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

amor tu miras cada vez que te pones esto elegante bota de montar

Faux parte superior de cuero con una silueta limpia y cierre con cremallera lateral

Ancho el eje acomoda terneros atléticos

Fácil encajar con el Simply Colección Comfort®

Tracción suela proporciona estabilidad extra

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

Bota para de montar Xandywc para mujer LifeStride, LifeStride, 5684 pantorrilla ancha Poliuretano negro 2c6e8e4Updated 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.

Bota para de montar Xandywc para mujer LifeStride, LifeStride, 5684 pantorrilla ancha Poliuretano negro 2c6e8e4

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

Bota de vaquero western Ariat Women s Heritage Roper Marrón angustiado Bota de montar Legivia de Call It Spring para mujer Caqui , Laredo Women s Miss Kate Western Boot Teal , Twisted X Boots Mujeres Hooey azteca patrón lienzo ocasionales zapatos Multi , Ariat Womens Heritage Contour Field Zip Tall Riding Negro Botas de trabajo de hojalata South Women by SW Work Boot Multi , Bota Muck Boot para mujer Arctic Sport Ii Tall Snow Boot Acai Purple , Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Negro , Bota occidental con flecos y tachuelas Roper para mujer Black / Crystal Stud , Noble Outfitters Womens Muds Stay Cool 6 Herradura de oro Noble Outfitters Womens Muds Stay Cool 6 Negro Bota de campo soberano de Mountain Horse Negro , Bota Ecuestre Subtlewc LifeStride para mujer Negro , Bota de punto de mujer Joan of Arctic Knit II de SOREL Negro , Incrustaciones y cristales de 14 pulgadas en color blanco Off Glitter de Corral, para mujeres Hueso Dublin Ladies Foundation Zip Paddock marrón , Bota de trabajo con punta de acero resistente al agua EH PO 11 BRN de John Deere para hombre marrón Bota con cordones Twisted X Boots WAL0007 All Around para mujer Sillín engrasado / rosa , Bota Ecuestre Rosaria-Wc LifeStride para mujer Black Wc , Twisted X Ladies Nude Print Driving Mocasines Bombardero / Desnudo , Bota Ecuestre Sam Edelman Penny 2 para Mujer Cuero negro Ariat Womens Volant S Tall Zip Tall Riding Negro Bota de campo Ovation Flex Sport para mujer Negro Bota de montar Xandy para mujer LifeStride Bronceado oscuro , Ariat Womens Volant S Tall Zip Tall Riding Negro Bota Ecuestre Xandy LifeStride para mujer Gris oscuro Bota de campo soberano de Mountain Horse Negro Bota de trabajo Darlin Darlin para mujer, color marrón oscuro, 10 B US Viejo negro Botas de vestir TuffyRider Belmont Ladies Negro 105 LD - ,

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 Calzado de running ft. Perpetua ft de memoria Fila Women s Castlerock, blanco, negro , PUMA Men s Roma Basic Sneaker Blanco / gris claro Zapatillas para correr ASICS Gel-Venture 6 para mujer Lima negra / carbón / neón , Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Cobblestone Anthracite 006 Zapato para caminar WW577 para mujer New Balance Blanco ASICS Kanmei Shoe Casual para mujer Negro / Negro / Blanco , Chaco Mujer Zcloud X2 Sandalia Deportiva Kelp Knit Supra Martine Run Skate Shoe para hombre Melocotón tropical / carbón de leña , Skechers Women s Spirit Walking Mint B (M) US menta , Jeffrey Campbell Sandalias Lindsay para mujer Negro Tierra Everwood Almendra , Easy Spirit Women s Motion Lace up Oxford Negro , Diadora N9000 Men Toe Snthetic Blue Sneakers Castillo de roca , Clásicos de base Slip-on (Tm) para hombres de Vans True White (Lienzo) , Zapatillas de paseo Skechers Go Walk Lite-15433 para mujer Gris pardo Zapatillas de tenis HEAD Revolt Pro 2.0 para mujer Blanco , Diadora Unisex EVO Aeon Beige Vaniglia Supra Vaider Skate Shoe Negro / Blanco-blanco / Negro Zapatillas de skate solar NIKE SB Check para hombre NIKE Goma antracita negra marrón J Slides Sandalia de mujer Naomi Slide Magenta , Sandalia Kai Dolce Vita para mujer Cuero Ivory , Sam Edelman Women s Bay Slide Sandal Peltre Metálico Cuero , Zapato Capezio Mary Jane Tap para mujer Negro Zapatos muy finos EK8881 de la danza de salón de señoras de las señoras con 1 \ , Bomba de vestido Ivanka Trump para mujer Liah Tela negra / multi Sandalia Birkenstock de cuero Florida para mujer Sfb Black Birko-flor Sandalia Birkenstock de cuero Florida para mujer Negro , Bota Freespirit Sbicca para mujer Coñac Botas Eric Michael Laguna para mujer Negro , Bota unisex New Rock M.1482X-S1 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 Bota para de montar Xandywc para mujer LifeStride, LifeStride, 5684 pantorrilla ancha Poliuretano negro 2c6e8e4 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