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

Goma suela

Patente parte superior de cuero

Anatómico corcho y látex plantilla extraíble

Ligero, poliuretano flexible suela

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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

Wolky caviar Comfort Jewel Wolky Cuero de caviar de marino 34d57ccUpdated 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.

Wolky caviar Comfort Jewel Wolky Cuero de caviar de marino 34d57cc

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

Sandalia original premier universal Teva W de mujer Beach Break Desert Sage , OluKai Kulapa Kai Carmesí / negro Drew Womens Cascade Nubuck azul marino , Sandalia Tanga Loea de mujer OLUKAI Negro Negro Sandalia Drew Angela para mujer Bone Smooth Leather , Sandalias KEEN Mujer Whisper-w Sterling Blue / Vestido Azul Drew Womens Cascade Denim Blue Nubuck Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave , Sandalia Yucatan ECCO Mujer Negro Sandalia Drew Element para mujer Nubuck negro , Revs New Classic Masaje de reflexología Sandalias para hombres & Mujer. Mira. Diseño. Terapia. Cojín Comfort Con plantilla de masaje & Absorción de impactos Único. Broncearse Sandalia KEEN Women s Exo-W Uneek Negro / gris acero , Sandalia Jacardia Merrell para mujer Armada Propet Women s TravelActiv Ss Sandal Negro rojo Chaco Z / Canyon 2 Negro acolchado , Sandalia KEEN Women s Venice H2 Cascade / Misty Jade Sandalia Terrell Post II Femenina de Merrell Atlantis , KEEN Sandalias Newport Eco-W para mujer Negro / Imán , Sandalia Kea Vent mujer Columbia Negro, pizarra , Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Acuífero , Sandalia de mujer Drew Alana Pewter-bronce Sandalia deportiva Coleman 2.Zerogrand Slide de Cole Haan, cuero negro Cuero negro Chaco Women s ZX2 Classic Athletic Sandal Bloom Peach , Sandalias Newport H2 de KEEN para mujer Tinte en espiral Chaco Women s Z2 Classic Athletic Sandal Florista , Sandalia de deslizamiento ultra Comfort NIKE para mujer Pure Platinum / Wolf Gray , Teva Terra Fi 4 - Mujeres La Manta Black , Sandalia para mujer Teva Terra-Float Travel Knit Legion Blue , Sandalias Swiftwater Mesh de Crocs 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 Sandalia de susurro de mujer Keen Negro / Gárgola , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Azul profundo / Cumbre Blanco-Negro Camiseta adidas Swift Run W para mujer Gris / Plateado Metálico / Blanco , Zapatillas de running New Balance 990V3 para mujer Armada Salewa Literas Literas para mujeres Negro / Pinky BZees Women s Wink Zapatos livianos de agua Heather Black , Zapatillas para caminar Skechers Go Step Lite-14750 para mujer Blanco negro Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Asfalto , Calzado de espuma fresca New Balance para mujer 1165v1 Púrpura , Tecnología Vionic Men s Orthaheel White / Navy Walker Bronce / negro , Zapatos OLUKAI Pehuea - Mujer Carbón de leña / Dark Shadow Zapatillas para correr W630V5 para mujer New Balance Rosado Zapatillas para caminar Go Step Lite-Petal 14723 de Skechers Performance para mujer Gris pardo Sandalias Chardon de Hush Puppies Chardon para mujer Negro , Sam Edelman sandalia Gio Slide para mujer Sonrojo , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Vino tinto / blanco , Sandalia Nora Ryka para mujer Gris oscuro , New Balance FuelCore Nergize V1 Fuel Core Cross Trainer para mujeres Silver Mink / Dusted Peach , Taipán de mujer de David Tate Verona Cordero blanco , Converse Chuck Taylor All Star Core Ox Blanco plateado Vans Unisex Old Skool Zapatillas clásicas de skate Blanc De Blanc , Zapato de skate DC Magnolia TX para mujer Azul blanco , Sandalias Birkenstock Unisex Arizona Tobacco Oiled Leather - 39 M EU / 8-8.5 B (M) US - , Sandalia de la diapositiva de las mujeres de Franco Sarto de las mujeres Armada Zapatillas de volante Mizuno Wave Lightning Z4 para mujer Azalea rosa / gris carbón , Bomba de Angel II para mujeres estilo suave Elegancia negra , Zapatos de paloma de oro Fiesta Party 1613 Noche de confort Sandalias de bomba de vestir, Salón de baile de mujeres Zapatos de baile (2.5 \ 3 \ Tacones) 1613l- Negro Giro GF23160 Cilindro para mujer Brillante rosa / negro Botín de tobillo Mazin Cayto para mujer Hush Puppies Marron oscuro Rockport Cobb Hill Women s Paulette Flat Gris

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 Wolky caviar Comfort Jewel Wolky Cuero de caviar de marino 34d57cc 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