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.

Cuero sintético / Nylon

Robusto parte superior flexible características soft faux gamuza con rápido secado de neopreno respaldo / transpirable inserciones de nylon tejido y resistente a la abrasión protector de dedos

Esto cordones al aire libre hasta sandalia atlética ofrece resistente al agua construcción / humedad wicking neoprene forro para todos confort diario

Ligero plantilla de espuma de EVA / acolchada respaldo garantizar amortiguación para tus pies; resistente lavable materiales ayuda resistir extendido use

Elástico cordones con cordones sistema / gancho y correa del talón del lazo mantén tus pies seguro y

duradero multi direccional suela de goma asegura confiable tracción / choque absorción sobre ambas superficies mojadas / secas

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

Northside Womens Savannah Sport Savannah Closed Toe Sandal Sport Tan 10485/ Coral 3aacd99Updated 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.

Northside Womens Savannah Sport Savannah Closed Toe Sandal Sport Tan 10485/ Coral 3aacd99

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

OluKai Ohana Sour Tart / Ice , Sandalia NIKE Benassi JDI para mujer Partícula Rosa / Plata Metálica , Drew Womens Cascade Nubuck marrón Drew Womens Cascade Sand Nubuck Chaco Women s Zx2 Classic Athletic Sandal Parcheado de ámbar Sandalia Teva Terra-Float Nova para mujeres Gris FABULICIOUS COCKTAIL-501RS - Diapositivas de plataforma para mujer, 5 \ Claro / Claro , Sandalia Uneek O2-w de mujer KEEN Black / Harvest Gold , Sandalia deportiva Zvolv X de Chaco para mujer Pino Raglan , Sandalias Columbia Sorrento Flip Athletic para mujer Salmón salvaje / ballena Sandalias KEEN Venice II H2-W para mujer Grape Kiss / Red Violet OluKai Kulapa Kai Jalea de guayaba / negro , Sandalia atlética de Rka Women s Glance Azul / verde azulado , Propet Women s TravelActiv Ss Sandal Azul negro , Propet Women s TravelActiv Ss Sandal Negro plata , Sandalia deportiva ECCO Terra 3S para mujer Dark Shadow / Emerald , Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva Sandalia original premier universal Teva W de mujer Beach Break Deep Wisteria Sandalia Yucatan ECCO Mujer Irritante Sandalias Newport H2 de KEEN para mujer Tinte en espiral , Zapatilla de deporte Sina 11 de Josef Seibel para mujer Negro , Sandalia NIKE Benassi JDI para mujer Rosa / Plata Chaco Mujer Zcloud X2 Sandalia Deportiva Angora Webb Sandalia NIKE Kawa Slide para mujer Blanco / rosa fuego / cometa azul / aluminio , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave Sandalia Drew Angela para mujer Dusty Multi Metallic Wolky Comfort Jewel Fantasía de serpiente blanca / negra posters , Sandalia Drew Angela para mujer Beige-multi , Corkys Rock Mujer Slip On Azul Multi ,

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 Dusty Sport mujer muy volátil Negro , OluKai Kulapa Kai Papaya / Negro Zapatilla Slip-On Hydro-Life W de Teva para mujer Gris oscuro Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Wolf Grey / Black-cool Grey-pure Platinum , Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Gris / sal marina , Zapatillas de running Kinvara 8 Saucony para mujer Rojo blanco , Zapatillas de senderismo adidas outdoor Terrex AX2R - Mujer Gris crudo / negro / amarillo semi-congelado , Zapatillas de running New Balance para mujer 590v3 Imán / vórtice , Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Negro 789 Speedo Women s Offshore Amphibious Pull-On Water Shoe Fucsia / Hawaiano , Chaco Mujer Zcloud X2 Sandalia Deportiva Scuba Eclipse , Zapatillas de running NIKE Lunarglide 9 para mujer Pure Platinum / Chrome ASICS Metrolyte II Shoe Women s Walking Negro / Plata / Blanco Chaco Mujer Zcloud 2 Sport Sandalia Angora de llama Sandalia Misty Fisherman de Ryka para mujer Turquesa / Amarillo , Zapatillas para caminar Ww840v2 New Balance para mujer Gris , Zapatillas Adidas de mujer, atletismo y rebote Cross-Trainer Utilidad Negro / Vapor Gris / Negro , New Balance 711v3 Heather Cross Trainer para mujeres Trueno / Poisonbery , Clásicos de base Slip-on (Tm) para hombres de Vans Carbón Sandalia con tacón Florence Block de Via Spiga para mujer Patente negra Sandalia con tachuelas Anica, de Cole Haan, para mujer Cuero desnudo , Boombah Women s A-Game - Juego de calas intermedias moldeadas - 8 opciones de color - Varios tamaños Azul marino / gris Drew Women s Summer Cuero de peltre Zapato de pista Md800v5 New Balance para mujer Oro rosa / negro Australia Luxe Collective Women s Slip-On de pan Arena Trotters para mujer Dea Slingback Pump Lavar florales Xelement 2505 Righteous Womens Black Zipper Botas para moto - 7 - Bota impermeable para excursionistas Zerogrand de Cole Haan para mujer Scooter Rojo Cuero-zinfandel Lana-niebla Slip clásico de mujer de SAS en Moca ,

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 Northside Womens Savannah Sport Savannah Closed Toe Sandal Sport Tan 10485/ Coral 3aacd99 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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