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

Importado

Sintético único

Radicalmente ligero

Responsivo Amortiguación de la entresuela de 5 gen

Goga energía máxima de retorno plantilla

suave tela superior y revestimiento

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

Skechers Performance On The B010EAM5S0 1374 Go 400 Vivacity Flip Carbón Flop Carbón de leña/ rosa fuerte 7518b54Updated 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.

Skechers Performance On The B010EAM5S0 1374 Go 400 Vivacity Flip Carbón Flop Carbón de leña/ rosa fuerte 7518b54

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 de mujer Drew Alana Verde / rosa Sandalias KEEN Venice II H2-W para mujer Grape Kiss / Red Violet , Sandalia de vestir Tillman de SoftWalk para mujer Blanco Chaco Mujer ZX3 Classic Sport Sandalia Azul inca Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre , Sandalia de susurro de mujer Keen Licor de café / amarillo Sandalia Drew Angela para mujer Negro , Encaje de quimera de Columbia para mujer Arrecife / Nivel del mar Xero Shoes Z-Trail Sandalia ligera - Descalzo-Inspirado Ruta de senderismo, Correr sandalias deportivas - De las mujeres Multi-marrón Sandalia NIKE Benassi JDI para mujer Blanco negro , Sirena de mujer Merrell Flip Q2 Athletic Sandal Aluminio Chaco Women s Z1 Classic Athletic Sandal Gaulteria Sandalia Merrell de mujer con gladiador de poste Dusty Blue , OLUKAI Hiona Sandal - Mujer Dark Java, dark Java , Sandalia NIKE Kawa Slide para mujer Hot Punch / Paramount Blue , Sandalia deportiva Tirra para mujer Teva Azul real Chaco Women s Z2 Classic Athletic Sandal Florista , Chaco Mujer Zcloud X Sandalia Deportiva Adobe disparado Wolky Comfort Jewel Gray Picasso Crash Sandalias Drew de mujer de Drew Cuero liso negro , Sandalia universal W Sanborn de Teva para mujer Negro Sandalia de mujer KEEN Rose Boysenberry / Red Violet , Sandalia deportiva Alise Les Filles para mujer Dusty Lilac Nappa Zapatillas de viaje Protic W3910 Vista Walker para mujeres Blanco suave , Earth Origins Sophie Negro , CLARKS Womens Wave Grip Sage Nubuck Sandalia de mujer Drew Alana Estaño / Bronce Sandalia deportiva Tirra para mujer Teva Gris oscuro , Sandalia de susurro de mujer Keen Cascada / Stone Blue

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 Calzado de running Run Runicic de Fila, para mujeres, resistente a los resbalones Negro / Estaño / Plata Metálica Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Hyper Orange / White / Total Crimson / Pink Blast CLARKS Womens Wave Grip Sage Nubuck , Zapatillas de running adidas Adistar Boost ESM para mujer Flash rosa / metal cero / gris claro , Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Juanetes caminando Whitney Womens Sneakers Athletic Zapatos Negro , Zapatillas Erika Propet para mujer Chili Red , NIKE Women s Tanjun Blanco negro , SOREL Womens After Hours Slingback Lu-negro , Zapatos OLUKAI Pehuea - Mujer Magenta / Magenta , Zapatillas deportivas NIKE Lunar Apparent para hombre Antracita Crimson Blk Gray Calzado deportivo de running New Balance Foam Cruz V1 de New Balance para mujer Dragon Fruit / Sea Salt , Zapatillas de deporte CL Nylon MH Reebok para mujer Eco azul / azul impresionante / verde azulado / añil / blanco / dólar , Nueve West Womens Xaeden Cuero natural claro , Aravon Women s Pc Three Strap Sandal Rio Red Leather Zapatilla Ascot de hombre UGG Musgo Verde Entrenador cruzado Gel-Fit Tempo 3 de ASICS para mujer Ciruela / Plateado / Rojo Rouge Skechers Women s Appeal Flex New Image Sneaker Rosa fuerte / negro , Vans Unisex Old Skool Zapatillas clásicas de skate rojo , Sandalia con tacón Florence Block de Via Spiga para mujer Patente de arena Sandalia para mujer Chandler de Carlos by Carlos Santana Rosa rubor , Birkenstock Arizona Sandalia de cuero con plantilla suave Plata , Zapatos de césped del escuadrón femenino Boombah - 14 opciones de color - Varios tamaños Negro / Rojo , Bota de montar de pantorrilla ancha Christine de Franco Sarto para mujer Negro , TOMS - Slip-On de lienzo clásico para hombre Llovizna Lana Gris / Faux Shearling Zapatillas de deporte Saucony Spitfire para mujer Citron / Púrpura / Naranja MICHAEL Michael Kors Mujeres Carrigan Bootie ciruela Bota de invierno impermeable SOREL de mujer Whitney Lace Negro Trotters Lucy Ballet para mujer Blanquecino Zapatillas de deporte lavables Walker Propet para mujer Sr Pewter , TOMS - Slip-On de lienzo clásico para hombre Universidad Negro ,

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 Skechers Performance On The B010EAM5S0 1374 Go 400 Vivacity Flip Carbón Flop Carbón de leña/ rosa fuerte 7518b54 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