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 textil correa

Importado

Espuma suela

Sintético suave correa se mueve naturalmente con tu pie

Solarsoft plantilla de espuma proporciona una sensación de felpa bajo los pies

Inyectado Entresuela de espuma Phylon es liviano y durable

ondulado surcos flexibles de suela deja que tu pie se mueva naturalmente

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

Sandalia Wolf deportiva para NIKE Kawa Slide para hombre Universidad Red 4196/ White/ Wolf Gray 74c74d5Updated 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.

Sandalia Wolf deportiva para NIKE Kawa Slide para hombre Universidad Red 4196/ White/ Wolf Gray 74c74d5

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

NIKE Women s Tanjun Marrón nocturno Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco Zapatillas de running Kinvara 7 Saucony para mujer Azul / negro / rosa , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Blanco negro Converse Chuck Taylor All Star Seasonal Color Hola Monocromo negro Zapatillas de running NIKE Unisex Flyknit Racer Citrus brillante / Laguna azul-negra NIKE Men s Roshe Run Wolf Grey / Sail-ghost Vela verde Sandalia deportiva NIKE Kawa Slide para hombre Medianoche azul marino / blanco , Zapatillas de running NIKE Juvenate para mujer Blanco negro Zapatillas de deporte Vans reflexivas ISO 1.5 para hombre Negro Under Armour Charged CoolSwitch de mujer Azul , Zapatillas de running Saucony Women s Triumph ISO 4 Negro / Aqua / Violeta Fulcro Karhu Fast5 para mujeres Lila / Hibisco Zapatillas para correr Nike Women s Flex Experience Rn 5 Wolf Gray / Racer Blue , Sandalias OLKAKAI Ohana - Hombre Dark Shadow / Dark Shadow , Converse Chuck Taylor All Star Seasonal Color Hola Zafiro rosa Zapatillas de running NIKE Juvenate para mujer Cool Grey / Cool Grey / Sunset Glow / White Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Azul / Calzado Blanco , Calzado casual tejido NIKE NIKE para hombre Negro / Cumbre negra Blanco , Zapatillas de deporte con adidas Pw Tennis Hu de adidas para hombre Armada , adidas Women s Distancestar W Blanco / Hi-res Verde / Real Purple , Sandalia deportiva NIKE Kids Kawa Slide (GS / PS) Clay Green / Barely Gray 300 , Reebok Classic Sneaker Negro / Negro / Carbono , Zapatillas antideslizantes Skechers for Work para mujer Eldred Blanco Zapatillas NIKE Women s Revolution 3 para correr Wolf Gray / Volt / Hyper Turquoise / White Zapatillas de running NIKE para hombre con versatilidad de tren gratis Total Crimson / Gym Rojo / Blanco / Negro Zapatillas deportivas 574S Sport New Balance para hombre Tormenta azul / blanco Zapatillas de running NIKE Air Huarache City para mujer Cloro azul Saucony Liberty ISO Zapatillas para correr para hombre S20410 Gris / Niebla ,

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 para caminar Reebok Women s Cloudride LS DMX Cráneo gris / gris plano / blanco / rosa venenoso Sandalia Teva Women s W Hurricane Xlt2 Sport Negro Under Armour Women s Charged Rebel Negro (001) / Rhino Gray Zapatillas de running Adidas Performance para mujer Ultra Boost Negro / Negro / Amarillo Zapatillas de running NIKE Air Huarache para hombre Hueso de Goma Ligera / Orquídea Amarillo NIKE Men s Roshe Run Gimnasio Rojo / Profundo Borgoña-blanco , NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Cool Grey / Black / Max Orange , New Balance Women s w990v4 Running Shoe Negro / Plata , Zapatillas para correr Mizuno Women s Wave Enigma 6 Deslumbrante Azul-ardiente Coral-plata , Zapatillas de deporte Arizona Trotters para mujer Desnudo , Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Plata / Metálico Sandalia Pavi de estilo suave para mujer Niebla blanca , Camiseta adidas Swift Run W para mujer Blanco / Blanco / Blanco , NAOT Sandalias Peyton Slide para mujer Cuero Latte Marrón / Cuero Espejo / Cuero Marrón Combo , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Turquesa , Drew Shoe Women s Tulip Comfort Negro , Zapatillas de tenis NIKE Womens Zoom Vapor X Negro / Solar Rojo-blanco , Zapatillas de running Golden Spike de Altra para hombre Azul Skechers Women s Appeal Flex New Image Sneaker Rosa fuerte / negro Zapatillas de deporte Xelero Genesis Comfort Comfort Therapeutic Extra Deep Sneaker Leather / Mesh Lace-up Blanco azul Sandalia de la diapositiva de Benicia de las mujeres de Dolce Vita Ropa de Ceniza , BareTraps Women s Cherilyn Slide Sandal Denim Blue Jessica Simpson Sandalia Kloe Slide para mujer Negro Trotters para mujer Dea Slingback Pump Oro FRYE Melissa Pull on Fashion Boot para mujer Negro , Zapatillas Nike Women s Flex Trainer 5 Wolf Gray / Racer Pink-sigilo Mona Flying Women s Cuero perforado Oxfords con cordones Zapatos para mujeres Wingtip Multicolor Zapatos Brougue Marrón 1 Fulinken Leather Women Oxford Flats Zapatos con cordones Boyfriends Botines Negro , Timberland PRO Mujer Resbalón castaña Calzado deportivo de mujer FitFlop Uberknit Walking Slip-On Negro / gris suave

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 Sandalia Wolf deportiva para NIKE Kawa Slide para hombre Universidad Red 4196/ White/ Wolf Gray 74c74d5 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