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 sintético

Goma único

100% Auténtico

Durable

Marca Nuevo

Original embalaje

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 Hombres Air Pegasus Zoom Pegasus 33 B07CK5S2BM Wolf Gray Hombres/ Blanco-cool gris 578fd7dUpdated 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 Hombres Air Pegasus Zoom Pegasus 33 B07CK5S2BM Wolf Gray Hombres/ Blanco-cool gris 578fd7d

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 New Balance 990V3 para mujer Blanco azul , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Negro / Blanco-corredor Rosa-gamma Azul , Adicción a las mujeres de Brooks 13 Ébano / Plata / Pagoda Azul , Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado , Zapatillas de running NIKE Airstream Sequent 3 para mujer Negro / Blanco-gris oscuro , Nike Men s Air Presto Essential Negro , Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Zapato de running ASICS para mujer Torrance Negro / Gris piedra Saucony Kineta Relay Runshield Running Shoe para mujer Negro / Menta NIKE Women s Revolution 4 Running Shoe Blue Pure Platinum Purple , Zapatillas para correr New Balance W840V3 para mujer Trueno Brooks Womens Hyperion Peacoat / Purple Wine / Aruba Blue , Saucony Women s Guide 10 Zapatillas para correr Plata / Berry , New Balance Women s w990v4 Running Shoe Gray / Castlerock Amortiguación 635v2 New Balance para mujer Plata Zapato deportivo ASICS Gel-Cumulus 18 para mujer Acuario / Flash Coral / Blue Jewel , Zapatillas de running NIKE Air Max Torch 4 para mujer Blanco Zapatillas para correr Saucony Women s Excursion Tr11 Gris rosado , Zapatillas de running adidas Ultraboost W para mujer Pizarra negra / negra / oscura Zapato de running ASICS GT-2000 4 para mujer Carbón de leña / Flash Coral / Flash Yellow , Zapato de running Saucony Zealot ISO 2 para mujer Azul marino / azul / sil , Zapatillas de running Saucony Women s Cohesion TR11 Gris / morado Zapatilla de running Pearl Road N2 v2 para mujer Pearl iZUMi Deep Lake / Lime Punch Calzado de running New Balance para mujer 530v2 de New Balance Reflejo / Luz Porcelana Azul Zapatillas de running Mizuno Wave Rider 20 para mujer Negro / Blanco / Rosa Glo , Zapatillas deportivas NIKE Womens Flex Contact Wolf Gris / Blanco-Pura Platino-Cool Grey Wolf Gray / Metallic Rose Gold Zapatillas de running NIKE Free Run para mujer - SU16-10 - Blanco Mango / Sunset Glow , Zapatillas de running Reebok Women s Run Supreme 2.0 Mystic Maroon / Poison Pink / White , Zapato de running Fuzex Rush para mujer de ASICS Medio gris / Bay / Flash Coral ,

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 deporte Onitsuka Tiger Mexico 66 Blanco negro KEEN Hombres Newport H2 Sandal Tinta / óxido de la India Zapatillas de running Adidas Performance Ultra s Boost M para hombre Gris (Gridos / Gridos / Grmetr) Zapato de running Indigo Ryka para mujer Azul / plata , Brooks Womens Levitate Azul / Menta / Plata Wolky Comfort Jewel Cuero de brandy , Zapatillas de running adidas Cloudfoam Super Flex para mujer Sun Glow / Easy Coral / Calzado blanco , NIKE Airs para mujer Zoom Pegasus 34 Low Top para correr con cordones Sneaker Wolf Gray / Thunder Blue Chaco Zcloud 2 Sport Sandal para hombre Seam Rust , Zapatillas de trail running TIMP IQ de Altra para mujer Carbón de leña / azul Zapatillas de running Inov-8 X-Talon 212 Trail Unisex Negro / amarillo / azul , New Balance WT510RS3 Trail Running Shoes para mujer Nimbus Cloud , NIKE Air Huarache Run Ultra Blanco / Blanco-Blanco-Negro (Mujer) , MT-onemix Carta de las mujeres de los hombres Print Air Cushion Mesh Running Sneakers Azul profundo / rojo Sandalias EVA Gizeh de mujer Birkenstock Scuba Coral Eva Zapato de running Saucony Zealot ISO 2 para mujer Té / Algodón , Bomba de cuña Dreamstride Women s Dreams Armada , Zapatillas de paseo Skechers Performance para mujer Go Walk 3 Slip-On Negro Chaco Zcloud 2 Sport Sandal para hombre Negro Sandalia con correa en T para mujer ECCO Sombra negra / oscura Sandalias de mujer Wolky Red Multi Zapatilla de pies Cush N de Prop & eacute; t para mujeres Corduroy negro Toalla de plataforma ECCO Women s Touch Plateau Platform Slide Blanco , Sandalia Aurora Slide de Marc Jacobs para mujer Oro , Bota de montar Andra para mujer, sin calificación Gris pardo , CL por Chinese Laundry Women s Via Wedge Bootie Whisky Súper Gamuza Bella Vita Womens Mel-Italy Oro Gentle Souls de Kenneth Cole Zapatillas de deporte Harper Fashion para mujer Hormigón Gabor Womens Gabor 72.722 Gris Zapatillas para caminar Slip-On Go Walk de Skechers Performance para mujer Agua

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 Hombres Air Pegasus Zoom Pegasus 33 B07CK5S2BM Wolf Gray Hombres/ Blanco-cool gris 578fd7d 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
716076

948048

3716312

7215981

9784975

12058707

12933766

14401161

18750487

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