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.

98% poliuretano / 2% metal

Importado

Suela sintética

Ola confort

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

Sandalia Easy Slide Slide para mujer Easy Street Easy Armada Armada 67cfb7aUpdated 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.

Sandalia Easy Slide Slide para mujer Easy Street Easy Armada Armada 67cfb7a

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

Sam Edelman Sandalia Glenn para mujer Desnudo natural , Sandalia Aerosoles Midday Women Slide Tela roja Sandalias Birkenstock de mujer con aplicaciones suaves de Granada Cuero aceitado negro , See by Chlo & eacute; Marrón medio Sandalia Aerosoles Midday Women Slide Combo verde claro Sandalia Ryana Marilyn Slide para mujer Blanco negro , ALMA NATURAL Sandalia Adalia Slide para mujer Puesta de sol , Sandalia para mujer Cait Donald J Pliner Almendra Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sandalia Dolce Vita Jene Slide para mujer Denim de Ceniza , Sandalia Aerosoles Midday Women Slide Combo floral , ALMA NATURAL Sandalia Adalia Slide para mujer Color de malva UGG Sandalia Cindi para Mujer Negro , Sandalia Sloan Slide de Naturalizer para mujer Negro Sandalia UGG Royale Slide para mujer Ribbon Red , CLARKS Leisa Cacti Slide Sandalia para mujer Cuero de peltre , Sandalias Hush Puppies para mujer Dachshund Slide Cuero marrón , Sandalia Sloan Slide de Naturalizer para mujer Gunmetal Sandalia Dolce Vita Jene Slide para mujer Light Blush Satin Sandalia de diametro de moda Manaan de 3 bandas para mujer de KAANAS Bronce , Sandalia de cuña OAHU de China Laundry Women Vintage Rose Suede , BareTraps Women s Cherilyn Slide Sandal Negro , Vince Camuto Sanella deslizante de mujer Azul / Natural , Birkenstock Arizona Sandalia de ante con suela suave cobre , Sandalia de la diapositiva de la cruz de Anica Criss Cross de las mujeres de Cole Haan Cuero negro Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sam Edelman Women s Slide Sandalia Cuero de gamuza de cabrito negro Sandalia Sbicca Fairy Slide para mujer Azul / Multi , Sandalias con tachuelas Gale para mujer Donald J Pliner Adular

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 de susurro de mujer Keen Beet Red / Honeysuckle , Zapatillas de running ASICS Gel-Scram 4 para mujer Negro / gris oscuro , Skechers Performance On The Go 400 Vivacity Flip Flop Carbón de leña / rosa fuerte Short NIKE Tempo para mujer Lt Photo Blue / Bright Crimson / Wolf Gray Zapatillas de deporte con diseño hidroeléctrico de mujer W Ahiga V3 de Helly Hansen Azul marino / Blanco apagado / Cayenne Skechers Sport Women s Breathe Easy Lovestory Mary Jane Flat Chocolate Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory , Zapato de running FUZETORA para mujer ASICS Azul oscuro / azul oscuro / rosa begonia , Carlos by Carlos Santana Sandalia Adalyn para mujer Brulée , Drew Womens Element Nubuck negro , Zapatillas de entrenamiento NIKE para mujer Duel Racer Running Negro-blanco-gris oscuro Zapato para caminar de devoción Ryka para mujeres Gris / azul claro / naranja Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Blanco óptico , Ahnu Women s Yoga Flex Cielo polar , nativo AP Mercury Liteknit Jiffy Black / Shell White / Natural Rubber Zapato de agua nativo de Lennox para hombres Regatta Blue / Shell White , Supra Vaider Skate Shoe Gris claro - Blanco Zapatos de golf FootJoy para mujer Enjoy Negro / púrpura , DAWGS Womens Arch Support Sandalias Loudmouth Z Betsy Ross , Nfinity Adult Evolution Cheer Shoes Blanco , Charles by Charles David Sandalia deslizante Goldie para mujer Oro rosa El arranque inicial MuckBoots MuckMaster original Negro Ariat Unisex Concord Chap Half Chaps Marrón claro , La Canadienne Womens Jennifer Cuero negro Bota de bota estilo LYLIANA FRYE para mujer Nubuck negro , Bota de invierno BEARPAW Lauren Tall para mujer Carbón , Bota de mujer Pleaser Electra-2020 / W / PU Patente blanca , VANELi Womens Hattie Tela vidriosa negra Skechers Women s Reggae Fest Willows Flat Malla de carbón ENMAYER Mujeres Moda retro negra Mocasines con punta redonda Resbalón en piso al aire libre Zapatillas con peludo y hebilla 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 Sandalia Easy Slide Slide para mujer Easy Street Easy Armada Armada 67cfb7a 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
1516200

2253796

2992217

6137251

6747527

8310482

9382457

12468574

14916179

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