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.

Sintético

Importado

Suela de goma

Plantilla extraíble

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 de running 15209/ Reebok Print Smooth running Ultk para mujer Negro/ polvo de asteroide/ blanco 9806870Updated 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 de running 15209/ Reebok Print Smooth running Ultk para mujer Negro/ polvo de asteroide/ blanco 9806870

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 NIKE Lunarglide 9 para mujer Cool Grey / Black-pure Platinum , Zapato de running ASICS Gt-2000 5 para mujer Rosa Brillo / Blanco / Púrpura Oscuro Zapatillas Nike MAYFLY WOVEN para mujer 833802 Negro / gris oscuro / negro Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Negro Calzado de running NIKE Kids Roshe One (GS) Pure Platinum Pink Blast 006 , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Black / Cosmo Pink / Winter Bloom New Balance W880gb7 para mujeres Azul claro NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Ocean Bliss / White , Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Blanco Zapato de running 840v4 New Balance para mujer Gris oscuro Zapatillas de running adidas Ultraboost W para mujer Gris / carbono / ceniza de perlas Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / Blanco-gris oscuro , Nike Hombres Air Zoom Pegasus 33 Acción Rojo / Negro-puro Platino-total Crimson , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Arctic Punch / White , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Negro / Fantasma , Brooks Womens Adrenaline GTS 18 Púrpura / rosa / plata , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Diva Blue / Flash Coral / Aqua Splash , Zapatillas de running NIKE Air Zoom Pegasus 34 para mujer Armería Marina / Mint Foam / Glacier Grey / Black , NIKE Women s Season TR 5 Cross Trainer Stealth / Pink Pow / Cool Gray / White , Zapatillas de running PUMA Women s Speed 500 Ignite Tobillo-Alta rojo , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco / Negro / Juego Real Zapatillas de running NIKE para mujer con elemento de cruz lunar Pink Pow / Polar / Black / White Zapatillas para correr Under Armour UA Speedform Gemini 2.1 para mujer Garza / Blanco / Luces púrpuras , Zapatillas de running Nike Zoom All Out Low para mujer negro EN la zapatilla de deporte Running Cloud para mujer Denim / Blanco Zapatillas de running NIKE para mujer Air Zoom Pegasus 34 (N) Mica Blue / White Racer Pink Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Polvo negro / asteroide / peltre / blanco

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 Deportiva Jordan Spires de Via Spiga Rico, amapola, gamuza Brooks Womens Revel Blanco / antracita / negro Zapatillas de running Flash para mujer New Balance Ozone Blue Glow / Vivid Ozone Blue / Silver , Zapatillas de deporte casual Gazelle de adidas para hombre Púrpura , Under Armour Speedform Velociti para mujeres Rhino Gray (076) / Glacier Grey , Zapatillas de running Topo Athletic Ultrafly - Mujer Púrpura / Lila Zapatillas de trail running Saucony Women s Grid Excursion TR9 Gris / Ciruela / Coral New Balance Ww1400v1 Trail Walking Boot-w para mujer Gris Zapato de agua Tetrex Rapid Crest para mujer Merrell Negro Taipán de mujer de David Tate Verona Cordero blanco Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Primer gris / verde azulado victoria / plata Zapatillas de running NIKE para mujer Air Presto Cargo Khaki / Cargo Khaki Zapato para caminar WW877 para mujer New Balance Plata , Zapatillas para caminar Skechers Go Step Lite-Swerve para mujer Armada Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Blanco negro Sanita Sanena Women s Deena Negro , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Naranja / Rosa , Drew Shoe Women s Tulip Comfort Ternero blanco , Zapatillas de tenis adidas Women s Barricade 2018 W Blanco / escarlata / escarlata , Zapatillas Superga 2750 Cotu para mujer Total Rose Mahogany , Zapatos de golf Sandbaggers Madison para mujer Lagarto Blanco , Sandalia UGG Royale Slide para mujer Negro , Zapatillas de volante Mizuno Wave Lightning Z4 para mujer Negro / real , Rieker Women s Elaine 58 por Remonte Boots Murmel Cristallino / murmel Bogota / schwarz Elastique Mocasines casuales Alpargata Deconstructed de TOMS para mujer Desierto Taupe Gamuza Zapatos de tacón alto con tachuelas y tachuelas Kaitlyn Pan en punta Patente fucsia francesa / Nude Trim / Gold Studs , Trotters para mujer Dea Slingback Pump Negro Negro , Bota impermeable Teva W Foxy para mujer marrón NIKE Kids Jr. Tiempo Rio IV (FG) Calzado de fútbol para suelo firme Negro / Blanco / Láser Naranja / Volt , ED Ellen DeGeneres Mujeres Lilliane2 Laguna / Lagoon Woven Dot / Kid Suede

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 de running 15209/ Reebok Print Smooth running Ultk para mujer Negro/ polvo de asteroide/ blanco 9806870 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