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

Superior Ajuste. la parte superior es transpirable, elástico y de apoyo para una comodidad similar a un calcetín. Flywire súper fuerte cables combinados con los cordones para bloquear tu pie.

Acolchado para entrenamiento. entresuela plegable es firme en el talón y más suave y más flexible en el antepié Esto te da la estabilidad que necesidad de entrenamiento más la amortiguación necesitas abreviar corre.

Estabilidad para levantar plataforma plana y talón de goma firme darte poderosa estabilidad para explosivo levantamiento. soporte completo te deja entrenar con confianza.

Ligero y duradero. material en alto desgaste áreas agrega durabilidad sin exceso peso.

Talón clip ayuda a reducir arrastre durante la pared ceremonias

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 entrenamiento Nike Womens Metcon 13767 3 Negro Womens/ 3 Blanco-volt-cloro Azul 0812250Updated 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 entrenamiento Nike Womens Metcon 13767 3 Negro Womens/ 3 Blanco-volt-cloro Azul 0812250

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 Mens Air Zoom Structure 19 Negro , Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey , Zapatillas de running ISO para mujeres de la guía Saucony Blanco azul , Skechers Performance Women s Go Run 400 Obstruct Negro / Aqua , New Balance Vazee Running Shoe-Breathe Pack Zapatillas de deporte de moda para mujer Tóxico , Zapatillas de deporte Reebok Print Run Dist para hombre Cool Shadow / White , Zapatillas de running adidas para mujer Cloudfoam Qt Racer W Malla negra / blanca / negra , Nike Hombres Air Zoom Pegasus 33 Citrus gris oscuro / blanco / negro / brillante Zapatillas Nike Air Max Essential para hombre Schwarz , Zapatillas de running Pure Cloudfoam de adidas para mujer Blanco / Blanco / Negro PUMA Men s Roma Basic Sneaker Blanco / Peacoat / avena Zapatillas de running W1540V2 New Balance para mujer Blanco azul Zapatillas de running ASICS Gel-Cumulus 19 para mujer Begonia Pink / Begonia Pink / Baton Rouge Zapatillas de running NIKE Zoom para mujer Negro Negro , Zapatillas de running Brooks Womens Glycerin 15 Neutral Max Cushion Evening Blue / Purple Cactus Flower / Teal Victory , Nike Hombres Air Zoom Pegasus 33 Universidad Rojo / Blanco / Negro , Calzado de running NIKE Kids Roshe One (GS) Foto Azul / Lobo Blanco Gris , Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul , Zapatillas de running Nike Zoom All Out Low para mujer Fire Pink Black Bright Mango 600 Zapatillas de running Under Armour Speedform Intake 2 para mujer Ivory (102) / Ghost Grey , Zapatillas de running Reebok Print 2.0 para mujer Gable Gray / Fire Spark / Blanco / Pure Silver , Zapatillas de running New Balance 990V3 para mujer Negro Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosa Fuego / Negro / Crema de Melocotón , Zapatillas de deporte para correr adidas Originals NMD_R1 para mujer Negro, rojo New Balance Women s w940v3 Running Shoe Plata NIKE WMNS Zoom Fly Women 897821-411 Blanco negro Zapatillas de running adidas Pureboost Xpose Clima para mujer Energy Blue / Energy Blue / Collegiate Navy Zapatillas de deporte Reebok Print Run Dist para hombre Negro / Aleación , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / antracita

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 NIKE Big Kids Air Max 90 Cuero Zapatos para correr, Ligero, Cómodo y Durable de grano completo y cuero sintético Blanco negro , Zapatillas de running Reebok Zpump Fusion 2.0 para mujer Aleación / Gris Estaño / Rosa Solar / Carbón / Blanco , Zapatillas Nike Women s Flex Trainer 5 Cool Grey / Black / Sunset Glow Sandalias Newport H2 de KEEN para mujer zen , Zapatillas de running NIKE Unisex Flyknit Racer Neo Turquesa / Brillante Glaciar Crimson Hielo , Wolky Women s Rio Sandalia Denim Crash Suede Sandalia Romika Women s Hollywood 04 Sport Castagne , Zapatillas de trail running Hierro V3 para mujer New Balance Coral vivo / Pigmento , Zapatillas de running ONEMIX para hombre y mujer Zapatillas de deporte para correr Gray Blcak Zapatillas de running para exteriores adidas outdoor Terrex Tracerocker W para mujer Gray Five / Black / Utility Black Skechers Women s Appeal Flex New Image Sneaker Negro Zapatillas de running Fresh Foam 1080v7 para mujer New Balance Ozone Blue Glow / Lime Glow Brooks Womens Adrenaline GTS 18 Blanco / morado / gris , Vionic Sage Dorrin Mujeres Baya , Sandalia de cuña WIGG672FLY de mujer FLY London Cool Graphite Trotters Mujer Simmy Mary Jane Flat Negro 1.STATE Womens Celvin Blanco Río Nubuck Drew Shoe Lagoon Diabético Terapéutico Mujer Profundidad Extra Sandalia Velcro Cuero Estaño , Wilson Womens Rush Pro 2.0 Gris / Neptuno Supra Vaider LC Sneaker Andorra blanco , Sandalia Sloan Slide de Naturalizer para mujer Gunmetal Jig Foo Latin Salsa Rumba Chacha Practice Ballroom Zapatos de baile para mujeres Negro y plata , Very Fine Dance Shoes 2707 (Grado de competencia) 3 \ Leopardo y oro Zapatos de paloma de oro 50 tonos de 1 \ Baile bajo del talón Zapatos de vestir: Mujeres Comfort Ballroom, Latín, tango, salsa, Swing, práctica, Theather Art por Fiesta Fiesta 8881- Terciopelo burdeos (pedido personalizado, 4-6 semanas de tiempo de procesTrotters Lucy Ballet para mujer Coñac , TOMS Clásicos de mujer (Armada) , BareTraps BT Barree Slip On Slipper para mujer Negro Bota vaquera Moonby para mujer de Michael Antonio Negro , Trotters para mujer Dea Slingback Pump Patente perlada de oro acolchado / negro , CLARKS Mujer Ashland Bubble Slip-On Cuero marrón ,

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 entrenamiento Nike Womens Metcon 13767 3 Negro Womens/ 3 Blanco-volt-cloro Azul 0812250 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
3759731

4470325

5299263

5584881

5637801

6461840

9843191

17397415

18922026

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