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.

Skechers Performance B071D2GL18 18035 Women Go s Go Run 400 Obstruct Negro b597937




Sintético
Importado
Caucho único
Radicalmente liviano y flexible
Responsive 5gen amortiguado entresuela
Altamente transpirable diseñado malla superior
Goga ejecutar la energía de retorno plantilla
Fabricante la garantía no puede aplicar
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


Skechers Performance B071D2GL18 18035 Women Go s Go Run 400 Obstruct Negro b597937 - 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

Skechers Performance B071D2GL18 18035 Women Go s Go Run 400 Obstruct Negro b597937

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.


Skechers Performance B071D2GL18 18035 Women Go s Go Run 400 Obstruct Negro b597937

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

Zapatillas de running Cortana 4 Saucony para mujer Blanco / rosa / azul Brooks Womens Hyperion Imperial Purple / Blue Jewel / Vida Nocturna Zapatilla Skechers para mujer Go Run 600-Obtain Carbón , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Negro / Negro / Carbono , Zapatillas de running New Balance 990v4 para mujer Lago Azul / Lago Azul , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Blanco / morado / rosa Zapatillas de running PUMA Women s Speed ​​500 Ignite Tobillo-Alta rojo , Amortiguación 460v2 New Balance para mujeres Azul plateado Zapatillas de running W775V2 para mujer New Balance Blanco negro , NIKE Mujeres Downshifter 7 Gris intenso / Pulso al atardecer / Atmósfera gris Zapatillas de running NIKE Women s Sock Dart BR Negro / blanco-glaciar azul Under Armour Women s Charged 24/7 Low NM Acero (035) / Gris Nublado Zapatillas de running adidas Cloudfoam Ultimate para mujer Orquídea Tinte / Orquídea Tinte / Aero Pink , EN Zapatilla Cloudster para mujer Blanco negro Zapatillas de running Nike Zoom All Out Low para mujer Negro / Gris Oscuro-antracita-blanco , Zapatillas de running NIKE Women s Sock Dart BR Sunset Glow / White-orchid Zapatillas de running adidas Alphabounce Beyond W, para mujer Ecru Tint / Ash Pearl / Ash Pearl Zapatillas de running adidas Ultraboost W para mujer Misterio Azul / Misterio Azul / Vapor Gris NIKE Mujeres Downshifter 7 Gris intenso / Pulso al atardecer / Atmósfera gris , Zapatillas de running NIKE Air Huarache para hombre Azul marino / difuso-obsidiana Brooks Womens Ghost 9 Negro / Sparkling Grape / Ceramic , Calzado deportivo New Balance para mujer Vazee Rush v2 Gris oscuro Zapatillas de running adidas Women s Edge Lux 2 Utilidad Negro / Negro / Blanco NIKE W Air Huarache City Low Women Ah6804-100 Blanco / Blanco / Blanco , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Espuma de mar / cielo cubierto New Balance Women s w990v4 Running Shoe Armada , Zapatillas de running Pureboost X de adidas Performance para mujer Negro / Negro / Gris Sólido Amortiguación 635v2 New Balance para mujer Plata , Zapatillas de running NIKE Womens Huarache Run Ultra PRM para mujer Negro / Azul Tinte-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 Sandalia deportiva Kankitta de Vince Camuto para mujer Cloudy Gray Combo , Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple , Zapatillas para caminar Reebok Mujer Zprint Her WS MTM Delirio Púrpura / Pigmento Púrpura / Primo Púrpura / Blanco , Zapatillas para correr Saucony Liberty para mujer Pizarra / Blush Sandalia para mujer Teva Terra-Float Travel Knit Paloma Salvaje Calzado de running NIKE Kids Flex 2017 RN (GS) Pure Platinum / Tough Red Zapatillas de running mujer HOKA ONE ONE Clifton 4 Castlerock / Asphalt Sandalia atlética de Rka Women s Glance Azul / verde azulado Sandalia Merrell Mujer Terran Strap II Negro Zapatillas de running NIKE Air Max 90 LX para mujer Gunsmoke / Gunsmoke , Zapato de running ASICS GEL-Venture 5 para mujer Gris plateado / turquesa / sacador de lima Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Negro Blanco 789 Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Grey Knit , Birkenstock Arizona Sandalia de cuero unisex Ante Mocha , Sandalias con hebilla de botón Ridgeport para mujer Rockport Bronce , Skechers Women s Go 4-14918 Calzado para caminar Carbón , Calzado de entrenamiento casual Comfort WX623v3 New Balance para mujer Negro Zapatilla Reebok para mujer ROS Workout TR 2-0 Cross-Trainer Niebla / Blanco / Meteorito / Puro Vans Herren Authentic Core Classic Sneakers Negro Negro Birkenstock Mules para mujeres Negro , Calzado Aria Pointe 121 para mujer Capezio Pétalo rosa Sandalia de plataforma Pleaser Stplash808 / C / B para mujeres Claro y Escarcha Alpargatas antideslizantes de lona clásicas para mujer de TOMS Negro sobre negro Sandalias de vestir Patti de Sam Edelman para mujer Candy Red Patent , Botas Amaya de mujer Hardy Mezcal marrón , Baabuk Wool Sneaker - Mujer Azul marino / blanco Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Café rojo , Bota de montar Charlotte de Propet para mujer Gamuza negra / negra Calvin Klein Womens Priscila Black Cow Silk / Neoprene Clásicos de base Slip-on (Tm) para hombres de Vans Negro 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.

Skechers Performance B071D2GL18 18035 Women Go s Go Run 400 Obstruct Negro b597937 - 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 Skechers Performance B071D2GL18 18035 Women Go s Go Run 400 Obstruct Negro b597937 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.

Carretera en funcionamiento
Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge 4585201
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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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