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.

Lona

Lona superior con TOMS puntera, y V elástica para fácil encendido y apagado

TOMS plantilla de gamuza clásica con cojín para comodidad

Látex inserción de arco para soporte adicional

One-piece suela para flexibilidad y durabilidad

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

TOMS 10812 - Suela de Gris lino B01LZ8N9OQ de lino clásica, cómoda y de ajuste fácil para mujer Gris claro fb8dea2Updated 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.

TOMS 10812 - Suela de Gris lino B01LZ8N9OQ de lino clásica, cómoda y de ajuste fácil para mujer Gris claro fb8dea2

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

Zapatillas Slip-On Roamer SAS para Mujer Mezclilla Trotters Liz Loafer para mujer Hueso , Trotters Liz Loafer para mujer marrón , Kmeioo Slip On Mocasines, mujeres Bordado mula Zapatos con felpa Terciopelo de piel de cordero Sin respaldo Toe Mule Slides Zapato negro sólido TOMS - Slip-On de lienzo clásico para hombre Acero gris microfibra monocromo , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Infinity Blue Abstract Leaf , Propet Women s Tour Walker Strap Sneaker Negro , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Negro / Tweed , CLARKS Mujer Ashland Bubble Slip-On Cuero marrón TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Lona azul marino , Propet Women s Tour Walker Strap Sneaker Blanco deporte Zapato Lites Bates para mujer Blanco , CLARKS Mujer Kinzie Light Loafer Flat Negro Mocasines casuales Alpargata Deconstructed de TOMS para mujer Black Metallic Woven , Zapatillas de deporte Slace Sport Sport D Lites Slip-On Mule Plata blanca TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Puntada azul marino de gamuza / manta Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Lona negra / blanca , Mocasín con cordones ocasionales de cuero KEESKY para mujer Zapatos sin cordones planos rojo , ENMAYER Mujeres Moda retro negra Mocasines con punta redonda Resbalón en piso al aire libre Zapatillas con peludo y hebilla Negro (vaca dividida) , TOMS - Slip-On de lienzo clásico para hombre Lona negra de poliéster , Propet Women s Tour Walker Strap Sneaker Blanco azul Zapato femenino Keli de Alegria Hola novia Ballet Tali de mujer Cole Haan Bow Bow Maple Sugar Leather Trotters Liz Loafer para mujer Negro acolchado en relieve Zapatillas de deporte Slace Sport Sport D Lites Slip-On Mule Gris blanco CLARKS Mujeres Sillian Bella Mary Jane Flat Gris sintético , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Ante negro TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Caoba roja , TOMS Clásicos de mujer Lienzo de rayas gris oscuro ,

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 entrenamiento de vuelo Jordan Nike para hombre Cool Gray / Cool Gray / Wolf Gray , Sandalia Uneek O2-w de mujer KEEN Black / Harvest Gold , Skechers Performance Women s The Go 400 Radiance Flip Flop Natural , Wolky Comfort Jewel 12000 Black Nubuck OluKai Ohana Burbujeante / negro Brooks Womens Revel Negro / antracita / Primer gris , Teva Terra-Float Active Lace - Mujeres Chispas de chocolate , adidas Performance Ultraboost X para mujer Trace Scarlet / White / Trace Orange NIKE Women s Tanjun Verde al aire libre / verde vintage Zapato de running X Mission 2 para mujer Salomon Mystic Purple / Light Grey / Granny Green , Saucony Triumph ISO 3 Running Sneaker para mujer Rosa / Berry / Plata Zapatillas de trabajo de seguridad Reebok para mujer con puntera de material compuesto Ateron Cross-Trainer Negro / Plata , Zapatillas de deporte Skechers Sport Women s Empire Inside Look Punto negro / blanco , Skechers Women s Walk Walk Lite-15375 Sneaker Negro / gris , Calzado para caminar Flare Drew Shoe para mujer Bone Calf / Bone Sporty Sandalias de pescador lisas Vintage Ariel de Dr. Martens para mujer Negro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Blanco azul Earth Origins Womens Gaven Gray Smu Entrenador Ryka Eva Nrg Cross para mujeres Amarillo negro , Vivien Flat de Bella Vita para mujeres Blush Kid Suede Leather , Birkenstock Mayari Black - 071091 Schwarz Lack Sam Edelman Sandalia Glenn para mujer Cuero nuevo y caído negro Bloch Dance Boost Mesh DRT Split Sole Dance Sneaker Negro , Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 1620- Tan Satin Mujeres de Modemoven Charol Bombas, Point Toe Tacones, Slingback Bombas, zapatos de noche, lindo Stilettos 10CM Leopard White Bomba de vestido Kalila Total Motion Kalport para mujer Ternero negro , 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 7001- Blue Velvet Bomba de vestido LifeStride Sevyn para mujer Denim oscuro , Zapatillas de deporte Easy Spirit AP1 Sport Azul-multi-floral , Zapatos De Bowling Dexter Keegan Blanco

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 TOMS 10812 - Suela de Gris lino B01LZ8N9OQ de lino clásica, cómoda y de ajuste fácil para mujer Gris claro fb8dea2 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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10120201

11589630

13435273

14732736

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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