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

Importado

Goma único

Sea casual fabuloso con estos querida slip-on zapatos de Earth & reg;.

Premium parte superior de cuero con recortes geométricos para una mayor atracción.

Fácil estilo sin cordones.

Guante suave forro de cuero.

Exclusivo plantilla de confort amortigua tu pie desde el talón hasta los pies. Arco reforzado soporte para todo el día comodidad. alivio del talón vaina. suela proporciona tracción y estabilidad. Mediciones: Peso: 8 oz Medidas del producto fueron tomadas usando tamaño 9, ancho B - Medio. tenga en cuenta que las mediciones puede variar por tamaño

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

Artemisa Dólar Artemisa amarillo Dólar suave suave 5ec15aeUpdated 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.

Artemisa Dólar Artemisa amarillo Dólar suave suave 5ec15ae

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

Drew Shoe La alegría de las mujeres Combo oliva , Zapatillas para caminar Nell Ryka para mujer Azul marino / rosa , Zapatillas para caminar Skechers Performance Womens Go Walk 3 Glisten Rosado Ballet de niña de flores Aerosoles femenino Serpiente de coral Calzado de senderismo Skechers Go s 15601 para mujer Blanco negro , Zapatillas deportivas sintéticas Drew Shoe para mujer Estampado floral fucsia Prop & eacute; t Zapatillas para mujer Wear N Wear Slip on LL Walking Sr White Navy , Calzado SCARPA Women s Crux Walking Tiburón / Resplandor azul Skechers Women s Go 4-14918 Calzado para caminar borgoña Propet Women s Mary Lou Mary Jane Sneaker Naval Blanca , Sandalia Vionic Shelley Slide para mujer menta , Prop & eacute; t Zapatillas para caminar TravelFit para mujer Baya Zapato de caminar de la correa de la andadera de la estabilidad de las mujeres de Propet Negro Easy Spirit Solana Oxford para mujeres Aceituna PUMA BTS x Collaboration Court Star (366202) Merrell Womens Barrado Cuero plateado Ros Hommerson Chelsea Mary Jane Slip On Shoes para mujer Azul / iridiscente Zapatillas de viaje Proather para mujer II Broncearse Zapatillas para caminar Omni Walker Grid de mujer Saucony Negro Sandalias SAS para mujer, Nudu Slide Oro SAS Womens Sunny Teal , Skechers Performance Women s On The Go City 3 Optimize Negro Sandalias Harper Huarache de SoftWalk para mujer Negro / peltre , Trotters Mujer Simmy Mary Jane Flat Negro Cobb Hill Rockport Mujeres Petra Mary Jane Flat marrón , Zapatillas para caminar Skechers Performance para mujer Go Fit TR Negro / rosa Zapatilla S Skechers para mujer E Z Flex 3.0 Swift N Sly Negro , Zapatillas para caminar Propell Mujer Travellite MJ Negro Zapatos de mujer RYKA Sky Walking Frost Grey / Aqua Sky / Iron Grey ,

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 ASICS Womens Gel-Fortitude 8 Gris medio / blanco / azul de porcelana Sandalia de susurro de mujer Keen Poseidon / Blue Danube , Nike Men s Air Presto Essential Negro / Foto Azul / Blanco Sandalias Newport H2 de KEEN para mujer Rosa negra / brillante , nike womens air max 1 JCRD zapatillas para correr 819808 zapatillas zapatos Hematita Metálica / Negro-gris oscuro Zapato de Trail Hellcat Boombah para hombres - 14 Opciones de color - Varios tamaños Negro / Naranja Zapatillas de trail running Saucony para mujer Calibre TR Trail Runner Negro | , Corredor Merrell para mujeres, agilidad Peak Flex Trail Patrulla de esquí Zapatillas de running MT-onemix Unisex Modern City Lightweight Running Jogging Gris oscuro Zapatillas adidas Outdoor Terrex Agravic Speed Shoe de mujer Gris crudo, Gris crudo, Gris ceniza , Skechers Performance Women s Define Wide Sneaker Negro , NIKE W Air Max 97 Ul 17 PRM Mujeres Ao2325-001 Piedra pómez clara / Antracita-fibra de vidrio , Zapato de caminar WW769V1 para mujer New Balance Chocolate Bungee , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino Blanco / Blanco-puro , Birkenstock Blue Footbed-Arch Soporte plano Ante Mocha , Zapatillas para caminar Ryka Women s Dash 2 Azul marino / gris / morado Zapatillas de entrenamiento Nike Air Zoom Strong 2 Metallic para mujer US) Negro Zapatillas de deporte Xelero Genesis Comfort Comfort Therapeutic Extra Deep Sneaker Leather / Mesh Lace-up Blanco azul , Zapatillas bajas con cordones All Star Canvas de Converse Chuck Taylor Blanco , Zapatillas para caminar con cordones Go Walk 4 de Skechers Performance para mujer Azul marino / verde azulado , KAANAS Santa Cruz Raffia Pom Pool Sandalia plana deslizante Natural Sandalias Sandalias Bashan moldeadas Sandalias Sandalias Jerusalem Negro Sandalia de la diapositiva de Anica de las mujeres de Cole Haan Cuero negro , Sandalias Peep Toe de Misu para mujer Latin Salsa Blackgo Practicar zapatos de baile para salón con tacón de 2.75 \ Negro , Ara Zapatillas Aster para mujer Taupe Metallic Lamfell CLARKS Mujer Maypearl Alice Tobillera Cuero negro Bota de ante FRYE Sacha Moto Shootie para mujer Negro , Bomba de vestido Kalila Total Motion Kalport para mujer Serpiente negra / múltiple , Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse Rieker Womens 70581 Mogano / Muskat / Nuss

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 Artemisa Dólar Artemisa amarillo Dólar suave suave 5ec15ae 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