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

Sandalias B071NZ9YTX de Mellow 13612 mujer Ros Hommerson Mellow Armada bd4723eUpdated 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.

Sandalias B071NZ9YTX de Mellow 13612 mujer Ros Hommerson Mellow Armada bd4723e

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 paseo Skechers Go Walk Lite-15433 para mujer Armada , Zapato para caminar Skechers Women s You Negro Prop & eacute; t Zapatillas para caminar TravelFit para mujer naranja Zapatillas para caminar adidas outdoor Terrex Voyager CW CP W para mujer Negro / Negro / Blanco Tiza Zapatillas para caminar ProFit TravelFit Slide para mujer Negro plata Saco de lona Drew Shoe Savannah Therapeutic Therapeutic Diabetic Extra para hombre Azul marino / ondulado / estiramiento , Dr. Comfort Lu Lu Profundidad terapéutica para diabéticos terapéuticos Mujer Velcro de cuero para zapatos castaña Zapato atlético para mujeres con neuropatía de Ped-Lite - Adele Negro Zapatillas de deporte de malla SAS para mujer Azul Easy Spirit Solana Oxford para mujeres Gris , Zapatillas para caminar al aire libre Go Skechers Performance para mujer Caqui , Bomba de cuña Lansing para mujer Trotters Desnudo cuero perfilado , Zapato de caminar sin tocar Skechers Performance para mujer Go Step Ante negro Skechers Women s Breathe Easy Sweet Jam Sneaker Rosado Calzado para caminar CUSH + WA365v1 New Balance para mujer Gris I-RUNNER Maria Puntera terapéutica para mujer con profundidad extra Calzado de cuero / malla con cordones Gris / Salmón , Drew Shoe La alegría de las mujeres Black Smooth Leather / Black Stretch Calzado de senderismo Skechers Go s 15601 para mujer Azul marino / blanco Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Blanco negro Zapato para caminar de devoción Ryka para mujeres Gris / Verde Claro / Rosa Oscuro Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Negro / rosa Skechers Performance Women s Define Wide Sneaker Negro , Zapatillas para caminar Cinch de mujeres Devotion Plus de Ryka Vino / Negro , Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr Black , Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On Vino Zapatillas de deporte Romy de Easy Spirit para mujer Blanco-fucsia Zapatillas para caminar Go Walk 4 Super Sock 4 de Skechers Performance para mujer Negro Hush Puppies Power Walker Sneaker para mujer Negro , Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Plata / Lima ,

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 NIKE Women s Air Max 1 Ultra Essentials Cumbre Blanca / Carmesí total Zapatillas de running adidas Alphabounce Beyond W, para mujer Ecru Tint / Ash Pearl / Ash Pearl , Columbia Montrail Women s Fluidflex X.S.R. Phantom Purple, Nappa Green , Zapato para caminar WW577 para mujer New Balance negro , CLARKS Sandalia Arla Kaydin para mujer Tela elástica negra Dyeables, Inc Womens Poppy Dress Pump Satén negro , Zapatillas para caminar ProFit TravelFit Slide para mujer Negro plata , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Acuarela floral Vionic Grace Samar - Sandalia Ajustable Mujer Natural , Nike Hombres Air Zoom Pegasus 33 Acción Rojo / Negro-puro Platino-total Crimson Earth Orígenes Nellie Azul parisino Skechers Womens Microburst - Perfect Note Ballet Flat Blanco negro , Chancleta Yoga Sling 2 Sanuk para mujer Carbón de leña / rayas naturales ALLROUNDER by MEPHISTO Zapatillas Mujer Fina Tex Sport Goma negra / nubuck gris medio Drew Smiles Slip para mujer Azul / Microdot , Nurse Mates Womens - Dash Gris oscuro Zapatillas Superga 2750 Cotu para mujer Armada Drew Shoe Women s Hope cuero terapéutico moda Oxfords Ante marrón SAS Womens Sunny Rojo / Arco Iris , Calzado para caminar Persad de la Mujer del Dr. Scholl s Shoes Infinity Blue Action Leather Diadora Unisex Intrepid NYL Blanco / negro / gris helada , Sam Edelman Women s Bay Slide Sandal Denim Blue / New Blue Calzas de la dinastía Azul marino / blanco , CLARKS Mujer Flores Dahlia Tobillo Botín Grey Suede / Leather Combi , Geox Kookean 5 Oxford para mujer Marina / arena CLARKS Mujer Caddell Hop Boot marrón , Bota chukka de mujer Timberland Londyn 6 \ Nubuck rosa oscuro , SOREL Women s After Hours Lace Up Sandalia de cuero y gamuza Harina de avena , Grandpro Hijas de Cole Haan Tela a rayas 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 Sandalias B071NZ9YTX de Mellow 13612 mujer Ros Hommerson Mellow Armada bd4723e 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