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.

Flexfilm

Importado

Caucho único

Eje mide aproximadamente parte baja del arco

EVERUN Plantilla para continuo amortiguación

interno sistema de correa abraza el mediopié

Ligero Superposiciones FLEXFILM proporcionar estructura y apoyo

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 running Kinvara 8 Saucony para running mujer Kinvara mujer Azul 49bd62bUpdated 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 running Kinvara 8 Saucony para running mujer Kinvara mujer Azul 49bd62b

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 de running Mizuno Wave para mujer con sombra Griffin / Blanco / Eléctrico , NIKE Women s Revolution 4 Running Shoe Wide Gunsmoke / Ocean Bliss / Dark Grey / White Zapato deportivo para acolchado 635v2 New Balance para mujer Silver Mink Pantalón largo clásico NIKE Golf para mujer Tech Essentials Golpe caliente / Lobo negro gris New Balance Women s w990v4 Running Shoe Negro / Plata , Zapatillas de running Pure Cloudfoam de adidas para mujer Cristal blanco / Crystal White / Talc Calzado de running NIKE RN 2017 2017 para mujer Gris , Brooks Womens Mazama Flor de Cactus Púrpura / Peacoat / Hot Coral , Zapato de running ASICS Gel-Quantum 360 cm para mujer Punto medio gris / glaciar / coral , Zapatillas de running adidas Aerobounce Pr W para mujer Blanco / Blanco / Core Black , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris negro , Zapatillas de running ASICS Gel-Cumulus 19 para mujer Aruba azul / azul de Aruba / azulejo turco Zapato de running 840v4 New Balance para mujer Gris oscuro , Zapato de running ASICS Gel-Quantum 360 cm para mujer Negro / Onyx / Rojo caliente Zapatillas de running NIKE Fs Lite 2 para mujer Antracita / Blanco / Negro , Zapatillas para correr New Balance Rushv3 para mujer Trueno Negro Zapatillas de running NIKE Mens Lunarglide 8 Multicolor / multicolor Zapatillas de running adidas Adistar Boost ESM para mujer Taupe / Mint , Saucony Women s Ride 10 Running Shoe Gunmetal / rosa , adidas outdoor para mujer Terrex Swift R2 GTX Grey Three / Gray Two / Chalk Coral , Zapatillas de running Nike Air Huarache Run Ultra Br para mujer Platino Blanco / Blanco-puro Zapatillas de running NIKE Air Zoom Pegasus 33 para mujer Gris oscuro / negro - blanco Zapatillas para correr W630V5 para mujer New Balance Rosado , Saucony Triumph ISO 3 Running Sneaker para mujer Azul marino Zapato deportivo para acolchado 680v5 New Balance para mujer Cielo azul Zapatillas de running NIKE Air Max Thea para mujer Blanco negro , Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco / Universidad Rojo / Negro , HOKA ONE ONE Bondi 4 Mujeres Corso azul / coral de neón Nike Men s Air Presto Essential Velvet Brown / Light Bone

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 Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente , KEEN Hombres Newport H2 Sandal Midnight Navy / Feather Grey , Wolky Comfort Jewel Bronce Picasso Crash , NIKE Roshe G Spikeless Golf Shoes 2018 Mujeres Blanco negro Chaco Mujer ZX2 Yampa Sandal naranja Reebok Classic Sneaker Blanco / gris claro Zapatillas de running NIKE Lunartempo 2 para mujer Aluminio / Negro / Paramount Blue / Hot Punch Zapatillas de running NIKE Air Zoom Vomero 12 para mujer Platino blanco / negro puro , Zapatillas de running adidas Alphabounce Beyond W, para mujer Gray Four / Carbon / Dark Solid Gray , Skechers Women s Go Step Lite-Indulge Loafer Flat Gris pardo Zapato para caminar Skechers Go Joy para mujer Carbón , Sandalias de cuero de Viv Viv de TOMS Silver Specchio Crocs Unisex Modi Sport Slide Pimienta / Perla Blanca , Zapato de malla Nohea de OLUKAI - Hombre Onyx / Onyx Easy Spirit Women s Traveltime Mule rojo , Calzado SCARPA Women s Crux Walking Camel / Poppy Red , Zapatillas de tenis ASICS Gel-Solution Speed 3 para mujer Flash Coral / Plum / Flash Coral , adidas Baseline K Sneaker Blanco / Blanco / Coral , Zapatillas para caminar Ryka Women s Devo Plus 2 Azul marino / verde azulado Zapatillas altas con cordones All Star de cuero Converse Chuck Taylor Trueno / Negro / Papiro , Sandalia FitFlop Strata Slide para mujer Coñac , Sam Edelman Women s Bay Slide Sandal Cuero blanco brillante , Bota de montar de pantorrilla ancha Christine de Franco Sarto para mujer Negro Bomba de mujer David Tate Dainty Negro , Bota de montar Charlotte de Propet para mujer Gamuza negra / negra , CLARKS Mujer Flores Dahlia Tobillo Botín Grey Suede / Leather Combi , UGG Jaida Glitter Oxford para mujeres Plata , Franco Sarto Womens Palmer Piel Metálica Antracita Zapatillas de running Nike Free Tr2 para hombre Ropa de cama / Ropa de cama / Blanco Zapatillas Slip-On Roamer SAS para Mujer Lino

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 running Kinvara 8 Saucony para running mujer Kinvara mujer Azul 49bd62b 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