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.

tejido y sintético

Fácil Deporte Spirit Efecta característica de sandalias gancho ajustable y cierres de bucle para un personalizado ajuste

personalizado Ligero Elon único.

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

Sandalias Efecta Efecta de Easy mujer Easy Azul Spirit Azul medio 1102a26Updated 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.

Sandalias Efecta Efecta de Easy mujer Easy Azul Spirit Azul medio 1102a26

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

Sandalia deportiva Nolita para mujer Anne Klein Luz Natural / Oro Metálico Sandalia para mujer Terrell Ari Wrap de Merrell Resquicio de esperanza , OluKai Ohana Pokeberry / Dark Java , Sandalia deportiva Tirra para mujer Teva Simplemente Taupe KEEN sandalia de Venecia para mujer Spray de oliva / surf negro Sandalias Bali KEEN para mujer Negro / Imán Sandalia Terrell Post II Femenina de Merrell Negro , Drew Womens Cascade Sand Nubuck Sandalia NIKE Kawa Slide para mujer Aurora Verde / Blanco / Pure Platinum Sandalia KEEN Women s Venice H2 Negro / Gris neutro Sandalia Uneek O2-w de mujer KEEN Black / Harvest Gold , Sandalia de susurro de mujer Keen Beet Red / Honeysuckle , Sandalias KEEN Venice II H2-W para mujer Skydiver / Blue Opal , Sandalia Terrell II de Merrell para mujer Armada Sandalia deportiva Nolita para mujer Anne Klein Tela azul marino Wolky Comfort Jewel Fantasía multicolor negra KEEN Hombres Newport H2 Sandal Tierra oscura / Acacia Wolky Comfort Jewel Choque de Multi Picasso OluKai Ohana Profundidades oceánicas / Negro Sandalia Terra-Float Livia de Teva para mujer Lima / Blanco , Zapato de vestir uniforme al aire libre Vulc N Vent Slip para mujer de Columbia Tiburón, fósil , Sandalia Deportiva Jordan Spires de Via Spiga Rico, amapola, gamuza , Teva Terra-Float Active Lace - Mujeres Negro , Sandalia deportiva Nolita para mujer Anne Klein Luz Natural / Oro Metálico Sanda Verra Teva para mujer Cuerda elástica , Sandalias KEEN Venice II H2-W para mujer Paloma / Turquesa Pastel , Chaco Women s Z2 Classic Athletic Sandal Eclipse estático , Wolky Women s Rio Sandalia Azul , Sandalias Newport H2 de KEEN para mujer Cáñamo / azul provincial ,

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 New Balance Foam Cruz V1 para mujer New Balance Blue Vapor / Camp Smoke Zapatillas de viaje Protic W3910 Vista Walker para mujeres Taupe liso , Zapatillas de running NIKE Unisex Flyknit Racer Blanco / Leggion azul , Zapatillas para correr Mizuno Women s Wave Rider 21 Patriot Blue / White , Sandalia deportiva Nolita para mujer Anne Klein Tela negra , Camiseta adidas Swift Run W para mujer Traza púrpura / Traza púrpura / Traza púrpura Mujer adidas Cf Racer Tr W Trace Purple / Orchid Tint / Aero Pink Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Racer azul-negro-blanco , Zapatillas de senderismo de malla Caldorado para mujer Montrail rojo , Zapatillas de senderismo Air Mesh de Ahnu para mujer con malla de aire Cielo polar , Zapatillas de senderismo acuáticas para mujer Astral TR1 Junction Basalto negro , Skechers Sport Relajación de mujer Breathe Easy Moneybags Sneaker Armada KEEN Women s Uneek 8MM Sandal Argel / Opalino Zapatillas de running adidas Aerobounce Pr W para mujer Core Black, Silver Met, Ftwr White , Zapatillas de running adidas Alphabounce Beyond W, para mujer Ftwr Blanco, Silver Met, Ftwr Blanco , STEVEN por Steve Madden Sandalia NC-Kore para mujer Cuero negro , CLARKS Mujer Gracelin Art Mary Jane Flat Cuero Tan oscuro Zapatillas de deporte vanguardista Speeds Varsity Vanguard para mujer Brooks Denim Blue Zapato de caminata Skechers Performance para mujer Go Walk 4 Pursuit Rosa SAS Mujer, Pampa Sandals Negro Nike Women s Sideline III Insert Cheerleading Blanco / puro platino / blanco Zapatillas Dr. Mary Comfort Flauta Lycra Elasticables Diamante Mary Jane Licra , 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 1670- Tan Pu 3N2 Mofo Turf Trainer Armada , Zapatilla de cuero Zoili de Nine West para mujer Cuero natural claro , Bomba de vestido animado de LifeStride para mujer Negro 1 , Botín con puntera anudada y floral Dingo Tooled Floral para mujeres, punta redonda - Di8951 Broncearse Zapatillas Nike Women s Flex Trainer 5 Antracita / Blanco / Gris Oscuro / Hipervioleta , Botines Joan Rain Wedge Chelsea Gloss Rain de mujer SOREL Colegiata Marina / Atmósfera , Vans Sk8-Hi Unisex Top informal Zapatos de skate, cómodos y duradero en Waffle de la firma Suela de goma Negro Negro

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 Sandalias Efecta Efecta de Easy mujer Easy Azul Spirit Azul medio 1102a26 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

For email to reach me, the word