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.

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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

Salomon Sense Ride Running Shoe - Mujer Ride Blue Bird Blazer azul/ Deep Lagoon/ Blazer azul marino b984901Updated 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.

Salomon Sense Ride Running Shoe - Mujer Ride Blue Bird Blazer azul/ Deep Lagoon/ Blazer azul marino b984901

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 de running adidas Galaxy Trail W para mujer Gris oscuro / gris Tech / naranja brillante , Salomon Women s Speedcross 4 W Trail Runner Bird Off Paradise , Zapatillas de running ONEMIX para hombre y mujer Zapatillas de deporte para correr Blanco negro Zapatillas de running Pearl NZUMi W s Trail N 2 para mujer Rosa eléctrico / gris Zapatillas para correr Saucony Women s Cohesion 10 Gris / negro / azul La Sportiva Akyra GTX Running Shoe - Mujer Pizarra / Azufre Zapatillas de trail running Altra King MT 1.5 - para mujer Negro Zapatillas impermeables Columbia para correr Trail Trail ATS Lite para mujer Negro, Steam Zapato de mujer Columbia de Redmond Trail Boulder, azul cielo Zapatillas de senderismo The North Face Ultra Fastpack II GTX para mujer Tnf Black / Bright Navy , Salomon Women s X-Mission 3 W-w Acuario MT-onemix retro ligero jersey superior amortiguador de choque imprimir zapatillas de deporte al aire libre Blanco negro , Reebok Women s Ridgerider 2.0 Trail Runner Lila Shadow / Coll. Zapatillas adidas outdoor Terrex Agravic GTX para mujer Gris Tres / Blanco / Coral Tiza Zapatillas para correr Under Armour UA Fat Tire 2 para mujer Premier Purple / Rhino Gray / Black Salomon Women s Speedcross 4 W Trail Runner Azul , Zapatillas de running Mizuno Women s Wave Kazan Trail Gunmetal / Yellow Cream , Calzado de running Trail para mujer New Balance para amortiguar 620v2 Granada / Outerspace Zapatillas Merrell Women s Vapor Glove 2 Barefoot Trail Running Black / Castle Rock , Zapatillas de trail running Xa Pro 3D GTX W para mujer Salomon Sombra , adidas outdoor Terrex AX2R Mystery Petrol / Mystery Petrol / Easy Coral Zapatillas deportivas - Merrell Women s Bare Access Flex Trail Runner Paloma Zapatillas de mujer Salomon X Ultra 3 W para excursionismo Dirigir , Zapatillas de trail WT610V4 para mujer New Balance Gris / morado , Zapatillas de running adidas outdoor Kanadia 7 Gore-TEX Trail para mujer Vapor rosa / Shock Slime / Rosa cruda , MT-onemix Women Men Awesome 3-en-1 zapatos ligeros Athletic Sneaker de Roma Verde claro / blanco I , Calzado de running XA Amphib W Trail para mujer Salomon Potente púrpura , Zapatillas Columbia Trail Trail WMNS de Plains Ridge para mujer Guijarro / Aqua Zapatillas de trail running 910 V4 New Balance para mujer Vivid Coral / Clear Sky

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 NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Blanco / negro / puro platino Zapatillas de running Alphabounce Cr de mujer adidas Originals Orquídea Tinte / Blanco / Oro Metálico , Zapatillas de running Mizuno Wave para mujer Wave Horizon Plata / rosa OLUKAI Womens Loea Sandal Mostaza / hueso , Wolky Comfort Jewel 12120 Offwhite Nubuck Zapatillas de agua Astral Brewess 2.0 para mujer Rosa Red , Sandalia de mujer Teva W Terra-Float Livia Tacion Aqua , Zapatillas ONEMIX para hombre, para mujeres Zapatillas ligeras, transpirables, para caminar al aire libre Zapatillas tipo calcetines Oro , Drew Smiles Slip para mujer Gris pardo , Sandalia de tacón de gamuza Salomé de mujer Anne Klein Azul claro Drew Smiles Slip para mujer Gris pardo , Drew Shoe La alegría de las mujeres Blanco , Sandalia P-Vinton de mujer amarilla Bronce Easy Spirit Women s Traveltime Mule Azul marino / ante blanco , Zapato San Antonio Mujer SAS, Zapatilla de deporte Freetime con cordones Blanco Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Negro Metálico , Zapato para caminar con cordones Go Step de Skechers Performance para mujer Gris pardo , Zapatillas Slip-on de mujer Skechers You Inspire Natural , nativo AP Mercury Liteknit Root Red / Shell White / Natural Rubber Calzado de entrenamiento casual Comfort New para mujeres wx857v2 de New Balance Blanco azul , Vans Unisex Old Skool Zapatillas clásicas de skate High-rise / Pewter Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Negro , Zapatillas para mujer Del Rey TOMS Arpillera Stuart Weitzman Landslid femenino Jeans ante Sandalia Otto Slide de mujer Yellow Box Aceituna , Mocasines de estilo de conducción Suzi de mujer Roper Tan 3 , NIKE Hyperdiamond 2 Keystone Softball Cleats para mujer Gris blanco , Zapatos OLUKAI Pehuea - Mujer Dusty Olive / Palm , Zapatillas de deporte lavables Walker Propet para mujer Gris Easy Spirit Women s Traveltime Mule Blanco 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 Salomon Sense Ride Running Shoe - Mujer Ride Blue Bird Blazer azul/ Deep Lagoon/ Blazer azul marino b984901 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