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

Zapato para Zapato caminar Skechers Women s 19926 You Negro Negro 7a65fbbUpdated 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.

Zapato para Zapato caminar Skechers Women s 19926 You Negro Negro 7a65fbb

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

Easy Spirit Deiny Women s Walking Negro Zapato para caminar New Balance Women s Womens 928v3 Walking Shoes Gris , La alegría de la mujer de la tierra Plata Zapatillas Sneachers Sport Women s Microburst Supersonic Fashion Sneaker Armada Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Plata Zapatillas Isla de las mujeres Softinos Blanco Zapatillas clásicas Vionic para mujer Walker Blanco / rosa Zapatillas de paseo Skechers Performance para mujer Go Walk 4 Kindle Slip-on Blanco negro , Zapatilla deportiva Kailee de mujer Ryka Frost Grey / Cotton Candy / Summer Grey , Zapatillas para caminar ProFit TravelFit Slide para mujer Oro / negro , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Negro Metálico Zapatillas para caminar ProFit TravelFit Slide para mujer Negro rojo , Zapato de caminata Skechers Performance para mujer Go Walk 4 Pursuit Azul marino / gris Zapatillas de deporte de cuero de Praga Drew Shoe para mujer Estaño , Zapato de agua Easy Spirit para mujer First Walker Azul-menta , New Balance WW577 Hook and Loop Walking para mujer Blanco , Zapato para caminar WW411v2 New Balance para mujer Blanco / rosa , Drew Shoe Athena para mujeres Blanco / Naranja , Sperry Womens Fathom Berry Pink , Zapatillas de deporte Arizona Trotters para mujer Cuero suave Nappa de grano completo negro Orthofeet Probado Alivio del dolor cómodo Fascitis plantar Diabético ortopédico Pies planos de juanetes Atlético femenino Zapatos - 910 , Botas de mujer Naturalizer Wanya Sage Nubuck Orthofeet más cómodo diabético tendón de Aquiles dolor de pies profundidad 940 zapatillas de mujer , Zapato para caminar con cordones Go Step de Skechers Performance para mujer Azul marino / blanco , Prop & eacute; t Olivia Oxford para mujeres Hueso , Zapato de mujer Farahh de Jessica Simpson Tela natural Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata blanca Zapatillas para caminar Go Walk 4 Propel de Skechers Performance para mujer Difuso: negro / negro , Drew Shoe Women s Tulip Comfort Hueso

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 para correr Altra Women s Torin 3.0 Lima , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Solar Rojo / Negro / Mango Under Armour Women s Charged 24/7 Low NM Acero (035) / Gris Nublado , Zapatillas náuticas para mujer Saucony Originals Broncearse , Zapatillas de running NIKE Huarache Run SE para mujer Plata Metálica / Plata Mate , Sandalia Savannah de Ryka para mujer Gris oscuro , Salomon Women s Speedcross 4 W Trail Runner Rojo tibetano / sangría / negro Zapatillas de running Gel-Moya para mujer de ASICS Azul negro , Ros Hommerson Calypso Slip para mujer Arena , Zapatillas de entrenamiento WX624v2 para mujer New Balance Blanco Zapatillas de running NIKE Lunar Skyelux para mujer Blanco / puro Platinum-off blanco , LifeStride Women s Camille Sandal Negro Diapositiva Encanta Teva para mujer Negro Sandalia Cecily con cuña Dansko para mujer Camello molido nubuck , Sandalia Sintética Mana de Nueve Mujeres del Oeste Medium Gold Synthetic , Skechers Women-14960 Sneaker Blanco Entrenador cruzado Fresh Foam Crush V1 de New Balance para mujer Dusted Peach / Sunrise Glo , Zapatillas de tenis 696v3 New Balance para mujer Pigmento / coral vivo , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Cumbre Blanco / Obsidiana / Universidad Rojo / Blanco , Zapato de skate casual Heathrow DC para hombre Negro / Turquesa / Blanco Supra Method Skate Shoe Azul marino / negro blanco , Converse Chuck Taylor All Star Seasonal Canvas High Top Sneaker Armada Sandalia Cato Dolce Vita para mujer Pelo de ternero leopardo , Sandalia Venice de Alegria para mujer Glimmer Glam Sandalia de mujer Lucienne Suede Slide de Nine West Suede verde claro , Elanor femenino de UGG castaña , 206 Cojín de mocasín de piel de oveja Pearson de mujer Castaño de gamuza , Vionic Womens Elevated Becca Wedge con cordones Negro Orthofeet Probado Pain Relief Tivoli Cómodo apartamento Pies Fascitis Plantar Diabetic Orthopaedic Mujeres extra anchas Botas de velcro , TOMS - Slip-On de lienzo clásico para hombre Lienzo de Shade Heritage

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 Zapato para Zapato caminar Skechers Women s 19926 You Negro Negro 7a65fbb 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