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.

100% poliéster

Nike Dri-Fit - Es un tela innovadora tecnología diseñada

Incorporado escritos - Para soporte y la comodidad cuando hacer ejercicio.

interno Llave de bolsillo - Ideal para llaves o pequeñas artículos.

Malla paneles - En las costuras laterales para la transpirabilidad y la comodidad cuando corriendo.

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

Short Short Blanco NIKE Tempo para mujer para Blanco/ Hyper Pink 75faf66Updated 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.

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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 ASICS Gel-Contend 4 para mujer Diva Azul / Plata / Orquídea , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Platino Blanco / Blanco-puro Zapatillas de running NIKE Air Huarache para hombre Hueso Ligero / Vela Zapatillas de running ASICS Gel-Cumulus 19 para hombre Gris / Negro / Directoire Azul , Zapatillas de running Reebok Print Smooth Ultk para mujer Negro / polvo de asteroide / blanco Zapatillas de running Kinvara 5 Saucony para mujer Blanco / Crepúsculo / Puesta de sol , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Platinum / Wolf Gray / Metallic Platinum / White , Zapatillas de running adidas Aerobounce W para mujer Indigo crudo / plata metálica / naranja de alta resolución , Zapatillas de running NIKE Fs Lite 2 para mujer Chalk Blue Racer Blue White Zapatillas para correr Nike Flex Experience RN 5 Sigilo / Mtlc Plata / Antracita Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Negro / Fantasma , Zapatillas running para mujer Saucony Liteform para mujer Negro Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Negro / sal marina , Calzado de running para mujer New Balance Cypher Castlerock / White Zapatillas de volante NIKE Air Zoom Hyperace para mujer Blanco negro Zapato deportivo ASICS Gel-Kayano 24 para mujer Blanco / azul violeta / amarillo de seguridad , Zapatillas de running NIKE Air Max Torch 4 para mujer Cool Grey / Hyper Violet / Hyper Punch , Zapatillas para correr NIKE Womens Sock Dart Negro / Vachetta Tan-negro-blanco Brooks Womens Adrenaline GTS 18 Grey / Rose Gold , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Diva Blue / Flash Coral / Aqua Splash New Balance W880gb6 para mujeres Gris azul , Zapatillas de running adidas Women s Edge Lux W Black / Vapor Gray Metallic / Orchid Tint Zapatillas de running ASICS Gel-Contend 4 para mujer Gris Medio / Mar Glaciar / Plata , Zapatillas de running Hupana 2 para mujer de HOKA ONE ONE - Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Jade Aqua-claro negro / blanco-claro Brooks Womens Hyperion Negro / Dewberry / Aruba Blue Zapato de running ASICS GT-1000 6 para mujer Azul Púrpura / Azul Índigo / Lima de Neón , Zapato deportivo para acolchado 635v2 New Balance para mujer Gris , Zapatillas de running NIKE Airstream Sequent 3 para mujer Sequoia / Metallic Platinum-white

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 Converse Chuck Taylor All Star Seasonal Color Hola Rojizo-Marron , NIKE Women s Revolution 4 Running Shoe Racer Pink / Black Hot Punch Wht Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Negro / Blanco-Hyper Punch , Zapatillas de running ASICS Gel-Kayano 23 para mujer Glacier Grey / Bay / White Zapatillas de running Adidas Performance Ultra s Boost M para hombre Energía / Energía / Negro Zapatillas NIKE Air Huarache Run SE (GS) Girls Fashion 904538 negro / negro-negro , Zapatillas de running adidas Runupt Runner Originals para mujer Ropa de cama / Ropa de cama / Ecru Tint , Zapatillas de running ASICS Gel-Cumulus 19 para hombre Carbon / Black / Hot Orange Zapatillas deportivas NIKE Lunar Apparent para hombre Jade claro de carbono / negro claro Earth Origins Sophie Azul marroquí , Sandalia Merrell para mujer Terran Ari Post Sport Negro , Zapatillas de running adidas Pureboost X Tr Zip para mujer Lino / Traza Caqui / Oro táctil Birkenstock Unisex Arizona Essentials Sandalias EVA Khaki Eva , Zapatillas de running ASICS Gel-Nimbus 20 para mujer Indigo Azul / Azul Indigo / Verde Opal , Zapatillas de running adidas Adizero Boston 6 W para mujer Noble Indigo / Blanco / Acero sin Procesar Prop & eacute; t Zapatillas para caminar TravelFit para mujer Lima / Azul Zapatillas de viaje Proather para mujer II Plata , Prop & eacute; t Olivia Oxford para mujeres Bronco Brown Skechers Sport Skech Air 2.0 City Love Fashion Sneaker Negro / rosa , Toni Pons Womens Vic Taupe Sedavi , BareTraps Women s Cherilyn Slide Sandal champán , Aveline Slide de Ted Baker para mujeres Blue Harmony Mephisto Sandalia Ivana para mujer Rosa viejo , Jig Foo Latin Salsa Rumba Chacha Practice Ballroom Zapatos de baile para mujeres Negro Bota occidental Roper Women s Cross Negro Ariat Womens Devon Pro VX Paddock Negro , SOREL Women s Out N Acerca de las diapositivas Gris claro / Alce Sandalias Peep Toe de Misu para mujer Latin Salsa Blackgo Practicar zapatos de baile para salón con tacón de 2.75 \ Negro , Bota de agua caliente Belleville 390 Broncearse , Bota a la rodilla Pleaser Seduce-2000 para mujer Red Stretch Patent ,

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 Short Short Blanco NIKE Tempo para mujer para Blanco/ Hyper Pink 75faf66 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

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