede & Oxford B01BD5VYQO Índigo 1297927 - 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

Prop & eacute; t Mujeres Lavable Lavable Walker Suede & Oxford B01BD5VYQO Índigo 1297927

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.


Prop & eacute; t Mujeres Lavable Lavable Walker Suede & Oxford B01BD5VYQO Índigo 1297927

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

Arcopedico Mujeres Ls Aceituna , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Negro-rojo , Tamaris Kela 23718 Oxford Flat para mujer Estructura Marina , Drew Shoe Women s Pepper Criss Cross cuero, caucho Casual Oxfords Negro Easy Spirit ON CUE Oxford para mujeres Blanco Blanco TOMS Kids 10009918 Alpargata-K Desert Taupe Suede Perforated , TOMS Kids 10009918 Alpargata-K Desert Taupe Gamuza / Lana , Zapatillas para mujer Del Rey TOMS Bloom Suede / mosaico de azulejos , Zapatillas Skechers Sport Mujer Imperio Moda Negro / Multi , Zapatillas de deporte de lona Converse Top Street Tonal Canvas para hombre Wolf Gray Easy Devitt Oxford Flat para mujer Negro UGG Mujeres Marji Snake Oxford Negro Mocasín de cuero con cordones para mujer Parodia de Caterpillar Negro , Aravon Betty-AR Oxford para mujeres marrón M4D3 Womens Slide Mule Sneaker Ternero blando blanco Mordenmiss Nuevo verano de primavera y verano de flores hechas a mano zapatos planos de cuero Mocasines Estilo 1 Negro , Zapatilla de deporte CRE205 / BGVY de mujer Demonia Vinilo de brillo negro Propet Women s TravelActiv Mary Jane Oxford Heather negro / azul marino , Softspots - Mujeres - Angie Gris pardo , Zapatillas de deporte de mujer Easy Spirit de Easy Spirit Gris Drew Shoe Women s Paradise II Velco Slip-on Taupe ternero Laberinto de mujeres de Franco Sarto Oxford Twilight Navy , Vestido Reef Iing Wingtip Oxford de mujer Caterpillar Carillón de viento / luz Prop & eacute; t Olivia Oxford para mujeres Bronco Brown , Zapatillas de deporte Easy Spirit AP1 Sport VID DE PROFUNDIDAD CLARKS Women s Candra Blush Flat Berenjena , Grand Tour Oxford de Cole Haan para mujeres Blazer azul / blanco óptico , Zapatillas de deporte lavables Walker Propet para mujer Hueso / Wht Moda mujer Skechers-Slide Thru Rosa claro
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 Under Armour Women s Charged Rebel Black (004) / Purple Rave Zapatillas de deporte NIKE Flex Training 6 Print para hombre Zapatillas de deporte atléticas Gris , Nike Air Max 90 Ultra BR para mujer Gamma Blue / Blue Lagoon , Zapatillas de running NIKE Air Zoom Vomero 12 para mujer Lava Glow / Black-racer Pink Zapatillas de running Mizuno Wave para mujer Wave Horizon Blanco negro , Zapatillas de running Lagetiva Wildcat Trail para mujer Gris / mantequilla , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris blanco Zapato KEEN Presidio MJ para mujer Negro Negro Cobb Hill Rockport Women s Angelina Dress Pump Almendra ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Negro / Curry , Easy Spirit Women s Traveltime Mule Suede claro natural / blanco , Sandalia Kinsley de Naturalizer para mujer Papaya roja / tostada , Zapatillas Hailee Ryka para mujer Azul / azul real / coral rosa , Zapatillas Hailee Ryka para mujer Frost Grey / eggshell Blue / electric Coral Zapatillas DC Skate Pure High-Top TX para mujer Lila , Supra Vaider LC Sneaker Carbón blanco Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Armada , Zapatillas de deporte PUMA Block Basket Trace Block para mujer Sterling Blue / Vanilla Ice , Under Armour Women s Performance Spikeless Blanco (100) / Brillo , Sandalia plana Sofie de Badgley Mischka para mujer Plata BareTraps Women s Jaylyn Slide Sandal Blanco Sandalia para mujer Qtee Slide de Anne Klein Rosa Claro / Natural Sintético , Star Wars: The Force Awakens Película Kylo Ren Cosplay Zapatos Sith Cosplay Boots Negro , Old Gringo Womens Nicolette Tobillo rojo , Bomba de Angel II para mujeres estilo suave Oro / Plata Cósmica , Bota de mujer de tacón alto elástico Kendra Stretch de Fix Textil negro Bota vaquera con punta en punta de cocodrilo de mujer Ferrini Sport Rust , Tiempo libre para mujeres de SAS Moca Splendid Women s SPL-tobey Tuxedo Oxford Nuez , Trotters Liz Loafer para mujer marrón

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.

Prop & eacute; t Mujeres Lavable Lavable Walker Suede & Oxford B01BD5VYQO Índigo 1297927 - 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 Prop & eacute; t Mujeres Lavable Lavable Walker Suede & Oxford B01BD5VYQO Índigo 1297927 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.

Oxfords
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.