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.

Importado

Ir rápido y se ve bien haciéndolo en el Brooks & reg; ¡2 zapatillas para correr!

Predecesor: Neuro.

Soporte Tipo: Neutro.

Amortiguación: Ligero, flexible respuesta.

Superficie: La carretera. 6 mm. malla superior con delgado sintético superposiciones. Sistema de hamaca se envuelve debajo del pie y aprieta a través de los ojales para una alineación mejorada y un mejor ajuste. Cierre con cordones. Lengua acolchada y collar. forro de tela ofrece un gran zapato sensación. plantilla de espuma. La placa crea un plataforma estable que promueve poderosos y alineados toe-offs. Mecanismo de engranaje permite el talón y la parte delantera del pie para moverse de forma independiente F

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

Brooks Azul Womens Neuro 2 Neuro Noche Azul/ Fiery China Azul/ Coral Fiery 2cd55e7Updated 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.

Brooks Azul Womens Neuro 2 Neuro Noche Azul/ Fiery China Azul/ Coral Fiery 2cd55e7

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 Boracay V3 para mujer Blanco negro Zapatillas de running deportivas New Foam Lazr V1 de New Balance para mujer Coral vivo / negro / blanco Munsell , New Balance W880gb6 para mujeres Gris azul Zapatillas de running adidas Women s Edge Lux 2 Gris Tres / Plata Metálico / Coral Fácil Zapatillas de running Saucony para mujer Stretch N Go Breeze Gris , Skechers Sport Skech Air Run High Fashion Sneaker Negro / rosa caliente Zapatillas de running adidas Originals Women s Edge Lux W Alta resolución Azul / Aero Azul / Blanco Zapatillas de running ISO para mujeres de la guía Saucony Gris / menta Brooks Womens Ghost 9 Peacoat Navy / True Red / Gold Zapatillas de running New Balance 990V3 para mujer Blanco azul Zapatillas de running NIKE RN Motion FK 2017 para mujer gratuitas negro Zapatillas de running ASICS Gel-Kayano 23 para mujer Silver / Pink Glow / Parachute Purple New Balance Women s w940v3 Running Shoe Plata Zapatillas de running NIKE RN 2017 para mujer gratis Negro / Blanco / Gris Oscuro / Antracita New Balance Women s w990v4 Running Shoe Gray / Castlerock , Zapatillas de running NIKE Air Max 2017 para mujer Negro / Blanco-antracita Zapatillas de running NIKE Lunar Skyelux para mujer Purple Shade / Bright Mango Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla del océano / Cirrus Blue / College Navy , Zapatillas de running adidas Solar Boost para mujer Gris / plata / rosa Zapatillas de running New Life Arishi v1 Luxe Fresh Foam de New Balance para mujer Saúco Calzado de running New Zealand Vazee Pacev2 para mujer Dark Denim / Lime Glow NIKE Mujeres Air Max 95 LX - , Zapatillas de running NIKE RN 2017 para mujer gratis Púrpura / Hot Punch-m , Zapatillas para correr ASICS Gel-Venture 6 para mujer Carbon Pink Zapatillas de running Saucony para mujer Echelon 6 Gris rosado , Zapatillas de running Fresh Life Arishi V1 para mujer New Balance Gris claro Zapatillas de running adidas Ultraboost W para mujer Negro / Negro / Choque Púrpura F16 NIKE Men s Benassi Just Do It Athletic Sandal Sepia Stone / Sepia Stone-summit White HOKA ONE ONE Hoka One - Zapatillas de running con altura de tobillo Instinct Speed para mujer Ultramarine / Aurora ,

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 Saucony Women s Triumph ISO 4 Blanco azul Fulcro Karhu Fast5 para mujeres Lila / Hibisco Zapato NIKE Roshe One femenino Hierro Púrpura / Blanco / Turbo Verde Reebok Classic Sneaker Sage Mist / Coal / Fierce Fucsia Inov-8 Mujer F-Lite 250 zapatos de fitness y visera de entrenamiento Bundle Blanco negro Zapatillas para correr WRL247WM New Balance para mujer Estratos , Zapatillas de running adidas Adizero Boston 6 W para mujer Azul Skechers Women s Walk Walk Lite-Daisy Loafer gris Zapatillas Slip-on de mujer Skechers You Inspire Blanco negro Zapatillas para caminar ProCon Zipiv TravelActiv para mujer Mora ASICS Womens Gel-Quantum 180 2MX Tela Low Top Slip On Running Sneaker Gris Piedra / Carbono / Rosa Concha Under Armour Women s Precision Mid Purple Rave (959) / Premier Purple , Melissa Shoes Womens Wonderful + Jason Wu Marrón / negro , Zapatilla deportiva Skechers You-14959 para mujer melocotón Zapatillas de deporte cuadradas Harrier de Gola para mujer Sol , Entrenador cruzado de trabajo 627v2 New Balance para mujeres Gris / rosa Vans Herren Authentic Core Classic Sneakers Blanco Vans Old Skool Unisex Adults Low-Top Trainers (Pared de llamas) Negro / Negro , Zapatillas de golf adidas Climacool Knit Cblack / D, mujer, de adidas Energy Blue Sandalia de cuña OAHU de China Laundry Women Vintage Rose Suede Sandalia Dolce Vita Jene Slide para mujer Tela blanca a rayas Sandalia para mujer Danica Slide de Cecelia New York Oro Sandalia Dolce Vita Hazle Slide para mujer Fuera de cuero blanco Birkenstock Arizona Sandalia de ante con suela suave Ante negro , Nike Kawa Slide Big Kids Negro / Brillante Lobo Carmesí Gris , LifeStride Women s See This Dress Pump Lux Navy Zapatilla Glamour de mujer verde Daniel Black Kidskin 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 \ 2719- Cuero plateado y malla de carne , FRYE Mujeres Pia Short Chelsea Boot castaña Allrounder by Mephisto - Calzado deportivo Kalibra para mujer Grafito nubuck / gamuza

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 Brooks Azul Womens Neuro 2 Neuro Noche Azul/ Fiery China Azul/ Coral Fiery 2cd55e7 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
8144272

9822121

13182386

13957855

15733579

15801347

15886772

16531731

18929073

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