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

Importado

Sintético único

Eje mide aproximadamente parte baja del arco

Fresco Entresuela de espuma

Engineered malla

Sin costura aplicación material

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 mujer Balance de correr de 19856 espuma fresca New Balance 1080v8 para mujer Blanco negro 5b1f686Updated 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 mujer Balance de correr de 19856 espuma fresca New Balance 1080v8 para mujer Blanco negro 5b1f686

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 adidas Ultraboost W para mujer Blanco / blanco / blanco cristal , Zapatillas de running NIKE Zooms Winflo 4 para mujer Ink / Deadly Pink-fire Pink Zapatillas para correr New Balance Rushv3 para mujer Poisonberry / Black , Amortiguación 635v2 New Balance para mujer Negro Zapatillas de running NIKE para mujer Lunarepic Low Flyknit 2 Colegio Navy / Black Concord Zapatillas de running Brooks Womens Ghost 10 Neutral acolchadas Pavo real negro / rosa / Living Coral New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running Kineta Relay para mujer Saucony Gris blanco Brooks Womens Mazama Diva Rosa / Antracita / Bluefish , Zapato para correr ASICS Conviction X 2 para mujer Océano profundo / azul Zapatillas de running New Balance 520v3 para mujer Negro / Gunmetal / Blanco Zapatillas de running W1260V5 para mujer New Balance Azul plateado Zapatillas de running Mizuno Women s Wave Hitogami 4 Naranja / gris , Pantalón largo clásico NIKE Golf para mujer Tech Essentials Black Crimson total Zapatillas de running Comfort Ride 575v2 New Balance para mujer Plata metalizada / azul , Zapatillas de running NIKE Air Huarache para hombre Olive Flax White 302 Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink , Short NIKE Tempo para mujer Negro, Racer Pink , Under Armour Women s HOVR Sonic NC Blanco (102) / Elemental , Zapatillas de running adidas Adizero Boston 6 W para mujer Utilidad Negro / Platino / Negro Zapatillas de running Nike \ Noble Red White Black 600) Zapatillas de running Reebok Hexaffect 5.0 MTM para mujer Polvo negro / asteroide / peltre / blanco Zapatillas de running Under Armour para mujer Toccoa Chambray Blue (400) / Overcast Grey NIKE Men s Benassi Just Do It Athletic Sandal Negro , Zapato de correr ideal para rejilla de mujer Saucony Gris blanco , Zapatillas de running ASICS GT-2000 6 para mujer Porcelana Azul / Porcelana Azul / Asics Azul Zapato deportivo ASICS Gel-Kayano 24 para mujer Porcelana Azul / Humo Azul / Blanco Zapatillas de deporte NIKE Youth Air Max 90 Ultra 2.0 Mesh Negro / Rojo Zapatillas para correr New Balance Rushv3 para mujer Pigmento / Alpha Pink ,

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 Sandalia atlética de Rka Women s Glance Azul / Menta / Coral Sandalia Omnium de mujer Teva Pizarra Zapatillas NIKE Women s Revolution 3 para correr Blanco / Metálico Gris oscuro / Gris lobo , Zapatillas deportivas 574S Sport New Balance para hombre Varsity naranja / blanco , Zapatillas de running Altra AFW1845F Provision 3.5 para mujeres Blanco , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Blanco negro Sandalia deportiva Terrell Ari Lattice de Merrell para mujer Helecho Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Lyl Blue / Lyl Bl / University Red / Wht Adidas Women s Adilette CF + Logo W Sandalia deslizante Vapor Gray Metallic / Vapor Gray Metallic / Black NIKE Men s Roshe Two Running Shoe Cool Grey / Stadium Green-mica Azul-blanco Sandalias EVA Gizeh de mujer Birkenstock Scuba Yellow , Under Armour Speedform Velociti para mujeres Lime Fizz / Fresco verde / blanco , Zapatillas de running Kineta Relay para mujer Saucony Púrpura / rosa SOREL Womens Ella T-Strap Paloma , J.Renee Womens Soncino Rhineston azul marino , Bomba Nikole para mujer Via Spiga Ruby Leather , L`Amour Des Pieds Womens Amadour Sandal Armada Atletismo de York Punto ligero de punto Entrenamiento liviano Sneaker, Unisex Zapato de running, duradero Tejer, malla, correr Sneaker Black Top , Zapatillas de deporte Fit Fitness GEL de ASICS para mujer Taupe / Cotton Candy / Coral Rose , Vans Unisex Old Skool Zapatillas clásicas de skate Blanco / goma Supra Vaider LC Sneaker Negro Acción / Blanco , UGG Mujer W Joni Lude Gray , Zapato de skate DC Mujer Trase V SE Blanco / Blanco / Rojo Atlético Así que Danca CH50 Character Shoe con 1.5 Heel Negro , 50 sombras de plata Zapatillas de baile: confort Boda del vestido de noche Bombas, salón de baile Zapatos para América, Tango, salsa, swing, Theather Art por 50 sombras (2.5 \ Tacones de 3 \ 2829- Blanco y Plata Horze Amara Easy Care Gamuza de ante con costilla Marron oscuro Propet Cush N Foot Slipper & Oxy Cleaner Bundle Corduroy negro Botas FitFlop Womens Mukluk Shorty II con borlas Negro , Mocasines casuales Alpargata Deconstructed de TOMS para mujer Tacota de ante negra Easy Spirit Women s Traveltime Mule Gamuza marrón oscuro

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 mujer Balance de correr de 19856 espuma fresca New Balance 1080v8 para mujer Blanco negro 5b1f686 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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