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.

Malla

Importada

Arriba la apuesta en tu correr diariamente con la dinámica y ASICS de apoyo y reg; Gel-Fortitude & reg; 8 zapatillas para correr eso dará usted un paseo suave ¡siempre!

Predecesor: GEL-Fortitude 7.

Soporte Tipo: Neutro.

Amortiguación: Ligero, flexible respuesta.

Superficie: La carretera. 10 mm. Altura de la plantilla: talón / puntera: 28 mm / 18 mmEl receptivo, amortiguado el diseño lo hace ideal para neutral corredores con piso pies o corredores que usan ortesis. Malla de ingeniería superior proporciona transpirabilidad ligera con menos costuras. Lace-up cierre. lengua y cuello.Transpirable forro de tela.Removable ComforDry & trade; características de la plantilla propiedades antiolor para un secador, más fresco feel.I.G.S & reg; (Orientación de impacto) Sistema) emplea componentes vinculados para impulsar el

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

ASICS Womens Gel-Fortitude 8 de Gris medio porcelana/ Womens blanco/ azul de porcelana ded713aUpdated 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.

ASICS Womens Gel-Fortitude 8 de Gris medio porcelana/ Womens blanco/ azul de porcelana ded713a

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 deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Blanco / Fresco Grey-metallic Silver Mochila para correr Newton Women s Motion 7 Fuschia / Teal 9 - Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Phantom / imán Under Armour Women s HOVR Sonic NC Negro (003) / Marfil , Zapatillas de running Under Armour Speedform Intake 2 para mujer Gris , Zapatillas de running Saucony Women s Breakthru 3 Azul / menta Zapatillas de running adidas Ultraboost W para mujer Gray Five / Carbon / Ash Pearl , Calzado de running para mujer New Balance Cypher Castlerock / White , Zapatillas de running NIKE Lunar Skyelux para mujer Golpe caliente / Marrón nocturno Saucony Women s Ride 10 Running Shoe Niebla / Rojo Under Armour Threadborne Fortis para mujer Gris Nublado (103) / Blanco , Zapatillas de entrenamiento NIKE Metcon Dsx Flyknit para mujer Negro / Cool Gray-hyper Punch , Zapatillas para caminar Skechers Go Run-Mojo para mujer Negro , Zapatillas para correr Pacev2 para mujer New Balance Silver Mink / Black , Brooks Women s Dyad 9 Primer gris / Capri Breeze / Plata , Zapatillas de running adidas Cloudfoam Super Flex W para mujer Misterio Azul / Blanco / Naranja Brillante , Zapatillas para correr Nike Zoom All Out Flyknit para mujer Blanco / Negro - Pure Platinum Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Hot Punch / Black-aluminium-university Blue NIKE WMNS Zoom Fly Women 897821-411 Equator Blue / Summit White-lagoon Pulse , Zapato deportivo ASICS Gel-Kayano 24 para mujer Gris medio / negro / Atlantis , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Cool Grey / Court Purple / Dark Grey , Zapatillas de running Pure Cloudfoam de adidas para mujer Blanco / Blanco / Negro , Zapatillas de running NIKE Air Zoom Vomero 12 para hombre Lobo Gris Puro / Negro-puro Platino-caliente Zapatillas de running NIKE Mens Air Zoom Structure 19 Luz retro / blanco-verde azulado Zapatillas de running Reebok Women s Supreme 3.0 MT Nube gris / Caribe trullo / polvo de asteroides , NIKE Air Max 90 Womens Style: 325213 Mujeres 325213-048 Lobo gris / blanco-gris oscuro , adidas outdoor para mujer Terrex Swift R2 GTX Grey Three / Gray Two / Chalk Coral Nike Men s Air Presto Essential negro , Zapatillas de running cross cross para mujer ASICS Cross Faire 2 para mujer Polvo azul / blanco / azul marino ,

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 OluKai Kulapa Kai Pale Jade Sandalias Birkenstock Arizona Mocha Birko-Flor Narrow Fit para mujer (8 mujeres EE. UU. - 39 N UE) - , Sandalias de encaje Ros Hommerson para mujer Oro , Calzado de running NIKE Kids Roshe One (GS) Hyper Cobalt / Deep Royal Blue / Varsity Maize , Sandalias Swiftwater Mesh de Crocs para mujer Nuez , CLARKS Sandalia plana Brizo Ravena para mujer Gris , Salomon Women s Speedcross 4 W Trail Runner Poseidón Zapatillas de running La Sportiva para mujer Trailido Bushido Fiordo , Zapatillas Columbia Trail Trail WMNS de Plains Ridge para mujer Silla de montar / pluma , Salomon Women s Speedcross 4 W Trail Runner Poseidón , Zapato para caminar feroz Ryka para mujer Gris oscuro Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow Zapatillas de running Saucony para mujer Echelon 5 Gris / rosa , FitFlop Womens Loosh Luxe & iquest; Cuero metalizado plateado Zapatillas OLUKAI Upena - Mujer Negro Sandalias Peep Toe de Misu para mujer Latin Salsa Tango Practice Bailarinas de salón con tacón de 3.3 \ Broncearse Zapatilla de pies Cush N de Prop & eacute; t para mujeres Merlot Corduroy Vans Auténtico (lavado de 2 tonos) marrón / blanco verdadero , Adidas W Adipower Boost Boa Golf Shoe para mujer Blanco / púrpura Glow / Mystery Ink , Zapatillas de golf Skechers Women s Go Golf Birdie Natural / Coral Zapatillas de tenis adidas Ligra 5 W para mujer Tinta noble / oro metálico / tiza púrpura Very Fine Dance Shoes 2707 (Grado de competición) 2.5 \ 2707- Satén marrón claro y oro claro , Noble Outfitters Womens Muds Stay Cool 6 Oro Zapatilla Glamour de mujer verde Daniel Hueso , Bomba de vestido Daisie de mujer Steve Madden Ante amarillo , Zapatillas de running NIKE Men s Train Speed 4 Max naranja / Hyper naranja / negro , Wolky Comfort Sneakers Kinetic 30120 Cuero blanco roto , TOMS - Suela de lino de lino clásica, cómoda y de ajuste fácil para mujer Chambray texturizado Oxford Tan / ajuste Ballet Tali de mujer Cole Haan Bow Bow Serpiente Roccia Natural Imprimir , TOMS - Slip-On de lienzo clásico para hombre Microstripe azul marino / recorte ,

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 ASICS Womens Gel-Fortitude 8 de Gris medio porcelana/ Womens blanco/ azul de porcelana ded713a 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