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.

Hecho por el hombre, malla

Importado

Caucho único

The GEL-Cumulus & reg; 20 zapatillas para correr es un ganador probado eso garantiza un viaje suave cada ¡tiempo!

Sucesor al Gel-Cumulus 19.

Soporte Tipo: Neutro.

Amortiguación: Alta energización Amortiguación.

Superficie: Road.Differential: 10 mm. Heel / Toe: 22 mm / 12 mm.Jacquard malla superior con FluidFit ™ tecnología combina multidireccional malla elástica y refuerzos sintéticos que se adaptan a la pie, proporcionando un excepcional ajuste. la construcción se reduce la probabilidad de fricción relacionada irritación. Laceración cierre con discreto ojetes que desembolsan tensión de encaje para confort óptimo. Características reflexivas materiales diseñados para mejorar la visibilidad con poca luz horas. Tonelada

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

Zapato de running 20 ASICS GEL-Cumulus/ 20 B01H3EBOKU para mujer Gris piedra/ negro eae2baaUpdated 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.

Zapato de running 20 ASICS GEL-Cumulus/ 20 B01H3EBOKU para mujer Gris piedra/ negro eae2baa

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 NIKE para mujer Lunarepic Low Flyknit 2 Blanco / negro / puro platino , Brooks Womens Adrenaline GTS 18 Blanco / Blanco / Gris Zapatillas de running NIKE Zooms Winflo 4 para mujer Lobo gris / blanco / antracita / Sunset Glow Calzado de running NIKE Kids Roshe One (GS) chispa costera azul / blanco-azul Brooks Women s PureGrit 5 Azul pavo real / Rosa virtual / Azul patriota Amortiguación 635v2 New Balance para mujer Gris , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-Gris Oscuro Zapatillas de running adidas Adizero Boston 6 W para mujer Noble Indigo / Blanco / Acero sin Procesar Zapatillas de running NIKE Roshe Two para mujer Caqui / Negro / Negro / Blanco Zapatillas de running adidas Supernova Sequence 9 W para mujer Súper Púrpura / Metálico / Plateado / Gris Medio Zapatillas de running NIKE Air Max Thea Jacquard para mujer Negro / Negro / Azul Racer / Blanco NIKE Mujer Air Max 90 Premium 361 Zapatillas de running Spinject para mujer Cali Coral / Ebony , Zapato deportivo ASICS Gel-Excite 4 para mujer Gris medio / blanco / verde hielo , Zapatillas de running NIKE Huarache Run SE para mujer Negro / oscuro / gris / adoquines , Brooks Womens Hyperion Peacoat / azul marino / cacatúa , Zapatillas de running ASICS GEL-Sonoma Trail para mujer Carbón de leña / menta / rosa fuerte Zapatillas de running a campo traviesa Fastwitch 8 para mujer Saucony Verde azulado / blanco , Zapatillas de running Altra Women s Intuition 4 Ocean / Teal ASICS Kanmei Shoe Casual para mujer Gris medio / gris medio / carbono , Zapatillas de running Skechers Go Run Ultra Road 2 para mujer Carbón Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Blanch Blue S75722 , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Dark Mulberry / Black Rose adidas Performance Ultraboost X para mujer Negro / Negro / Misterio Ruby Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Microchip / Blanco / Carbón metálico , New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running adidas Ultraboost W para mujer Unity Ink / Unity Ink / Dark Navy , Zapatillas de running NIKE RN Commuter 2017 para mujer VAST GREY / WHITE Zapatillas de running ASICS Gel-Cumulus 19 para mujer Azul Púrpura / Negro / Flash Coral ,

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 Zapatillas de running NIKE Juvenate para mujer Chalk Blue / Loyal Blue Zapatillas de running Trail para acolchado WT410v5 de New Balance para mujer Negro Calzado de running XA Amphib W Trail para mujer Salomon Clima tormentoso Zapatillas de trail running Inov-8 Arctic Claw 300 Negro / azul / plata / lima Zapatos de agua para mujer Dreamcity Zapatos deportivos atléticos para caminar Rosa roja , Zapatillas de running NIKE RN 2017 para mujer gratis Vela / Plata Metálica / Gris Pálido / Blanco Cumbre , Drew Shoe Bayou Diabético terapéutico para mujeres Sandalia de profundidad extra Calzado de cuero Ninguno Azul / Microdot Sandalia de vestir Desi Easy Street para mujer Negro Patente / Negro , Zapatillas de entrenamiento cruzado PUMA para mujer Fierce Core Puma Black-paradise Pink Zapato de agua Easy Spirit para mujer First Walker Gris nativo AP Mercury Liteknit Clay Pink / Shell White / Natural Rubber Zapatillas ALC-Diva Cross-Trainer para mujer Avia Gris hierro / Niebla fría gris / Plumeria , Supra Scissor Skate Shoe Negro / Blanco-negro , JiYe Performance Sports Shoes Zapatillas de deporte de baloncesto para hombre Black Sliver Skechers Women s Empress-Wide-Awake Rosado Sandalias de mujer Mia de Sandalias de Jerusalén rojo , Birkenstock Mujer Mayari Birko-Flor Sandal Magenta Haze Birko-flor Sandalia Mila Slide de mujer Naturalizer Gamuza rosa Sandalias veganas Arizona Arizona Birkenstock Sfb Navy Birko-flor , Sandalia Merrell Terrano Ari Slide para mujer Dusty Olive , Zapatillas de voleibol Wave Bolt 5 para mujer de Mizuno Negro / real Dublín Ladies Wye Boots , Bomba de cuña Blare para mujer J.Renee Tela negra con brillo de arlequín Trotters Lucy Ballet para mujer Cerezo negro Zapatillas de salón de baile femenino de Very Fine CD6017 con HP 2.75 \ Satén negro CLARKS Womens Quartz Sandalia Negro , Bota western para mujer FRYE Sacha Moto Shootie Cognac-78001 Sam Edelman Women s Addison Sandalia de tacón Desnudo clásico , Bota de moda Polley Daze para mujer de Dr. Martens Multi , Clásicos de base Slip-on (Tm) para hombres de Vans (Checker) Racing Rojo / Blanco ,

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 Zapato de running 20 ASICS GEL-Cumulus/ 20 B01H3EBOKU para mujer Gris piedra/ negro eae2baa 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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