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.

Textil y sintético

Importado

Caucho único

Eje mide aproximadamente parte superior del arco

Estos los zapatos son específicamente diseñado para un anatomía de la mujer, con un cuidado esculpido último. Y hemos reforzado el brazo superior de el sensifit, para sujeta tu pie en lugar de técnico senderos y ayuda desciendes con más confianza.

Una chasis asimétrico diseñado para acomodar una mujer \\ u2019s la anatomía maximiza estabilidad y amortiguación, mientras minimizas fatiga en accidentado terreno.

Dos diferentes tipos de caucho son estratégicamente colocado en la suela para durabilidad y más agarre. Lengüetas agresivas en el área del talón agarra el terreno, dándote más controlar cuando estás moviéndose a lo largo de un clip saludable.

Salomon 2 años limitados garantía

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

Zapatillas de mujer mujer X Ultra Mid 3 19932 de de Salomon para mujer Mallard azul 68e6a74Updated 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.

Zapatillas de mujer mujer X Ultra Mid 3 19932 de de Salomon para mujer Mallard azul 68e6a74

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

Wonderheel 7 \ Belleville Botas de Combate y Vuelo Impermeables, 790 Broncearse , Botines de tacón de aguja de mujer Sammitop 8 cm Zip Up Botas cortas de invierno con piel Serpiente , Botín de ante Dakota de Therafit Shoe para mujer Negro Bomba de cuña Gumdrop French Sole FS / NY para mujer Cartizze negro , Blundstone Hombres 1464 Nubuck Boots Gris Nubuck Botín de hebilla de mujer Brinley Co Gris Bota alta mística de la rodilla de Franco Sarto para mujer Cuero negro Botín Naomi para mujer de Steve Madden Taupe Suede Arcopedico 1361 Shocks 5 Botas para mujer Brown Starry Night Bota de media caña Georgia GB00133 Barracuda Gold , Roper Women s Liza Southwestern Liza - Mocasines ocasionales para conducir Rosado Bota de mujer FRYE Natalie Short Lug Engineer Carbón de leña suave engrasado Bota de mujer FRYE Natalie Short Lug Engineer Cepillo completo de grano Cognac Bomber / Neon Blue de conducción de mocasines de conducción Twisted X - calzado de cuero auténtico para exteriores #Vans Classic Blue White Canvas Zapatillas Slip-On sin cordones Unisex Coach Womens - Botas de punta de tobillo a puntera de Montana Saddle / Natural Multi Calf / Shearling Mix Arcopedico 1361 Shocks 5 Botas para mujer Noche estrellada azul marino Bota de tacón de aguja con punta única de Liliana DB54 mujer ante punta del muslo Dril de algodón lavado Blundstone Mujeres 1352 Chelsea Boot Shiraz , Jeffrey Campbell Botines de ante Cromwell para mujer Negro apenado , Pleaser INFINITY-2020 9 \ Black Patent / Blk Botas de cuero FLY London Sven 731 para mujer Alfombra de gasolina Botas de mujer de Dr. Martens Women s Flora Arcadia de plata negra , FRYE Bota de Combate con Cremallera Frontal Julie para Mujer Forest Soft Oiled Suede , Zapatillas sin cordones Slip-On Sport Sport Roper para mujer, punta redonda, marrón marrón , Sandalias de peonía Easy Spirit para mujer Azul , Zapatos posteriores de Dr. Scholl s Women para mujeres Estuco de microfibra Prop & eacute; t Blizzard de mujer Zip de invierno con cremallera Marrón / nylon ,

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 para mujer New Balance Cypher Mulberry / Angora Tacones Pleaser para mujer Infinity 901 Claro / Claro Brooks Womens Adrenaline GTS 18 Azul marino / verde azulado / menta , Zapatillas para correr Mizuno Women s Wave Paradox 3 Negro eléctrico verde Colchoneta de amortiguación 620v2 New Balance para mujer Silver Mink / Gunmetal Zapato de agua amazónica de mujer Baffin Rojo oscuro Zapatos de tierra Camino Cuero negro premium Zapatillas de running definitivas Ryka para mujer Gris blanco , Zapatillas de running adidas Pureboost X TR, para mujer Lino / Traza Caqui / Oro táctil , Zapatillas de baño Skechers Performance para mujer Go Step-Seashore Natural CLARKS Haley Stork Loafer para mujer Armada Sandalias de cuero de Viv Viv de TOMS Cuero de Miel / Emossed Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Pure Platinum / Dark Grey Propet Women s Mary Lou Mary Jane Sneaker Naval Blanca , Ballet de niña de flores Aerosoles femenino Serpiente gris , Birkenstock Papillio Daytona Lux Sandalia para mujer Ramo floral Zapatillas de entrenamiento NIKE Flex Supreme Tr 6 para mujer Negro / Negro / Antracita , Calzado de senderismo Go Joy 15602 de Skechers para mujer Negro / gris , Nike Tech Xtreme Cadet Glove Negro / Negro / Gris Fresco / Volt Zapatilla de deporte S-Inspira de Skechers para mujer Negro , Zapatillas de golf adidas W Climacool Knit para mujer Tiza Púrpura / Azul / Coral Sandalia Bobbi Slide de Dolce Vita para mujer Lt Blue Denim Sam Edelman Sandalia Glenn para mujer Cuero de Neymar Natural Naked Tumble Sandalias Jerusalem Sandalia Nuri Slide rojo , Pantuflas Luxe Taupe de Powerstep para mujer Gris pardo , Apretón de jóvenes y adultos Zapato de grifo blanco y negro para espectadores , Zapatillas de ciclismo Pearl IV iZUMi W Select para mujer Negro Negro Bella Vita Womens Tex-Italia Gamuza de coñac , Propet Women s Alice Oxford Negro , ara Laurel Loafer Mujer plana Taupe Snake Combo

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 Zapatillas de mujer mujer X Ultra Mid 3 19932 de de Salomon para mujer Mallard azul 68e6a74 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
2115519

3405326

4654320

6567065

8047706

9001241

11027786

15626708

16509062

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