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

Sintético único

Mixto material superior para transpirable soporte

corte bajo diseño para la libertad de movimiento y más rápido transiciones

Memoria plantilla de espuma de tecnología para una comodidad lujosa

Ligero, superposiciones en relieve

Cmeva entresuela y suela para cojín

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 de running Reebok running Women 19380 s B01N3P1G8I Supreme 3.0 MT Negro/ carbón/ plata/ blanco 12e349eUpdated 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 de running Reebok running Women 19380 s B01N3P1G8I Supreme 3.0 MT Negro/ carbón/ plata/ blanco 12e349e

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 Brooks Womens Glycerin 15 Neutral Max Cushion Plata / gris / rosa , Zapato deportivo para acolchado 635v2 New Balance para mujer Plata / Amarillo , Zapatillas de running adidas Duramo Lite W para mujer Gris Medio / Metálico / Plateado / Claro / Gris , Zapatillas de running New Balance Boracay V3 para mujer Supercell / Ozone Blue / Bleached Sunrise Zapatillas de running NIKE Lunartempo 2 para mujer Negro / Blanco / Antracita Zapatillas de running NIKE Air Max Flair para hombre Negro / blanco negro Zapatillas de running Under Armour Charged Bandit 3 para mujer Glacier Grey (100) / Rhino Gray , Zapato de running FUZETORA para mujer ASICS Azul oscuro / azul oscuro / rosa begonia Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro / Plata / Tinta Peacoat Zapatillas para correr Nike Women s Free Rn Distance 2 Pure Platinum / Black / White Zapatillas de running Nike Zoom All Out Low para mujer Pure Platinum / Cool Grey Zapatillas de running NIKE Mens Air Zoom Structure 19 Gris blanco Zapatillas de running NIKE Air Max Jewell para mujer negro Under Armour Women s Drift 2 Sigilo gris (008) / rosa islandesa , Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Cobre rosa / sal marina / chicle Under Armour Women s Charged Bandit 3 Negro (001) / Glacier Grey , Zapatillas de running NIKE Air Zoom Vomero 13 para mujer Racer Blue / Blue Tint , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco , Under Armour Women s Thrill 3 Apollo Grey (962) / Blanco , Brooks Womens Ghost 10 GTX Negro / Azul pavo real / Malvavisco Zapato deportivo ASICS Gel-Excite 4 para mujer Negro / Blanco / Rosa Knockout Calzado de running NIKE Kids Roshe One (GS) Blanco / puro platino / blanco , Zapatillas de deporte PUMA Mens Fierce Core Lightweight Training Dahlia roja Calzado deportivo New Balance Vazee Urge v1 para mujer Rosa / negro , NIKE Women s Tanjun Coral Stardust / Sail NIKE Women s WMNS M2K Tekno, Negro / Negro-Phantom-Volt Negro / Black-Black-Phantom-volt , New Balance Women s w990v4 Running Shoe Armada , Saucony Women s Guide 8 Running Shoe Azul / azul marino / amarillo Zapatillas de running NIKE para mujer Epic React Flyknit Negro / gris oscuro

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 Sandalias Hush Puppies Zendal Slide X-Band para mujer Cuero Taupe / gamuza , OluKai Kulapa Kai Jalea de guayaba / negro Sanda Verra Teva para mujer Surf Taupe Multi Zapatillas trail running Inov8 Trailroc 270 para mujer y visera de entrenamiento Coral / Negro Zapatillas para caminar Ryka Women s Devo Plus 2 Blanco azul , Zapato para caminar Skechers Women s You Negro Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Lt Gray , Sandalia plana Nanci de Naturalizer para mujer Cuero negro Zapatillas para caminar Travelite para mujer Propet Blanco , FLY London Wege Mousse - Negro (Cuero) Sandalias para mujer Cuero negro) Zapatillas Ektelon T-9 Roadster Mid Indoor Court Blanco negro , K-Swiss Vendy II Sneaker para mujer Plata blanca , Vans Auténtico Escarabajo / True White , Zapatillas DC Beach Trase TX Skate Vino , Earth Origins Womens Gaven Gray Smu , Vince Camuto Sanella deslizante de mujer Azul / Natural , Sandalias Jerusalem Sandalia Arad Slide marrón Sandalia mujer Jessica Simpson, diadema Crizma Blanco negro Birkenstock - Sandalias de gamuza con aplicaciones de suela suave Arizona para mujer Terciopelo gris terciopelo suave plantilla , Sandalia Bobbi Slide de Dolce Vita para mujer Lt Blue Denim , Sandalias Sandalias Bashan moldeadas Sandalias Sandalias Jerusalem Gris Sandalia plana Kip de Trotters para mujer Equipaje Sandalia del vestido de la inversión de LifeStride de las mujeres Rojo clásico Zapatos de danza del salón de baile de las señoras muy finas EK9691 con 1.3 \ Satén marrón claro , Bomba de vestido de consuelo para mujer LifeStride Negro Party Party Velvet Vestido de noche bomba Colecciones, Comodidad Zapatos de boda: Salón de baile de mujeres Zapatos de baile, medio Tacón, Tango, Suave, Estándar Terciopelo rojo 7001 , Sandalias Nuria de Toni Pons para mujer rojo Bota de moda BEARPAW para mujer Abigail Nuez dura , Zapatillas de deporte lavables Walker Propet para mujer Sr Pewter Dr. Martens Women s Bennett Oxford Broncearse ,

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 de running Reebok running Women 19380 s B01N3P1G8I Supreme 3.0 MT Negro/ carbón/ plata/ blanco 12e349e 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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10987098

12184790

14656726

17235816

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