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

Caucho único

Eje mide aproximadamente parte baja del arco

EVERUN plantilla

Engineered malla

Tejida soporte del talón

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 Omni de running Saucony para mujer running Omni 16 Teal B0161ZOSF0/ Citron 3258bd6Updated 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 Omni de running Saucony para mujer running Omni 16 Teal B0161ZOSF0/ Citron 3258bd6

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 New Balance 1340v3 para mujer Blanco / morado Zapatillas para correr Topo Athletic Runventure 2 - Mujer Frambuesa / Negro , Zapatillas de running Under Armour Speedform Intake 2 para mujer Azul , Zapatillas de running adidas Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco Zapatillas de running ASICS Gel-Contend 4 para mujer Plata / Campanula / Carbono , Zapato deportivo para acolchado 680v5 New Balance para mujer Silver Mink , Camiseta adidas Swift Run W para mujer Blanco / Blanco / Blanco , New Balance Women s w990v4 Running Shoe Negro / Plata , Saucony Women s Ride 10 zapatillas para correr Púrpura Citron Zapatillas de running Kineta Relay para mujer Saucony Fuschia / cerveza , Zapatillas para correr Zante V3 para mujer New Balance Thunder / Artic Fox Skechers Sport Skech Air Run High Fashion Sneaker Verde carbón Zapato de running 840v4 New Balance para mujer Gris claro , Zapatillas Reebok Women s Runner 2.0 MT Track Negro / carbón / ácido rosa / blanco , HOKA ONE ONE Zapatillas de running Challenger ATR 4 para mujer Negro / Azalea , Zapatillas de running ASICS GEL-Foundation 12 para mujer Plata / Pistacho / Pink Glow , En la ejecución de calzado de mujer en la nube Todo blanco Calzado de running EM N3 de Pearl iZUMi para mujer Shadow Gray / Argel Blue adidas Alphabounce Rc w para mujer Core Black / Sun Glow / Footwear Blanco Zapatillas de running adidas Questar Ride para mujer Carbono / Negro / Gris Zapato de running ASICS para mujer Torrance Carbon / Pink Peacock / Black Zapatillas de running New Balance 520v3 para mujer Negro Zapatillas de running Mizuno Wave para mujer Wave Horizon Azulejo azul-amarillo , Mizuno Women s Wave Prophecy 5-w Jacinto Violeta / Royal Purple Zapatillas de entrenamiento Nike Womens Metcon 3 Negro / Blanco-volt-cloro Azul , Zapatillas de running adidas Alphabounce Beyond W, para mujer Gray Four / Carbon / Dark Solid Gray Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul Calzado deportivo New Balance para mujer Vazee Rush v2 Negro / rosa , Zapatillas de running NIKE Air Max Thea para mujer brillo fucsia / brillo fucsia ,

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 KEEN Clearwater CNX para mujer Anémona / acacia Sandalia deportiva Tirra para mujer Teva Simplemente Taupe Sandalia Hurvane XLT de Teva para mujer, Hazel Black, 9 US Lago Black / Gray , Zapatillas de running Adidas Performance Ultra s Boost M para hombre Pizarra negra / negra / oscura Zapatillas de running NIKE Unisex Flyknit Racer Legion Blue / Black-gamma Blue Zapatillas Reebok Women s Runner 2.0 MT Track Nube gris / polvo de asteroides / peltre NAOT Womens Shore - Fuente de Triángulos morados Sandalia KEEN Clearwater CNX para mujer Anémona / acacia Zapatillas de deporte Onitsuka Tiger Mexico 66 Blanco / Borgoña Zapatillas para correr NIKE Women s Air Vapormax Flyknit Plum-m Zapatos OLUKAI Pehuea - Mujer Pokeberry / Pokeberry Sandalia de tobillo con hebilla para mujer Merrell, para mujer Negro MONTAÑA BLANCA Miso Mule cuero de mujer Marron oscuro Birkenstock Mujer, Ashley Low Heel Comfort Sandal Iridescent Black Birko Flor , Zapatillas de senderismo Zen Pro para hombre SCARPA Carbón de leña / tónico Zapato de malla Nohea de OLUKAI - Hombre Gamuza Mocha-Mocha , Linea Paolo Felicia | Bordado de gamuza negro SOREL Womens Ella Lace-Up Natural Linea Paolo Felicia | Cuero perforado negro , Deflector de gravedad Alivio del dolor comprobado Desafío G de las mujeres Mighty Walk - Mejor Zapatos para el talón Dolor, dolor en el pie y fascitis plantar Blanco, morado Vans Unisex Old Skool Zapatillas clásicas de skate (Cuero Premium) Negro / Blanco Verdadero Sandalia Bobbi Slide de Dolce Vita para mujer Lino de arena Sandalia Cato Dolce Vita para mujer Cuero rosa dorado , Birkenstock Arizona Sandalia de ante con suela suave Cuero Habana Aceitado , Sandalia de diapositivas de ocio para mujer de Seychelles Ante amarillo Bota de media caña Georgia GB00084 Negro , Zapato deportivo Fiat Fusion para mujer Cocodrilo eléctrico Zapatilla Glamour de mujer verde Daniel Bone Kidskin , Salvatore Ferragamo Womens Pisa 70 Suede de arcilla , TOMS Kids 10009918 Alpargata-K Negro brillo iridiscente

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 Omni de running Saucony para mujer running Omni 16 Teal B0161ZOSF0/ Citron 3258bd6 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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3160460

5597491

7787253

9099702

9368786

11818502

17352207

19016558

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