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.

Poliéster

Importado

Goma suela

con doble correa poliéster jacquard correas superiores alrededor del pie y a través del entresuela para un personalizado ajuste

Ajustable y durable de alta resistencia correas del talón de las correas

moldeado por inyección hebilla de bloqueo de escalera

topográfico surcos texturizados en plantilla para tracción

No marcado ecotread 25% reciclado compuesto de goma con 3 mm z-inspirado asas

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 deportiva Chaco Zvolv 11226 Chaco X2 para mujer Sandalia Azul volcánico 401bfbdUpdated 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 deportiva Chaco Zvolv 11226 Chaco X2 para mujer Sandalia Azul volcánico 401bfbd

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

Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre Sandalias Newport H2 de KEEN para mujer Rama de Malaquita / Plata , Sandalia Uneek O2-w de mujer KEEN Etherea / Whitecap Sandalias Uneek EXO-W de mujer KEEN Negro / gris acero , Sandalia Uneek O2-w de mujer KEEN Acero Gris / Vapor Sandalia Savannah de Ryka para mujer Gris blanco , Tacones Pleaser para mujer Infinity 902 Negro Negro Sandalias KEEN Venice II H2-W para mujer Paloma / Turquesa Pastel , Sandalia de susurro de mujer Keen Sombra Oscura / Cerámica , Sandalia Drew Angela para mujer Dusty Multi Metallic , Earth Origins Sophie Negro , Sandalia deportiva Bondi de Donald J Pliner para mujer Magenta , Wolky Comfort Jewel 12500 nubuckleather rojo Sandalia de mujer Drew Alana Pewter-bronce , BZees Sandalia con cuña de beso para mujer Blue Multi Print MAVIRS Brand Women Sandalias 2018 Verano Correa de tobillo de flecos Stilettos 12CM Extreme High Heels Zapatos azules negros Tamaño EE. UU. 5-15 Multicolor Drew Womens Ariana Dusty Gold , Sandalia Drew Element para mujer Denim Blue Nubuck , Sanda Verra Teva para mujer Cuerda elástica , Sandalia de susurro de mujer Keen Cascada / Stone Blue , Wolky Comfort Jewel naranja , Zapatillas de viaje Protic W3910 Vista Walker para mujeres Negro Sandalia de mujer Bellini Nickel Broncearse , Chaco Mujer ZX1 Classic Sport Sandalia Pep Pine , KEEN Hombres Newport H2 Sandal Brindle / Sunset Chaco Women s Zx2 Classic Athletic Sandal Fusion Rose , Nike Benassi Duo Ultra Slide Racer Pink / Black / Sunset Glow Sandalia Drew Element para mujer Nubuck negro Sandalia deportiva Tirra para mujer Teva Gris oscuro ,

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 Encaje de quimera de Columbia para mujer Arrecife / Nivel del mar , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Dark Gray Mtlc Silver Blk , Zapatillas de running Spira Phoenix para mujer con resortes Carbón / Negro / Tsai Rojo Zapatillas de running 430v1 New Balance para mujer Gunmetal / Seafoam , Wolky Comfort Jewel 20570 Cuero rojo , Camiseta adidas Swift Run W para mujer Blanco / gris, uno / hielo, menta , Nautilus ESD Soft Toe Atlético CT Mujer Oxford Gris , Calzado deportivo G-Defy Gamma-Ray para mujer Gravity Defyer Gris, rojo Sandalia Sharene de Calvin Klein para mujer Blanco suave , Bomba de cuña Lansing para mujer Trotters Gamuza marrón oscuro Zapato de running ASICS para mujer Torrance Gris piedra / helado , Zapato de running ASICS GT-1000 6 para mujer Gris / blanco / gris , Zapatillas deportivas NIKE Lunar Apparent para hombre Índigo ligero de carbono / obsidiana neutral , Vans Old Skool Unisex Adults Low-Top Trainers Noche parisina , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco Bomba de vestido Lexre Twin Gore de mujer Aravon Ante negro , Drew Shoe Women s Tulip Comfort Cobre Metálico , Sandalia de cuña Ollie Wedge Chinese Laundry para mujer Brillo dorado / arena Sandalia FLY London Women s Whin176fly Slide Lead Borgogna Koolaburra de UGG Women s W raychel Slide Sandal Ánfora , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 1659- Silver Stardust & Silver Leather Mephisto Helen 3478 castaño Braun Zapatillas de ciclismo Pearl iZUMi W X-Alp Enduro v5 para mujer Negro / gris oscuro , Zapatos planos de tacón bajo YDN para mujer con tacón bajo, deslizamiento en zapatillas Bowknot Zapatos antideslizantes de verano en deslizamiento Gris Bella Vita Womens Mel-Italy Blush-gamuza , MICHAEL Michael Kors Mujeres Carrigan Bootie Dark Gamuza Deportiva Caqui , Bomba de vestimenta Cambredge para mujer Jessica Simpson Black Soft Nappa , Cole Haan Women s Althea Bootie Java Easy Spirit AP1 2E, punta redonda, lona, Oxford rojo TOMS Clásicos de mujer Lienzo rojo

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 deportiva Chaco Zvolv 11226 Chaco X2 para mujer Sandalia Azul volcánico 401bfbd 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