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.

Malla

Caucho suela

Camuflaje-imprimir malla y sintético superior

extraíble Plantilla acolchada EVA

Goma suela

Seis ojos cierre de encaje

WaveSpring Tecnología

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 deportivas Spira Spira 6470 XLT Camo Limited Edition para mujer B06ZY87KCY Rosa/ Blanco 0b77971Updated 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 deportivas Spira Spira 6470 XLT Camo Limited Edition para mujer B06ZY87KCY Rosa/ Blanco 0b77971

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

NIKE Mujeres Air Zoom Pegasus 33 Negro / gris frío / gris lobo / blanco Zapatillas de running Adidas Performance Ultra s Boost M para hombre Amarillo solar / amarillo solar / negro , Converse Chuck Taylor All Star Seasonal Color Hola Polvo Púrpura / Blanco / Negro , Zapatillas para correr Nike Women s Flex Experience Rn 5 Antracita / Rosa explosiva / Negro , Converse Chuck Taylor All Star Seasonal Color Hola Días por delante , Zapatillas Nike Women s Flex Trainer 5 Clearwater / Volt / Bl Lgn / White , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco / Blanco / Azul Zapatillas de running New Balance Coast V3 para mujer Negro / Imán , Zapatillas Drew Shoe para mujer Paige Ternero negro Converse Chuck Taylor All Star Seasonal Color Hola Yute , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Blanco / blanco / crudo Calzado casual tejido NIKE NIKE para hombre Cool Gray White Black 004 , Zapatillas NIKE Femenil Flex 2016 Rn Running Pink Blast / Blanco / Negro / Eléctrico Verde Zapatillas de deporte casual Gazelle de adidas para hombre Carga de oliva / goma de carga de oliva , Zapatillas de running Reebok Zprint 3D para mujer Nube gris / Rebel Berry / Poison Pink / Pewter , NIKE Men s Lunar Fingertrap Cross Trainer Antracita / Negro Under Armour Women s Charged Rebel Black (004) / Purple Rave , Carro de mujer Brooks Fucsia NIKE Kids gratis Rn (grande) Hyper Violet / Black / White / Hyper Turquoise Zapato NIKE Roshe One femenino Concord / Ember Glow-sail , Converse Chuck Taylor All Star Seasonal Color Hola Azul Calzado casual tejido NIKE NIKE para hombre Cool Gray White Black 004 Converse Chuck Taylor All Star Seasonal Color Hola Blanco negro Zapatillas de running NIKE Roshe One para mujer Crimson total / Crimson total , Zapato de running ASICS GEL-Venture 5 para mujer Gris plateado / turquesa / sacador de lima Zapatillas deportivas 574S Sport New Balance para hombre Gunmetal / Navy Zapatillas de running Saucony Women s Triumph ISO 4 Vizi Rojo / Blanco Zapatillas Drew Shoe para mujer Paige Ternero de hueso , Zapatillas de running New Balance Fresh Foam Lazr V1 Melocotón espolvoreado / Melocotón espolvoreado ,

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 Saucony Women s Ride 10 Running Shoe Gunmetal / rosa Zapatillas de running adidas Mana Bounce 2 W Aramis para mujer Blanco / Metálico / Plateado / Negro , Zapatillas de running adidas Aerobounce Pr W para mujer Aero Blue / Aero Blue / Noble Indigo Propet Women s TravelActiv Ss Sandal Negro plata , Sandalia atlética Salomon Women s Crossamphibian Swift W Higo , Zapatillas de trail running Xa Pro 3D W para mujer Salomon Azul Medieval Zapatillas de running New Balance para mujer 490v4 Silver Mink / Spectral Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Blanco , Zapatillas de running NIKE Air Max 90 para mujer Blanco / negro / polvo / rojo solar , P W Minor Leisure Mujer Terapia Casual Extra Profundo Calzado Cuero Cordones Negro , Sandalias FitFlop Womens Cova con cordón cerrado Negro , Zapatillas de running Mizuno Wave para mujer con sombra Deslumbrante azul / blanco , Drew Shoe Naples Diabético terapéutico para mujeres Depth Extra Leather Shoe-On marrón , Sandalia para mujer con correa de tobillo Naot para mujer Lustre negro, cuervo negro, serpiente negra , Naot Footwear Women s Prescott Nubuck azul aceitoso / cuero dorado / nubuck esmeralda aceitosa Zapatillas de running adidas Aerobounce Pr W para mujer Blanco / Blanco / Core Black Apis Mt. Negro , Easy Spirit Women s Traveltime Mule Patente negra / plateada Zapatillas Cross-Trainer Ryce para mujer Enhance 3 Azul / naranja , Onitsuka Tiger Asics Mexico 66 Negro Negro Diadora N.9000 Arrowhead Saltire Navy , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Cumbre Blanca / Obsidiana Drew Women s Summer Armada Orthaheel by Vionic Indulge Gemma Mujeres punta redonda lona gris Slipper Tan Terry , JOE N JOYCE Slippers Zuecos Zapatos Cuero sintético estrecho - Para hombre y mujer Azul oscuro Zapatillas de deporte Moc de Acorn Women s Spa Support Natural Zapatillas Sidi Women s Carbon Carbon Road Blanco , Botas para mujer InterestPrint Botas con cordones con diseño exclusivo Multi 14 , Zapatillas de deporte Trolide Derby Trainer Fashion Rockport para mujer Stone Multi Leather Steve Madden Womens Kyra Gray Multi

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 deportivas Spira Spira 6470 XLT Camo Limited Edition para mujer B06ZY87KCY Rosa/ Blanco 0b77971 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
1832385

2777156

3455722

7035657

10537925

11213716

14746922

15107650

16910513

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