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

Zapatillas destello Zapatillas de running AMPLICA Coral para mujer ASICS Azul índigo/ Azul índigo/ Coral destello 9d4dac7Updated 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.

Zapatillas destello Zapatillas de running AMPLICA Coral para mujer ASICS Azul índigo/ Azul índigo/ Coral destello 9d4dac7

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 Originals Women s Tubular Dawn Originals Trace Cargo Zapatillas adidas Ultraboost X All Terrain LTD Shoe para mujer Negro Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Blanqueado Lila / Rosa Blast-white Calzado de running Filaity Memory Finity de Fila Negro, blanco, plata metalizado Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Negro / rosa pavo real / ciruela Caspia , Calzado de running Brooks Womens Ariel 16 Overpronation Stability Mood Indigo / Capri Breeze / Grisalle , Zapato de running ASICS para mujer Torrance Negro / Gris piedra Zapatillas de running Nike Zoom para mujer Cool Grey / Black / Fire Pink / Dk Grey , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Púrpura / rosa / verde azulado , Zapatillas de running con amortiguación W560v7 para mujer New Balance Gris negro , Calzado deportivo de punto Chelsea con memoria de Fila para mujer Blanco / Blanco / Highrise Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / Blanco / Rosa , Zapatillas de running Reebok para mujer Negro / Rosa ardiente / Blanco NIKE Women s Season TR 5 Cross Trainer Stealth / Pink Pow / Cool Gray / White , En running para hombre Cloudflow Malibu / Neon Sneaker - 12 D (M) US - Zapatillas de deporte clásicas para uso general Summer Utility WL696 para mujer de New Balance Playa Arena / Libélula Pantalón largo clásico NIKE Golf para mujer Tech Essentials Negro / Racer Rosa / Antracita / Cool Gray Zapato deportivo para acolchado 680v5 New Balance para mujer Silver Mink , Under Armour Women s Dash 2 Apollo Gray (101) / Blanco , Zapatillas de running NIKE Mens Lunarglide 8 Racer azul / negro Zapatillas de running ASICS Gel-Kayano 23 para mujer Púrpura / Plata / Menta Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Negro / Blanco / Antracita , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Negro / Blanco / Antracita Zapatillas de running adidas Alphabounce Beyond para mujer Gris / gris / naranja claro , NIKE Free RN - 831509601 Fuego rosa / azul resplandor / luz violeta / rosa explosión Zapatillas de running NIKE Air Max Torch 4 para mujer Negro / Plata Metálico / Rosa Flash , Zapatillas de running ISO para mujeres de la guía Saucony Blanco azul adidas Performance Ultraboost X para mujer Trace Scarlet / Crayon White / Trace Orange Zapatillas de running Mizuno Wave para mujer con sombra Griffin / Blanco / Eléctrico ,

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 Sandalia Yucatan ECCO Mujer Polvo blanco , Sandalia de mujer KEEN Rose Blue Opal / Provincial Blue Sanda Verra Teva para mujer Negro. NIKE WMNS Zoom Fly Women 897821-411 Provence púrpura / negro , Zapatillas de running adidas Women s Cf Superflex W Blanco / negro / negro , Zapato de agua Helly Hansen para mujer Aquapace 2 Penguin / Bright Bloom / Carboncillo Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Gris azul , Converse Chuck Taylor All Star Core Ox Monocromo negro , Zapatillas de entrenamiento NIKE Flex Supreme TR 5 Cross para mujer Blanco / Plata Mate / Melón Brillante Entrenador campeón Gusto Cross Trainer Malla azul claro , Plataforma de satén mono de ante PUMA para mujer Silver Pink-whisper Zapatillas DC Mujeres Chelsea TX SE Sneaker Gris / gris / gris , Zapatillas de tenis adidas Barricade Classic Wide 4E, Negro / Plateado Metálico / Blanco, 6 M US Blanco / negro / gris medio Vans Auténtico Negro / Blanco verdadero Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco NIKE Blazer Mujer Low SE Premium Birkenstock Mujer Mayari Birko-Flor Sandal Plata , Kea de las mujeres de Flexus Ante rojo , Sandalias planas de piscina Walking Cradles para mujer Abrigo blanco de cachemira / corcho , Sandalia Enroelia Slide de ALDO para mujer Blanco , Bernardo Women s Bridget Slide Sandal Espresso Glove , 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 1643- Black Weave Pu y Black Mesh Bota de moda Emma Tall BEARPAW para mujer Arce Blondo Women s Velma Waterproof Riding Boot Negro Ariat Womens Volant S Tall Zip Tall Riding Negro , Zapatilla de mula de bosque de mujer bellota Búho de chocolate TOMS Clásicos de mujer (Negro) Bomba Hazel para mujeres Sam Edelman Tela de Jute Glam LFL by Lust for Life Sloop Oxford Flat para mujer Rosa , Nueve siete de cuero genuino de las mujeres punta estrecha tacón grueso elegante negocio hechos a mano bombas zapatos Verde ,

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 Zapatillas destello Zapatillas de running AMPLICA Coral para mujer ASICS Azul índigo/ Azul índigo/ Coral destello 9d4dac7 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

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