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.

Gamuza, sintético

Importado

Sintético único

Eje mide aproximadamente Parte baja del arco

Espuma Plantilla Lite 1

La diseño original de Scam construido en un más delgado último

Mínimo lengua y cuello relleno

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 skate Etnies The mujer para mujer The Scam Etnies W Blanco 2bacad9Updated 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 skate Etnies The mujer para mujer The Scam Etnies W Blanco 2bacad9

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

Vans Old Skool Unisex Adults Low-Top Trainers (Cuadros primario) Negro / blanco , Zapatillas DC Mujeres Chelsea TX SE Sneaker Azul oscuro Vans Herren Authentic Core Classic Sneakers Negro Negro , Vans Auténtico Jeans / Bianco , Vans Herren Authentic Core Classic Sneakers Ají picante Clásicos de base Slip-on (Tm) para hombres de Vans Black Off White Checker Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Legion Verde / Blanco-negro , Zapato de skate casual Heathrow DC para hombre Estaño Clásicos de base Slip-on (Tm) para hombres de Vans (Denim C & l) Negro / Goma , Vans Unisex Old Skool Zapatillas clásicas de skate Floral hawaiano Supra Vaider LC Sneaker Blanco oliva , Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Cumbre Blanco / Obsidiana / Universidad Rojo / Blanco , Clásicos de base Slip-on (Tm) para hombres de Vans Tablero de ajedrez negro / de peltre Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Goma negra / clásica Vans Old Skool Unisex Adults Low-Top Trainers Gris / Violeta Vans Old Skool Unisex Adults Low-Top Trainers (Suede del cerdo) Noche parisiense / Blanc De Blanc , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco , Supra Flow Run Evo Skate Shoe Negro / Dk Gris-blanco-m Supra mujer Skytop Sneaker Black-fade / Black Vans Unisex Brigata Zapatillas de skate, náuticas Zapato de barco inspirado Clásico, Cómodo y duradero en Waffle original Suela (Cuero / Cuadros) Esttebl / Pttngsl Clásicos de base Slip-on (Tm) para hombres de Vans Clásico Negro / Blanco , Vans Old Skool Unisex Adults Low-Top Trainers Cabernet / True White Supra Vaider LC Sneaker Rojo blanco Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre Azul / gris / rojo Lakai Riley Hawk Skate Shoe Textil de medianoche Supra Vaider LC Sneaker Blanco oliva oscuro , Supra Skytop Skate Shoe Blanco Blanco Vans Unisex Era Zapatillas de skate, clásicas Low-Top cordones Estilo en Durable Doble costura Lienzo y Original Suela de gofre (Gold Mono) Port Royale Vans Auténtico (lavado de 2 tonos) marrón / blanco verdadero ,

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 adidas Alphabounce Em W para mujer Energy Blue / Energy Aqua / Mystery Petrol Zapatillas de volante Nike Women s Zoom Hyperface 2 Blanco negro Zapatillas de running retro Gel-Lyte III para mujer ASICS Cacatúa / Cacatúa , Chaco Women s Z2 Classic Athletic Sandal Swell Melocotón Zapatillas deportivas 574S Sport New Balance para hombre Azul marino / Vintage Indigo , NIKE Mujeres Air Zoom Pegasus 33 Pure Platinum / blanco-negro-voltio Camel Women & rsquo; s Zapatillas de moda Al aire libre con cordones Zapatillas de trail running con cojín de aire Ligero y transpirable Púrpura Skechers Performance Women s The Go 400 Radiance Flip Flop Azul marino / blanco Zapatillas de running NIKE Lunar Skyelux para mujer Blanco / puro Platinum-off blanco , Ballet de niña de flores Aerosoles femenino Negro exótico Vivien Flat de Bella Vita para mujeres Negro Propet Women s Fisherman Sandal ciruela Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Vintage Verde / Negro , Vionic Rest Farra - Sandalias de apoyo para mujeres Patente de Marina , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Lt Gray , Zapatos de entrenamiento Nike Free TR 6 para mujer Blanco / Brillante Melón / Wolf Gray Prop & eacute; t Olivia Oxford para mujeres Hueso Clásicos de base Slip-on (Tm) para hombres de Vans Blanco , Sandalia FitFlop Glitzie Slide para mujer Paloma Salvaje Sandalias de plataforma deslizante con plataforma Forella de Gentle Souls para mujer Flor negra , Sandalia para mujer FRYE Ally Criss Cross Slide Blanco , Sandalia Vince Westcoast para mujer Vince Caucho magenta , Sandalia de cuero Lucili de Nine West para mujer Negro / Cuero Multi , 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) 2713 Oro Zapatillas de mujer ASP Haflinger Gray Multi Nacido - Mujeres - Karava Combo negro Joules Women s Rockingham Botines Dark Leopard Rubber , Timberland PRO Hightower Hightower 6 \ Café turco Drew Shoe Women s Pepper Criss Cross cuero, caucho Casual Oxfords Negro , Alegria Womens Madi Interlockin Choco

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 skate Etnies The mujer para mujer The Scam Etnies W Blanco 2bacad9 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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4631592

5142387

6757027

9861688

12899514

15001051

16705141

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