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 y malla

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

Zapatilla Blanco para mujer Saucony A8 tipo | A8 Blanco | 95d4d2cUpdated 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.

Zapatilla Blanco para mujer Saucony A8 tipo | A8 Blanco | 95d4d2c

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

Zapato de running ASICS para mujer Torrance Azul humo / Azul humo / Begonia rosa , Calzado de running NIKE RN 2017 2017 para mujer Negro / Fuchsia Glow / Hyper Magenta , Zapatillas de running New Balance de mujer Fresh Foam Kaymin v1 Fiji / Coral vivo Under Armour Women s HOVR Sonic NC Grafito / Plata Metálica / Grafito , Zapato de running ASICS Gt-2000 5 para mujer Grey medio / blanco / rosa , Zapatillas de running New Balance 520v3 para mujer Negro , Calzado de running New Balance Foam Zante v2 para mujer New Balance Gris / rosa , Calzado de running Kindred para mujer Ryka Gris / rosa Zapatillas para correr Brooks Cascadia 12 para mujer Peacoat / Flor de la pasión / Bluebird , Zapatillas para correr inteligentes para mujer Runtopia Reach 2.0, equipadas con chip de seguimiento de datos Ash Black Zapatillas de running ASICS GT-2000 6 para mujer Indigo Blue / Indigo Blue / Smoke Blue Zapatillas de running NIKE Air Max Flair para hombre Wolf Gray / Cool Grey-black PUMA Men s Roma Basic Sneaker Blanco / Blanco / Goma HOKA ONE ONE Bondi 4 Mujeres Corso azul / coral de neón , Zapatillas de running Mizuno Wave para mujer Inspire 14 Dapple Gray / Clover , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Turq / Black-volt-rio Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Zapatillas de running ISO para mujeres de la guía Saucony Pizarra / Melocotón , Zapatillas de running adidas Performance Women s Lite Slip-On Marrón / marrón / rosa Zapatillas de running adidas Ultraboost W para mujer Blanco / blanco / blanco cristal Zapatillas de running NIKE Zooms Winflo 4 para mujer Ink / Deadly Pink-fire Pink , Zapatillas para correr New Balance Rushv3 para mujer Poisonberry / Black Amortiguación 635v2 New Balance para mujer Negro , Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Colegio Navy / Black Concord , Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running Kineta Relay para mujer Saucony Gris blanco , Brooks Womens Mazama Diva Rosa / Antracita / Bluefish , Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul ,

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 Fresh Life Arishi V1 para mujer New Balance Luz Porcelana Azul Sanda Verra Teva para mujer Atlántico Norte Brooks Womens Adrenaline GTS 18 Negro Negro Zapatillas de running ASICS Gel-Cumulus 19 para mujer Negro , adidas MUJERES ORIGINALES CLIMACOOL 02/17 ZAPATOS # BY9289 Urban Trail / Calzado Blanco Teva Terra-Float Active Lace - Mujeres Paloma Salvaje Sandalia Omnium de mujer Teva Pizarra , Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Stealth Gray / Glacier Grey / Pink Chroma Zapatillas adidas Outdoor Terrex Agravic para mujer Sun Glow, White, Super Blush , Calzado de running NIKE Lunarcharge Br Tobillo-High para mujer Armería blanco / claro azul-blanco , Zapatillas para correr Nike Downshifter 6 para mujer Negro Negro Zapatillas de paseo Skechers Go Walk Lite-15435 para mujer Mezclilla Sandalias Chardon de Hush Puppies Chardon para mujer Gris hielo , Zapato para caminar Eden Propet para mujer Blanco Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Negro / Blanco / Antracita David Tate Wish Sandalia para mujer Negro Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Champagne / Metálico Sneachers Sport Premium-Premix Slip-On Sneaker para mujer Naval Blanca Bota Deia para mujer de TOMS Desert Taupe Suede Radial Perf / Farren Naturalizer Reilly Espadrille Sandal de las mujeres Tórtola , Vans Herren Authentic Core Classic Sneakers Blanco / apagado blanco , Cole Haan Zerogrand Genevieve Perf Trainer para mujeres Ch Argento Cuero Perforado / Blanco Óptico Zapatillas de tenis ASICS Gel-Challenger 11 para mujer Negro / Negro / Rosa Caliente , Sandalia Cato Dolce Vita para mujer Lino de arena Sandalia Ryana Marilyn Slide para mujer Azul blanco , 50 tonos de oro Colección de zapatos de baile, Noche de confort Vestido de boda bombas, Zapatos de salón Para América, Tango, Salsa, Swing, Theather Arte por 50 sombras (2.5 \ Tacones) 5004- Gold Sparkle Zapatillas Bellamo de Ted Baker para mujer Gris Zapatillas de running NIKE Men s Train Speed 4 Blanco negro , Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Azul Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma (Mte) peltre / tela escocesa ,

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 Zapatilla Blanco para mujer Saucony A8 tipo | A8 Blanco | 95d4d2c 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