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.

Caucho

Predecesor: Flex 2016 RN.

Soporte Tipo: Neutral a underpronation (Supinación).

Amortiguación: Ligero, flexible respuesta.

Superficie: La carretera.

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

NIKE Men Negro Flex s Flex 2017 RN 15384 Antracita Gris Negro 780dfeeUpdated 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.

NIKE Men Negro Flex s Flex 2017 RN 15384 Antracita Gris Negro 780dfee

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

Calzado deportivo New Balance para mujer Vazee FuelCell V1 Saúco / Plata Zapatillas de running adidas Questar Ride para mujer Carbono / Negro / Gris , Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul , adidas Performance Ultraboost X para mujer Carbono / Cristal Blanco / Núcleo Negro , adidas Women s Edge Lux 2 Sol / Plata / Naranja Zapatillas para correr ASICS Gel-Venture 6 para mujer Pavo real de carbono / negro / rosa , Zapatillas de running NIKE Zoom para mujer Negro / Blanco Gris Oscuro Camiseta adidas Swift Run W para mujer Blanco / Blanco / Blanco , ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Phantom / Black / White , Zapatillas de running NIKE Women s Air Max 1 Ultra Essentials Cumbre Blanca / Carmesí total Nike Hombres Air Zoom Pegasus 33 Acción Rojo / Negro-puro Platino-total Crimson Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Gris / sal marina Zapatillas de running Under Armour para mujer Toccoa Grava (001) / Orange Peel , Zapatillas de running NIKE Air Max Invigor para mujer, blancas / plateadas (9.5) - Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Blanco , Zapatillas de correr Nike Air Max 1 Essential para mujer Negro / Dove Grey / Pure Platinum , Zapatillas de running Mizuno Wave para mujer Inspire 13 Malibu Azul / Rosa Mujer adidas Cloudfoam Qt Racer Negro / Blanco / Carbono Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Rosa cromo / rosa cromo Zapatillas de running New Balance Foam Cruz V1 para mujer New Balance Linaza Zapato de running ASICS GT-1000 6 para mujer Negro / Blanco / Atlantis , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco , Zapatillas de running adidas Cloudfoam QT Racer para mujer Negro / Negro / Blanco , NIKE Men s Benassi Just Do It Athletic Sandal Blanco / Variedad Royal White Zapatillas de running Brooks Women s Glycerin 16 Road Ébano / verde / negro Zapatillas adidas Ultraboost X Shoe para mujer Zapatillas de running Kineta Relay para mujer Saucony Gris / Heather / Coral Skechers Sport Skech Air Run High Fashion Sneaker Carbón de leña / verde Short NIKE Tempo para mujer Acción Verde / Negro / Lobo Gris ,

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 Ohana Paraíso / negro , Zapatillas de running Newton Kismet para mujer Azul Zapatillas de running Under Armour Speedform Intake 2 para mujer Gris , Zapatillas de running Kineta Relay para mujer Saucony Negro Negro , Sandalia Drew Angela para mujer Beige-multi , Zapatillas de running Merrell para mujer All Out Rush Trail Púrpura / Azul Horizonte , Zapato para caminar Ryka Women s Sky Gris pizarra / Plata cromo / Azul Robin Orthofeet Plantar Fascitis Ortopédica Pies planos diabéticos Juanetes caminando Whitney Womens Sneakers Athletic Zapatos Blanco Aerosoles Women s Obstachle Course Gladiator Sandal Serpiente negra BareTraps Sandalia Jacey para mujer Rosa roja , Under Armour Speedform Velociti para mujeres Glacier Grey (003) / Ultra Blue Calzado para caminar Flare Drew Shoe para mujer Bone Calf / Bone Sporty , Toni Pons Womens Noa-Ch Black Chia Skechers Womens Be-Light - Florescente Armada Zapatillas NIKE Zoom para hombre Stefan Janoski Skate Jungla / Blanco Vans Old Skool Unisex Adults Low-Top Trainers Mantequilla de manzana / True White , Vans Auténtico (Gumsole) Negro Hush Puppies Mujeres Power Walker Ii Loafer Cuero blanco NIKE Blazer Mid Metro GS - 325060003 Violeta Supra Vaider LC Sneaker Cool gris-blanco , Zapatos de golf de ocio FootJoy para mujer Gris / rosa Sandalia para colgar edredones de mujer Anne Klein Tela azul / blanca mediana , KAANAS Sandalias planas de alpargatas tejidas de Martinica para mujer Natural-1 , J Slides Sandalia de mujer Naomi Slide Oro rosa , Sandalia Thoalle Slide de ALDO para mujer Brown Print Salón muy fino Salsa de tango latino Zapatos de baile para Mujer SERA701BBX Flate Heel + plegable Paquete de cepillo Gamuza rosa , Zapatilla Smoking Boxers Platform para mujeres de Soludos - Arena Beige , Bota de moto Leggylady para mujer Stuart Weitzman Nerón , Botas de suela de cuero antideslizante de 6 \ Ls60 negro , Trotters Liz Loafer para mujer Beige-oro ,

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 NIKE Men Negro Flex s Flex 2017 RN 15384 Antracita Gris Negro 780dfee 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
5072997

8706696

9176183

9301154

10657861

10796514

12538044

15500829

18431411

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