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.

Cuero, sintético cuero o textil

Importado

Espuma único

Completamente superior plegable y talón para fácil embalaje

Airy malla superior para la comodidad del clima cálido y estilo

En forma de panal almohadillas de espuma de la malla superior para mayor comodidad

Longitud completa entresuela es durable suficiente para servir como suela

Caucho suela con inspiración de Waffle diseño para la tracción y durabilidad

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 NIKE Juvenate para Zapatillas mujer Hueso Ligero/ Hueso Ligero/ Negro/ Blanco d298281Updated 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 NIKE Juvenate para Zapatillas mujer Hueso Ligero/ Hueso Ligero/ Negro/ Blanco d298281

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

Calzado de running NIKE Women s Air Relentless 6 Green Glow / Mtlc Silver / Bright Mango Zapatillas de senderismo adidas AX2 CP para mujer Vista Gris / Negro / Super Blush S17 , Under Armour Women s Charged Rebel True Ink (918) / Penta Pink Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Antracita / Tarta , Zapatillas de running ASICS Gel-Scram 4 para mujer Gris medio / laguna Converse Chuck Taylor All Star Seasonal Color Hola Azul Zapatillas de running Spira Scorpius II para mujer gris Nfinity Vengeance Cheer Shoe (par) Blanco , Zapatillas de deporte Onitsuka Tiger Colorado Eighty Five Negro Negro Under Armour Women s Charged Rebel Gris Nublado (102) / Éxito NIKE Women s Tanjun Black Metallic Gold Converse Chuck Taylor All Star Seasonal Color Hola Blanco óptico , Vibram Furoshiki Phulkari Sneaker para hombre y mujer Flor azul Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Marrón / Marrón verde azulado NIKE Women s WMNS Zoom Fly, negro / blanco Barely Gray / Sequoia-light Pumice Zapatillas de deporte adv W de adidas Originals para mujer Blanco / Blanco / Gris Uno Zapatillas de deporte Onitsuka Tiger Colorado Eighty Five Blanco Blanco , Reebok Classic Sneaker Blanco negro , NIKE Men s Roshe Run Gimnasio Rojo / Profundo Borgoña-blanco Zapatillas deportivas 574S Sport New Balance para hombre Verde encubierto / blanco , Under Armour Women s Speedform Slingride 1.1 Negro (001) / Blanco New Balance W860pg7 para mujeres Gris / Purpl , Converse Chuck Taylor All Star Seasonal Color Hola Polvo morado , Converse Chuck Taylor All Star Seasonal Color Hola Negro / Garza / Negro Zapatillas de running Adidas Performance Ultra s Boost M para hombre Medium Heather gris / Medium Gray Heather / Dark Heather gris NIKE Men s Roshe Run Negro Negro , Sandalias OLKAKAI Ohana - Hombre Negro Zapatillas Drew Shoe para mujer Paige Ternero negro , Zapatillas deportivas 574S Sport New Balance para hombre Coco tostado / blanco ,

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 HOKA ONE ONE Recuperación ORA para mujeres Negro , Zapatillas de running NIKE Air Huarache para hombre Hyper Turquesa / Blanco Wolky Comfort Jewel 30500 Cuero rojo , Sandalias mujer Teva Womens W Original Universal Premier-Leather Sport Medianoche negro Calzado de running Pearl iZUMi para mujer Em Road H3 v2 Cerise / madreselva , Sandalia Shaye Sport mujer Donald J Pliner Platino , Converse Chuck Taylor All Star Seasonal Color Hola Lila congelada Zapatillas de running adidas Women s Rockadia W Trail Trace Púrpura / Blanco / Core Black , KEEN Hombres Newport H2 Sandal Brindle / Sunset , PUMA Men s Roma Basic Sneaker Blanco negro , Zapatillas sin cordones Flex Merit para mujer Merrell Rosa roja , Zapato para caminar carismático Ryka para mujer Negro / Rosa / Gris Vionic con Orthaheel Tide Sport Sandalia para mujer Azul claro , SJP de Sarah Jessica Parker Mujer Meteor Sneaker Caramelo , Botín para mujer Pleaser Asp1018 / Bpu / M Negro , Cobb Hill Rockport Women s Angelina Dress Pump Negro , Zapatillas de tenis adidas Barricade Classic Wide 4E, Negro / Plateado Metálico / Blanco, 6 M US Blanco / Core Black / Mid Gray , Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy de arena Sandalias Sia de Tommy Hilfiger para mujer Armada , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2713- Ultra Silver Zapatillas de deporte Trinity Sneaker Dance de Bloch Blanco LOVELY BEAUTY Lady s Ballroom fabricmps Zapatillas de baile Color de piel , Ferrini Ladies Maverick Square Toe Boots Broncearse Bota Ecuestre Lanesha de Lucky Brand para mujer Tabaco Zapato clásico Haflinger para mujer Orquídea , Arcopedico Mujeres L8 Negro , Blondo Womens Liam impermeable Bootie Grey Suede Baabuk Wool Sneaker - Mujer Azul marino / blanco , Drew Shoe Heather Synthetic Oxfords para mujer brandy , TOMS - Slip-On de lienzo clásico para hombre Espiga de carbón ,

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 NIKE Juvenate para Zapatillas mujer Hueso Ligero/ Hueso Ligero/ Negro/ Blanco d298281 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

For email to reach me, the word