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.

Hecho por el hombre

Importado

Sintético suela

Plataforma mide aproximadamente 0.5 pulgadas

All- Confort

espuma revestimientos, soporte de arco con espuma de memoria, calcetín antimicrobiano, liviano y suela flexible.

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

ALMA NATURAL Sandalia de pescador Beatrice Beatrice para pescador mujer de Rosa 6ee1d5bUpdated 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.

ALMA NATURAL Sandalia de pescador Beatrice Beatrice para pescador mujer de Rosa 6ee1d5b

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

Earth Origins Sophie Escarlata , Sandalia de cuero Velope de nueve mujeres del oeste Luz natural Skechers sandalia ancha de deporte 600-brilliancy para mujer Armada , Sandalia KEEN Women s Uneek Slide-w Negro Negro , KEEN Hombres Newport H2 Sandal Imán / Capuchina , KEEN sandalia de Venecia para mujer Spray de oliva / surf negro Sandalia Juniper Mujer Chaco Negro , Sandalia deportiva Nolita para mujer Anne Klein Luz Natural / Oro Metálico , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Black Smooth , Sandalia Yucatan ECCO Mujer Shadow White , Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia gris Sandalia Yucatan ECCO Mujer Multicolor , Skechers Performance Women s The Go 400 Radiance Flip Flop Negro Chaco Mujer Zcloud X Sandalia Deportiva Dahlia Eclipse , Sandalia Yucatan ECCO Mujer Negro / topo / negro Sandalia Yucatan ECCO Mujer Biscaya Earth Origins Sophie Women s Sandal Negro , Wolky Comfort Jewel Black Croco , Zapato de vestir uniforme al aire libre Vulc N Vent Slip para mujer de Columbia Tiburón, fósil , Sandalias KEEN Mujer Whisper-w Vino de uva / vino de uva , Sandalia NIKE Benassi JDI para mujer Habanero Rojo / Habanero Rojo Adidas Women s Adilette CF + Logo W Sandalia deslizante Black / Vapor Gray Metallic / Trace Maroon Sandalia deportiva Big Water para mujer Columbia Cielo azul, blanco Sandalia Drew Element para mujer Nubuck marrón , Sandalia deportiva Alise Les Filles para mujer Dusty Lilac Nappa , Chaco Women s Z2 Classic Athletic Sandal Mandarín de mimbre , Tacones altos Mujeres Chanclas Sandalias de verano Plataforma Cuñas Zapatillas Chica39; s Fashion Beach Shoes rojo Chaco Mujer ZX1 Classic Sport Sandalia Pep Pine Chaco Zcloud 2 Sport Sandal para hombre Pirámide Verde ,

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 Brooks Womens Revel Maadular Verde / Menta / Blanco Zapatos Go Walk Go Step Original de Skechers para mujer Negro Negro , Wolky Comfort Jewel 30100 Cuero blanco , Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco Lanzamiento de Brooks Womens 4 Electric Brooks Blue / Toreador / Blanco , Sandalia Drew Angela para mujer Negro , Zapatillas Drew Shoe para mujer Paige Ternero blanco Chaco Women s Z2 Classic Athletic Sandal Glaciar negro El corredor de trail impermeable Xa Pro 3D Cs para mujer de Salomon Sombra , Zapatillas de trail running Salomon Women s Sense Mantra 3 W Cosmic Purple / White / Coral Punch Zapatillas de running NIKE Huarache Run SE para mujer Negro / Negro-blanco Zapatillas de running Saucony para mujer Peregrine 5 Trail Gris / Menta / Púrpura , Hush Puppies Men s Gus Oxford Cuero blanco deportivo Bota Cami Alegria para mujer Tile Me More Dusk Sandalia Pepita de mujer Fisher Negro Negro Skechers Women s Go Walk Joy-15615 Sneaker Multi , Sandalias SAS para mujer, Nudu Slide Oro Zapatillas de deporte Fit Fitness GEL de ASICS para mujer Taupe / Cotton Candy / Coral Rose , Reebok Zapatillas de deporte Royal Aspire 2 Fashion para mujer Us-black / Rose Gold / White Zapatillas de skate Vans Unisex Authentic Authentic Decon (Cuero Premium) Wild Dove / True White , Skechers para mujeres Reggae Fest-Marlin-Fisherman Open Back Mule Ajuste relajado y espuma de memoria a / C Zapato de agua Carbón , jiandick Womens Rhienstone Chunky Heels Noche vestido de novia Sandalias de diapositivas Oro BareTraps Sandalia para mujer Flossey Slide Mezclilla , Sandalia Victoria Slide de mujer SCHUTZ Perla Zapatillas de ciclismo Pearl XZUMi W X-ALP Drift IV para mujer Rojo Rouge W X-ALP ENDURO IV Negro / Paloma Zapatillas de fútbol Copa Mundial de Adidas Performance para hombres Negro / Blanco / Negro Zapatilla Lenox de mujer TOMS Jersey de brezo gris hierro forjado Zapato femenino Keli de Alegria White Tooled Sam Edelman Loraine Loaine para mujer Black / Multi Jacquard ,

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 ALMA NATURAL Sandalia de pescador Beatrice Beatrice para pescador mujer de Rosa 6ee1d5b 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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