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

Talón Altura: 1 1/4

Frente Altura: 1/2

Ajuste: Fiel al tamaño

Características de este artículo incluyen: Pies diabéticos, Cuidado y bienestar de los pies, Heel Pain & Heel Spurs, Medicare Aprobado, Ortesis

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 Verde Fucsia de B06ZYHFBSJ mujer Drew Alana Verde/ Fucsia edeccf0Updated 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 Verde Fucsia de B06ZYHFBSJ mujer Drew Alana Verde/ Fucsia edeccf0

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

Sandalia deportiva Chaco Zvolv X2 para mujer Azul volcánico Sandalia NIKE Benassi JDI para mujer Blue Glacier Blue Chlorine Blu Drew Womens Cascade Nubuck negro Adidas Women s Adilette CF + Logo W Sandalia deslizante Blanco / Negro / Choque Rosa , CLARKS Womens Wave Grip Smokey Brown Sandalia deportiva Tirra para mujer Teva Simplemente Taupe , Sandalia NIKE Kawa Slide para mujer Hot Punch / Paramount Blue Wolky Comfort Jewel Caviar negro ALMA NATURAL Sandalia de pescador Beatrice para mujer Chocolate OLUKAI Pua Sandal - Mujer Deep Red / Bean Sandalias KEEN Venice II H2-W para mujer Grape Kiss / Red Violet Earth Origins Sophie Azul marino Skechers Performance On The Go 400 Vivacity Flip Flop Blanco / azul claro , Sandalias Terrell Ivy Lattice Sport de Merrell para mujer Negro , Sandalia Drew Element para mujer Denim Blue Nubuck , Sandalia para mujer Teva Terra-Float Travel Knit Legion Blue , Sandalia Archie Fisherman del Dr. Scholl Nubuck negro Sandalia Terrell Ivy Post Sport Merrell para mujer Sodalita , Sandalia deportiva Nolita para mujer Anne Klein Luz Natural / Oro Metálico Sandalias Newport H2 de KEEN para mujer Tinte en espiral Sandalia Savannah de Ryka para mujer Azul Medieval / Azul Suave Sandalia deportiva Tirra para mujer Teva Grey Gradient , CLARKS Womens Wave Grip Sage Nubuck , Sandalia deportiva Chaco Zvolv 2 para mujer Eclipse estándar , Earth Origins Sophie Frost Grey , Wolky Comfort Jewel 70100 canales blancos Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Black Sport , MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 melocotón , Propet Women s TravelActiv Ss Sandal Negro plata ,

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 Wolky Rio Sandal Círculos negros , KEEN Hombres Newport H2 Sandal Imán / Capuchina , Zapatillas de trail running Icebug Women s Mist RBX9 Shell / Orquídea Zapatillas de running Reebok para mujer Print Ultk Cool Shadow / Stark Grey / Desert Glow / White / Silver Zapatillas de senderismo Merrell Moab 2 GTX para mujer Negro Zapatillas de viaje Proather para mujer II rojo , P W Minor Leisure Mujer Terapia Casual Extra Profundo Calzado Cuero Cordones Negro , Calzado de caminata Allrounder by Mephisto para mujer Firelli Tinte naranja quemado lavado Drew Smiles Slip para mujer Negro / Microdot , Sandalia Teagan de mujer Adrianna Papell Negro , Zapatillas para caminar Skechers Go Step Lite-Wander para mujer Lite-gris , Sandalias de vestir de piña de mujer de Jack Rogers Blush / Gold , Botín para mujer Pleaser Asp1018 / Bpu / M Faux blanco, blanco , Skechers Sport Women s Burst City Scene Zapatillas de moda Blanco negro ASICS para mujer Gel-Fit Yui 2 SE Negro / Limelight / Plata , NIKE Air Huarache Run Ultra Dessert Sand / White-Dessert Sand (Mujer) Sam Edelman Women s Slide Sandal de mujer Almendra / Sedona Naranja , Sam Edelman Sandalia Bernice Slide para mujer Marfil moderno SCHUTZ Sandalia Yaslin Slide para mujer Negro , Sandalias Jerusalem Sandalia David Slide marrón , Sam Edelman Women s Slide Sandalia Yute / Caramelo Dorado , Zapato de voleibol Boost W Volley Response 2 de adidas Performance para mujer Limo semi solar / plata metalizada / negro 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 16612- Escala de grises y malla negra Salón de baile de mujeres Zapatos de baile Tango Zapatos de boda de la salsa Sera7017EB Cómodo-Muy Fino 3 \ de 5] Tan satinado , Skechers BOBS de Women s Keepsakes High-Snow Cat Slip on Slipper Negro Negro Bernie Mev Women s Gem Wedge Pump Soportar , Blundstone 510 - Tallas para mujeres, Black Premium Lthr, (A prueba de la intemperie, Resistencia al deslizamiento), W / Black Elastic Sangre Zapatillas de running NIKE Men s Train Speed 4 Jade / Midnight Turquoise / Black / Volt , Mocasín Juliet de Old Friend Women castaña Zapatos OLUKAI Pehuea - Mujer Stormy Blue / Stormy Blue ,

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 Verde Fucsia de B06ZYHFBSJ mujer Drew Alana Verde/ Fucsia edeccf0 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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2581850

4024169

5074935

9525877

9636084

10936137

13061954

18361215

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