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.

Sintético y malla

Importado

Caucho único

Descenso la montaña en la luz y veloz Brooks & reg; ¡Zapato de trail-running!

Predecesor: Ninguno.

Soporte Tipo: Neutro.

Amortiguación: Ligero, flexible Amortiguación.

Superficie: Sendero. 6 mm. Transpirable malla, textil, y sintético superior materiales. resistente al agua (DWR) acabado superior ayuda a evitar los elementos.Saddle características ajustables ojetes que tiran desde el talón hasta envolver el pie medio para un ajuste seguro Cierre con cordones. Lengua de peluche y collar. forro de tela ofrece un gran zapato sensación. plantilla de espuma. Entresuela de ADN BioMoGo entrega dinámico, amortiguación adaptable.Progresivo Diagonal Rollbar (PDRB) guid

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

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

Brooks Womens Mazama Mazama Diva Rosa Womens Rosa/ Antracita/ Bluefish 9126998Updated 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:

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.

Brooks Womens Mazama Mazama Diva Rosa Womens Rosa/ Antracita/ Bluefish 9126998

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

Calzado de running NIKE Kids Roshe One (GS) Negro / Blanco-Gris Oscuro , Zapatillas de running Fresh Foam 1080v6 de New Balance para mujer Gris azulado , Zapatillas de running Reebok para mujer ZigWild TR ll Aleación / gris / nítido azul / negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Azul marino / gris Zapatillas de running adidas Adizero Boston 6 W para mujer Noble Indigo / Blanco / Acero sin Procesar Zapatillas de running NIKE para mujer Lunarstelos Golpe caliente / negro / blanco / azul medio , Zapato deportivo para acolchado 635v2 New Balance para mujer Pigmento Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado , Zapatillas de running ASICS Women s Roadhawk FF SP Cereza / flor / abedul Zapatillas de running Gobiv2 para mujer New Balance Poisonberry / Black Zapatillas de running New Balance 990V3 para mujer Gris / rosa Skechers Women s Go MEB Razor 2 Carbón Zapatillas de golf adidas W CC Ballerina II para mujer Destello rosa / corriendo blanco / corriendo blanco Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Max naranja / negro / blanco Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Púrpura / rosa / verde azulado , Zapatillas de running Alphabounce W para mujer adidas Performance Easy Orange / White / Light Onix NIKE Men s Flex 2017 RN Antracita / Clear Jade / Black / Blustery Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Cool Grey / White / Volt / Blue Glow , Zapatillas de running NIKE Mens Lunarglide 8 Blue Glow Zapatilla para mujer Saucony tipo A8 Blanco | , Zapatillas de running New Balance Urgev2 para mujer Negro / Champagne Metálico , Zapatillas de running W630v5 para mujer New Balance Thunder / Vivid Ozone Blue / Metallic Silver Zapatillas de running NIKE RN 2017 para mujer gratis Gimnasio Azul / Azul Glaciar / Órbita Azul Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Wolf Gray / Purple Earth , Zapatillas de running ASICS GEL-Kayano 22 Lite Show para mujer Aqua Splash / Silver / Pink Glow Zapatillas de running NIKE Flex Run 2015 para mujer Prsn Violet / Blk / Fchs Glw / White Brooks Womens Adrenaline GTS 18 Púrpura / rosa / plata ONEMIX Zapatillas para correr Air Cushiong para mujer, Zapatillas deportivas deportivas ligeras Gris / Naranja Zapatillas de running NIKE Air Max Thea para mujer Sigilo / Blanco / Club Púrpura

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 de running NIKE para mujer Lunarepic Low Flyknit 2 Electro Verde / Negro / Azul Mediano , Chaco Women s Z2 Classic Athletic Sandal Brocade Pine , Sandalias de encaje Ros Hommerson para mujer Plata Zapatillas de running NIKE para hombre con versatilidad de tren gratis Total Crimson / Gym Rojo / Blanco / Negro , Sandalia Savannah de Ryka para mujer Azul marino / verde , Northside Womens Savannah Sport Closed Toe Sandal Tan / Coral , Sandalia Savannah de Ryka para mujer Gris blanco , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-negro Jambu Women s Spain Flat Tierra oscura , Bota Deia para mujer de TOMS Negro Negro Zapatillas de running New Balance 990v4 para mujer Angora / Angora , Hush Puppies Men s Gus Oxford Broncearse , BCBGeneration Women s Becca Sandal Escarlata , Zapatillas de running NIKE Roshe Two para mujer Caqui / Negro / Negro / Blanco , Sandalias de mujer Merrell Terran Cross II Negro , Sandalia universal Teva para mujer Mosaico Deep Teal , Skechers Women-14960 Sneaker Gris blanco , MOZO Zapatillas para mujer The Maven Sneaker Lunar Rock Camper Mujer derecha Nina K200619 Mary Jane Flat Multi , Skechers Women s Go Walk Revolution Ultra Sneaker Negro / Multi , Skechers Zapatillas con cordones de espuma viscoelástica D Lites para mujer Mayor ventilador gris / blanco , Zapato de entrenamiento Cross-Trainer Vazee Wx77v1 para entrenamiento New Balance para mujer Deep Porcelain Blue / Bleached Sunrise , Cobb Hill Rockport Mujeres Petra Mary Jane Flat Negro Clásicos de base Slip-on (Tm) para hombres de Vans True White Botines moldeados A-Game para mujer Boombah - 15 opciones de color - Varios tamaños Negro / real Twisted X Ladies Tough Basket Driving Mocs marrón , Botas Tumbling para mujer de Dr. Scholl s Shoes Cobre marrón LL Gisele-7 Lyte-2 muslo alto gamuza ante material puntiagudo punta estrecha botas de tacón de aguja negro Microsuede negro TOMS Clásicos de mujer Negro sobre negro Easy Spirit Women s Traveltime Mule Suede claro natural / 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.

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 Brooks Womens Mazama Mazama Diva Rosa Womens Rosa/ Antracita/ Bluefish 9126998 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.

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

Geometry Construction Reference

Thirteen elementary straightedge
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and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

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