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.

Cinta de poliéster

Importada

Sintética único

El logo incrustado en el sandalia puede ser diferente de lo que está representado en el sitio.

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

Flip-flop Teva Women s Zoey Mush II Zoey Women Teva Coral a1822c2Updated 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.

Flip-flop Teva Women s Zoey Mush II Zoey Women Teva Coral a1822c2

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

Wolky Comfort Jewel 12000 Black Nubuck Sandalia de mujer KEEN Rose Negro / Gris neutro Sandalia de cuña Cape May de mujer Jambu Impresión de hielo , Sandalia Savannah de Ryka para mujer Gris / Purpl , Sandalia Kea Vent mujer Columbia Iceberg, Fawn Chaco Mujer ZX1 Classic Sport Sandalia Aguacate estático , OluKai Kulapa Kai Coral / Negro , CLARKS Sandalia Saylie Jade para mujer Arena Adidas Women s Adilette CF + Logo W Sandalia deslizante Shock Pink / Collegiate Navy / Aero Pink , Sandalia de mujer KEEN Rose Brindle / Shitake Skechers Performance On The Go 400 Vivacity Flip Flop Carbón de leña / rosa fuerte , Sandalia Teva W Terra Fi 4 para mujer Rocío Atlántico Norte , Sandalia deportiva Teva W Hurricane 4 para mujer Negro , Sandalias KEEN Venice II H2-W para mujer Negro / gris acero , Sandalia original Teva Women s Original Miramar Fade Coral Sand Multi Sandalia deportiva Tirra para mujer Teva Buena Powder Blue , CLARKS Womens Wave Grip Negro sintético Sandalia Terrell Post II Femenina de Merrell Negro Sandalia Yucatan ECCO Mujer Biscaya Fabulicious Women s Caress 401Dm sandalias de diapositivas Claro Rhinestone negro / peltre , Sandalias de encaje Ros Hommerson para mujer Microtouch negro , Merrell Women s Around Town Print Sandalia atlética Fiery Red , Sandalia de Gladiador de Piedra Lunar de Ahnu para Mujer Lace Dark Plum , Chaco Mujer Zcloud X Athletic Sandalia Island Tango Sandalias Swiftwater Mesh de Crocs para mujer Jean azul , Sandalias de encaje Ros Hommerson para mujer Plata Sandalia deportiva ECCO Terra 3S para mujer Arona / Papaya Earth Origins Sophie Negro Sandalia Terrell II de Merrell para mujer Negro ,

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 Speedform Slingwrap para mujer Negro (001) / Glacier Grey NIKE Men s Flex 2017 RN Negro / Metálico Cool Gray / Light Carbon , Zapatillas para correr Saucony Women s Cohesion 10 Gris / verde azulado / cidra Zapatillas de running NIKE Lunar Skyelux para mujer Pure Platinum / Stealth / Summit White , Zapatillas de montaña La Sportiva Tempesta GTX para mujer Stone Blue / Mint Sandalias con tachuelas de cuero para mujer de Royou Yiuoer Sandalias con tira en T naranja Bomba de plataforma FLY London Yadi718fly para mujer Sand Cupido / Mousse , Zapatillas de entrenamiento Nike Womens Metcon 3 Blanco / Refleja Plata , EN Zapatilla Cloudster para mujer Magenta / Berry Zapatillas de running Nike Air Zoom Vomero 11 para hombre Negro / Blanco / Antracita , MICHAEL Michael Kors Zapatillas Allie Trainer para mujer Almirante Sandalia Terra FI Lite de Teva para mujer Agua , Drew Shoe Women s Tulip Comfort marrón , Zapatillas para caminar Travellite Propet para mujer Negro Sandalia Palila de Marc Fisher para mujer Argento Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos Negro New Balance WW577 Hook and Loop Walking para mujer Hueso , Zapatillas para tenis Junior All Propulse Fury de Babolat Kid rojo Converse Unisex Chuck Taylor All-Star Top casual Sneakers en Classic Estilo y color y lienzo duradero Uppers Blanco negro , Sam Edelman Women s Bay Slide Sandal Negro / Negro / Marfil Punteado Brahma Cabello , Sandalia Maci de mujer Dansko Negro Full Grain , Athena Alexander Womens Soraya Negro , Zapatillas de voleibol WOMS BK-SL Wave Lightning Z para mujer Mizuno Negro / Plata Mizuno (MIZD9) Zapatillas femeninas de 9 picos Advanced Finch Elite 3 Clep Softball Cleat Real / Blanco , Cobb Hill Rockport - Aubrey Dress Pump para mujer Estaño Bomba Hazel para mujeres Sam Edelman Cordón botánico negro desnudo , Pantuflas Matilda de Katy Perry para mujer Uva Zapato clásico Haflinger para mujer Natural , Propet Women s TravelActiv Mary Jane Oxford Heather negro / gris UGG Mujer W Joni Lude Gray

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 Flip-flop Teva Women s Zoey Mush II Zoey Women Teva Coral a1822c2 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

For email to reach me, the word