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 y Sintético

Goma único

A Max Unidad de aire en el único ofrece un cómodo sentir bajo los pies

One-piece superior ultra-breathab

Encaje Arriba

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 Air de running NIKE Women s Air Cumbre Max 1 1 Ultra Essentials Cumbre Blanca/ Carmesí total 6d91232Updated 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 Air de running NIKE Women s Air Cumbre Max 1 1 Ultra Essentials Cumbre Blanca/ Carmesí total 6d91232

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 para correr ASICS Gel-Venture 6 para mujer Negro / Pixel Rosa , Zapato deportivo ASICS Gel-Kayano 24 para mujer Azul Púrpura / Regatta Azul / Blanco Zapatillas de running NIKE Air Max Flair para hombre Negro / antracita-negro Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Amortiguación 635v2 New Balance para mujer Gris Brooks Women s Ravenna 8 Peacoat / Purple Cactus Flower / Fusion Coral , Under Armour Women s Drift 2 Negro (001) / Blanco , Zapatillas para correr Saucony Women s Triumph 11 Blanco / Vizicoral / Navy Zapatillas de running Pearl NZUMi W Em Road N 2 V3 para mujer Ibis Rose / White , adidas Running Womens Supernova Sequence 9 Gris sólido / plata mate / blanco , Zapatillas para correr W630V5 para mujer New Balance Plata , Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Diva Blue / Flash Coral / Aqua Splash , Zapatillas de running NIKE Air Huarache para hombre Stealth White 003 , Zapatillas de running Reebok Print Smooth Ultk para mujer Nube Gris / Gable gris / Asteroid Dust / Polar Blue / Black / Fire Spark , NIKE Mujer en temporada TR 4 Print Cross Trainer Fuschia Glow / Bold Berry / Black / Artesanal Teal NIKE Men s Benassi Just Do It Athletic Sandal Blanco / puro platino / negro / blanco Zapatillas de running Kineta Relay para mujer Saucony Púrpura / rosa Zapatillas de running Nike Air Zoom Elite 8 para mujer Racer Azul / Blanco-azul Ráfaga de color rosa brillante Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / Blanco , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Estratos , Zapato de running de punto fuzeX para mujer ASICS Flash Coral / Negro / Amarillo de seguridad , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Azul claro Zapatillas de running New Balance 990V3 para mujer Negro , Zapato de running GEL-Fortitude 7 para mujer ASICS Azul índigo / plata / ciruela , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Utilidad Verde / Maroon Zapato deportivo ASICS Gel-Excite 4 para mujer Blueprint / Silver / Mint Zapatillas de running New Balance 520v3 para mujer Gris , Zapatillas de running ASICS Gel-Flux 4 para mujer Negro / Plata / Carbono ,

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 NIKE RN 2017 para mujer gratis Platino blanco / negro puro , Under Armour Charged CoolSwitch de mujer Azul Zapatillas adidas Response TR Shoe para mujer Gris medio / Calzado blanco / gris oscuro Zapatillas de running New Balance para mujer 590v3 Imán / vórtice , Columbia Rogue F.K.T. Steam / Dark Raspberry , NIKE WMNS Zoom Fly Women 897821-411 Platino blanco / negro puro Zapato para caminar Ryka Women s Sky Gris pizarra / Plata cromo / Azul Robin , Drew Shoe Women s Paradise II Velco Slip-on Ternero negro Crocs Flip Crocband unisex Volt Green / White Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Riviera / Microchip Gray , Zapatilla de deporte con cordones de cuero para mujer Xelero Blanco / Lt Blue Mesh , Lakai Limited Footwear Mens Griffin Textil gris / negro , Zapatillas para caminar RYKA Women s Devotion Plus Blanco / plata cromo / almendra helada PUMA Tsugi Shinsei Velour para mujer Zapatillas ligeras de moda casual Cordobés , Zapatillas tenis NIKE Air Zoom Ultra para mujer Gran gris / negro / blanco , Zapatillas de deporte Xelero Matrix Comfort Therapeutic Extra Deep Sneaker Shoe Leather con cordones Negro Under Armour Womens UA Jet 2017 Negro / Blanco / Blanco LI-NING Zapatillas de baloncesto Samurai III Wade Culture para hombre Zapatillas de deporte Mujeres negras , 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 7015- Red Velvet , Bota de montar Xandywc para mujer LifeStride, pantorrilla ancha Gris oscuro Pikolinos Womens Hamilton W2E-8642 brandy , Bota de invierno clásica UI GI para mujer castaña Blundstone EVA Rustic Black Boot Bota de invierno SOREL Womens Slimboot Negro / Parrilla Gentle Souls de Kenneth Cole Zapatillas de deporte Harper Fashion para mujer Hormigón , Zapatillas Merrell para mujer Applaud Chill-W Helecho Bota de Combate Julie para mujer FRYE Sand Soft Oiled Suede Botín de tobillo Cyra Catelyn de Hush Puppies para mujer Gray Shimmer , Botín de tobillo Mazin Cayto para mujer Hush Puppies Negro LIBERTY NEGRO Botas a rayas con punta de mujer Vegas Strip - Lb71124_X Marron oscuro

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 Air de running NIKE Women s Air Cumbre Max 1 1 Ultra Essentials Cumbre Blanca/ Carmesí total 6d91232 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