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.

Tela o Textil

Importado

Sintético único

BZEES Tecnología en la nube,

Gratis Camas de pies de espuma, Estiramiento dinámico Parte superior,

Aire Suela infundida, Lavable en la lavadora, Air Dry

Antimicrobiano y OdorControl Tecnología,

Memoria Foam Arch, extraíble calcetín, comodidad Cone Gel. para que te sientas sin peso energizado y gratis

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

BZees B01MQKEUEM BZees Sandalia Deportiva Jive Mujer para Mujer Gris bbf5fdeUpdated 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.

BZees B01MQKEUEM BZees Sandalia Deportiva Jive Mujer para Mujer Gris bbf5fde

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

Chaco Women s ZX2 Classic Athletic Sandal Edredón Melocotón Sandalia Deportiva Jordan Spires de Via Spiga Paja de gamuza Sandalia de mujer Drew Alana Blanco negro Sandalia Savannah de Ryka para mujer Coral / gris Teva ALP Premier - Mujeres Armada Sandalias de tirantes Sunstone para mujer Merrell Vertiver Sandalia Terrell II de Merrell para mujer Dark Slate , Sandalia de deslizamiento ultra Comfort NIKE para mujer Mango brillante / negro Sandalia para exteriores ECCO Women s Offroad Lite Rainer Sandal Outdoor Blanco / Sombra Blanco , Sandalia KEEN Women s Venice H2 Negro / Gris neutro , Chaco Women s Zx3 Classic Athletic Sandal Festoon azul , Teva Terra Sandal para hombre Madang Olive Webbing Sandalia Drew Angela para mujer Bone Smooth Leather , Wolky Comfort Jewel Black Croco Sandalia Tanga Loea de mujer OLUKAI Dark Java / Dark Java Sandalia Drew Angela para mujer Dusty Multi Metallic HOKA ONE ONE Recuperación ORA para mujeres Negro Sandalia Archie Fisherman del Dr. Scholl Nubuck azul marino Chaco Mujer ZX1 Classic Sport Sandalia Burst Blue , Sandalia deportiva Columbia Montrail para mujer Molokini III Negro, Steam , Sandalia deportiva Tirra para mujer Teva Mar de Bering , Sandalia Juniper Mujer Chaco Negro Chaco Mujer Zcloud 2 Athletic Sandal Candy Grey Chaco Mujer Z Eddy X2 Sport Sandalia Motivo negro Sandalia Original Lux Original Teva Mujer Niebla de mar , Chaco Mujer Zcloud X2 Sandalia Deportiva Runa Teal , Wolky Comfort Jewel 30800 Cuero azul oscuro Sandalia Archie Fisherman del Dr. Scholl Moonstone / Pink Nubuck , Teva Terra Sandal para hombre Negro ,

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 Zapatillas de running Mizuno Women s Wave Catalyst 2 Gris / menta , Orthofeet Probado Pain Relief Plantar Fascitis Ortopédica Cómodo diabético Plano extra ancho Feet Naples Womens Sandalia Azul , Sandalia Yucatan ECCO Mujer Shadow White , Sandalia original Teva W de mujer Malena Fade Dark Purple / Multi , 361 Zapatillas de trail running 361-Overstep 2 para mujer Ébano negro Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Negro / Microdot , Romika Tahiti 03 Sandalia de vestir para mujer Negro , Skechers Women s Appeal Flex New Image Sneaker Negro , Zapatillas para correr Mizuno Women s Wave Sky Piedra gris - Plata Drew Shoe Women s Paradise II Velco Slip-on Gris pardo , Drew Shoe Women s Hope cuero terapéutico moda Oxfords Negro , Zapato para caminar WW411v2 New Balance para mujer Negro Birkenstock Arizona Sandalia de ante con suela suave Cuero engrasado Skechers Work Zapatillas Marbleton resistentes a los resbalones D Lites para mujer Negro Zapatillas de deporte Skechers Sport Women s Empire Inside Look Negro / Multi ¡Pensar! Efecto Ametista / Kombi Zapato deportivo ortopédico para mujeres Vionic Kona Gris / Lima Vans Auténtico Vestido azul / azul / rojo , Zapatillas de golf ECCO Women s Classic Hybrid II Blanco / Capri Breeze Sandalia de cuero Lucili de Nine West para mujer Cuero negro Sandalia para mujer Qtee Slide de Anne Klein Light Pewter Synthetic , Bota de montar Andra para mujer, sin calificación Negro Under Armour Women s Drift 2 X MNSWR Raisin Red (500) / Beige Canvas Old Gringo Womens Nicolette Tobillo rojo Bálsamo de mujer Sheep Touch Blusa de piel con doble cara Closed-Back Slippers Castaño , Mordenmiss Nuevo verano de primavera y verano de flores hechas a mano zapatos planos de cuero Mocasines Estilo 1 Rojo SoftMoc Women s Bali Hi Moincasin Plain Bootie castaña , Zapatillas SAS Women s Tour con cordones Zapato de confort activo Plata Ovation Ladies Synergy Front Zip Paddock Boots marrón , Clásicos de base Slip-on (Tm) para hombres de Vans Black Off White Checker

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 BZees B01MQKEUEM BZees Sandalia Deportiva Jive Mujer para Mujer Gris bbf5fde 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