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.

malla

Mesh Upper, Suela de goma

Universal Sistema de flejado Con dos ganchos y Cierres de lazo

Acolchado Plantilla de EVA

Antimicrobiana Protección con Microban patentado

Teva Shoc Pad en el Talón para mejorar Absorción de impacto

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

Sandalia Hurvane XLT de mujer, Teva para Hazel Púrpura mujer, Hazel Black, 9 US Púrpura 06e9c79Updated 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.

Sandalia Hurvane XLT de mujer, Teva para Hazel Púrpura mujer, Hazel Black, 9 US Púrpura 06e9c79

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 casual Slip-On de Bernie Mev para mujer Pantalones , OluKai Ohana Sour Tart / Ice , Teva Terra-Float Active Lace - Mujeres Azul , OluKai Ohana Pewter / Black Sandalias KEEN Venice II H2-W para mujer Paloma / Turquesa Pastel Chaco Women s Zx2 Classic Athletic Sandal Danubio violeta Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Negro Wolky Comfort Jewel Fantasía Roja Multicolor Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso Sandalia Uneek O2-w de mujer KEEN Aura Astral / Libertad , Skechers Performance Women s The Go 400 Radiance Flip Flop Negro , OluKai Ohana Pewter / Black , Sandalias con tira en el tobillo INFINITY 935 para mujer Pleaser Negro Negro Sandalia deportiva Hammond de Spring Step para mujer Armada , Sandalia Omnium de mujer Teva Pizarra , Chancleta Eezay adidas para mujer Blanco / negro / negro , Wolky Comfort Jewel Dos tonos blancos Sandalia de susurro de mujer Keen Poseidon / Blue Danube , Sandalias para mujer Teva Women s W Hurricane XLT2 Kerne Deep Wisteria Earth Origins Sophie Platino , Earth Origins Sophie Frost Grey , Sandalia Teva Women s W Hurricane XLT Infinity Sport de Teva Bosque ártico HOKA ONE ONE Recuperación ORA para mujeres Blanco negro Calzado casual Slip-On de Bernie Mev para mujer rojo , Sandalia de deslizamiento ultra Comfort NIKE para mujer Mango brillante / negro , Rihanna X Puma Leadcat Fenty Sandalias de piel para mujer Orthofeet 872 Women s Comfort Diabetic Extra Depth Sandal OLUKAI Luana Sandal - Mujer Blanco negro , Drew Womens Cascade Nubuck 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 Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue KEEN Hombres Newport H2 Sandal Tinta / óxido de la India , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Hyper Magenta / Royal Pulse , Chaco Mujer Zcloud Sport Sandal Charlie Peach Salomon Women s Speedcross 4 W Trail Runner Bird Off Paradise Zapatilla de deporte Trail running wt690 para mujer de New Balance Teal / rosa Vasque Men & fnof; _Ts Constant Velocity II Trail Running Shoes Vestido Blues / Neutral Gray Zapatillas de mujer Salomon X Ultra 3 GTX para mujer Corona azul Zapatillas de running New Balance 1340v3 para mujer Blanco / morado , Sperry Top-Sider H2O Escape Bungee para mujer Gris , Calzado de espuma fresca New Balance para mujer 1165v1 Gris blanco , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Arctic Punch / White , Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White , Salvatore Ferragamo Women s Groove Bianco / Gancio Oro Spazzolato , Sandalia de mujer David Tate Violet Negro , Sandalia Crystall de FitFlop para mujer Peltre metálico Sam Edelman Women s Slide Sandalia Yute / Caramelo Dorado , Vans Unisex Old Skool Zapatillas clásicas de skate Clásico Negro / Blanco Vans Old Skool Unisex Adults Low-Top Trainers (35u) (Tablero de ajedrez) Negro / Rojo Diadora Unisex I.C. Pompeian Red / Nautical Blue Vans Unisex Old Skool Zapatillas clásicas de skate Lux Leather Porcini negro , NIKE W Air Foamposite One Womens Aa3963-200 Partícula Beige / Partícula Beige Birkenstock Mules para mujeres Gris pardo Sandalias Sandalias Bashan moldeadas Sandalias Sandalias Jerusalem Azul Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar 7010-verde , Magic-MC New Summer Zapatillas Cork Slides Pisos con 2018 Nuevas mujeres ocasionales Deslizamiento en toboganes de playa Zapatillas Flip Flops Talla extra 04 U-lite para mujeres Zapatilla de lana de piel caliente Casual Slouch de invierno Piso plano y al aire libre Loafer, pantuflas de lana para mujeres Negro , Ellie Shoes Women s 711 Coco Platform Sandal Claro y negro Giesswein Unisex Brixlegg Tierra , T.U.K. Negro ,

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 Sandalia Hurvane XLT de mujer, Teva para Hazel Púrpura mujer, Hazel Black, 9 US Púrpura 06e9c79 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