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.

Textil y sintético

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

Importado

adidas

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

Zapatillas Zapatillas de Blue running adidas Noble Aerobounce Pr W para mujer Aero Blue/ Aero Blue/ Noble Indigo 6f08261Updated 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.

Zapatillas Zapatillas de Blue running adidas Noble Aerobounce Pr W para mujer Aero Blue/ Aero Blue/ Noble Indigo 6f08261

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

Zapato de running ASICS para mujer Torrance Negro / Gris piedra Zapato deportivo ASICS Gel-Kayano 24 para mujer Gris medio / negro / Atlantis , Saucony Women s Ride 10 zapatillas para correr Púrpura Citron , Zapatillas de running MBT MBT para hombres MBT Blanco azul , Zapatillas de running W775V2 para mujer New Balance Gris / morado , WMNS Zoom Fly Blanco negro , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta Zapatillas de running Reebok para mujer Collegiate Royal / White , Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Menta , Zapatillas de running Nike Air Huarache Run Ultra para hombre Blanco Blanco , NIKE Womens Air Zoom Wildhorse 4 Running Shoe 880566-007 Negro / Volt Green Abyss / Menta-obsidian-vapor Green Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa , Zapatillas de running Reebok Women s Run Supreme 2.0 Negro / Rosa solar , Zapatillas de running NIKE Lunartempo 2 para mujer Chalk Blue Black Racer Blue 400 Zapatillas para caminar Reebok Women s Cloudride LS DMX Cráneo gris / gris plano / blanco / rosa venenoso , Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Hyper Violet / Blue Cap Zapatillas de running adidas Women s Edge Rc W Gris Dos / Plata Metálica / Energy Aqua , Zapatillas NIKE Kids Revolution 3 (GS) para correr Mango brillante / Plata metalizada / Gris lobo , Zapato deportivo ASICS Gel-Excite 4 para mujer Aruba Azul / Plata / Acuario , Zapatillas de running Mizuno Wave para mujer con sombra Griffin / Blanco / Eléctrico , Zapatillas de running adidas Questar Ride para mujer Carbono / Negro / Gris Zapatillas de running NIKE Runallday para mujer Negro / Leche Azul Gris Oscuro , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Shadow Brown / Metallic Pewter / Rush Maroon , Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue Zapatillas de running NIKE RN 2017 para mujer gratis Negro / Blanco / Gris Oscuro / Antracita , Zapatillas para correr ASICS Gel-Venture 6 para mujer Negro / Verde hielo / Naranja Calzado deportivo NIKE CK Racer para mujer Negro / vela negra Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral , Nike Air Max 90 Ultra BR para mujer Gamma Blue / Blue Lagoon

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 Yucatan ECCO Mujer Titanio , Zapatillas de running NIKE Mens Lunarglide 8 Rio Teal / Volt-clear Jade-medianoche Turquesa 361 Zapatillas de running para mujer 361-Enjector Sparkle / Phlox_1339 Zapatillas de running de tierra firme para entrenamiento de campo Primus Swimrun FG de Vivobarefoot para mujer Negro / Naranja , Zapatillas de agua para mujeres Astral Porter Azul marino / blanco Zapatillas de running Nike Kaishi para hombre Lobo gris / negro-blanco , Skechers Women s Spirit Walking Mint B (M) US menta , CLARKS Mujer Caddell Shine Sandal Nubuck sintético de arena Sandalia de tiras de lectura Comfortiva para mujer Negro La sandalia Beglad de mujer FLEXX Cognac Vacchetta Sandalia Imogene Gladiador de Naturalizer Negro KEEN Women s Uneek 8MM Sandal Rododendro / Gris neutro Sandalia de henna de mujer Sam Edelman Amarillo / Multi , Cobb Hill Rockport Women s Angelina Dress Pump Burdeos , Zapatilla S Skechers para mujer E Z Flex 3.0 Swift N Sly Armada Zapatillas para caminar WW669V1 New Balance para mujer Gris , Calzado Lakai Limited para hombre Bristol Gamuza dorada / azul Zapatillas de deporte Omni Walker de rejilla Saucony para mujer Plata blanca , Plataforma de campo Leisa para mujeres de CLARKS Cuero de arena , Sandalia Dolce Vita con diadema para mujer Tejido negro , UGG Mujer W Joni Niebla de Lavanda , Carlos by Carlos Santana Sandalia mujer Slide Liliana Armada , Sam Edelman Sandalia Bernice Slide para mujer Nuevo Nude Leopard Brahma Hair Vince Camuto Sandalias Ellanna Slide para mujer Negro Calzas de la dinastía Blanco negro Botín de ante TOMS Sevilla para mujer Dark Amber Suede / Volantes , Trotters Lucy Ballet para mujer Blanquecino Cole Hoan Zerogrand Wing-Tip Oxford para mujer Avena Nubuck Vionic Women s Action Emerald Lace up Broncearse , Arcopedico Mujeres Ls White Sparkle ,

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 Zapatillas Zapatillas de Blue running adidas Noble Aerobounce Pr W para mujer Aero Blue/ Aero Blue/ Noble Indigo 6f08261 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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