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

Caucho único

Eje mide aproximadamente parte baja del arco

I.G.S. (Orientación de impacto) Sistema) Tecnología: Filosofía de diseño ASICS que emplea enlaces componentes para mejorar el pie marcha natural de golpe de talón a despegar.

FluidRide Entresuela: FluidRide proporciona lo último combinación de recuperarse y propiedades de amortiguación con peso reducido y excepcional durabilidad.

Rearfoot y antepié GEL Sistemas de amortiguación: Atenúa el shock durante el impacto y fases de despegue, y permite el movimiento en múltiples planos como las transiciones del pie a través de la marcha ciclo.

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 de Negro running ASICS hombre Gel-Cumulus Fantasma 19 para hombre Negro/ Negro/ Fantasma 6f68934Updated 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 de Negro running ASICS hombre Gel-Cumulus Fantasma 19 para hombre Negro/ Negro/ Fantasma 6f68934

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 Kinvara 5 Saucony para mujer Plata / Vizi Coral / Citron , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Niebla del océano / Cirrus Blue / College Navy , Zapatillas de running ZanteV2 Breathe de New Balance para mujer Blanco / Hola Lite , Zapatillas de running Wstro para mujer New Balance Negro / gris / Lime Glow , Zapatillas de correr de espuma fresca New Balance 1080v8 para mujer Baya del Saúco / Coral Vivo , Zapatillas para correr Under Armour UA Speedform Fortis 2 para mujer Cubierto Gris / Blanco / Cristal , Zapatillas de running ASICS GT-2000 6 para mujer Gris medio / plata / carbono Calzado de running NIKE RN 2017 2017 para mujer Black / Hot Punch / Aurora Green , Zapatillas de running NIKE Air Zoom Vomero 12 para mujer Lava Glow / Black-racer Pink , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Rosado Zapatillas de running Fresh Foam 1080v6 de New Balance para mujer Gris azulado Zapatillas de running Questr Cc W de mujer adidas Aero Blue S, Aero Blue S, Colegiata Marina , Zapatillas para correr Nike Women s Free Rn Flyknit 2017 Deep Borgoña / Pure Platinum Calzado deportivo New Balance Vazee Urge v1 para mujer Blanco negro , Zapatillas de running Saucony para mujer Hurricane ISO 2 Azul / Plata / Citron Zapatillas para correr Nike Zoom All Out Flyknit para mujer rojo , Zapatillas de running NIKE Air Max 90 para mujer Negro / Negro / Gran Gris Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Blanco Aurora Verde 104 Zapatillas para correr Mizuno Women s Wave Rider 21 Trébol / blanco , Brooks Womens Adrenaline GTS 18 Blanco / negro / gris Zapatillas de running NIKE Flyknit Racer NIKE para hombre Oreo-negro / blanco , Zapato de running ASICS GEL-Cumulus 20 para mujer Negro / Flash Coral , Zapatillas de running ASICS Gel-Nimbus 19 para mujer Carbón / Blanco / Flash Coral , Zapatillas de running Under Armour Speedform Slingshot para mujer - SS16 Negro / Aluminio , Nike Men s Air Presto Essential Gris oscuro / plata metalizada , Saucony Women s Guide 10 Zapatillas para correr Plata / Berry Zapatillas de running NIKE Airstream Sequent 3 para mujer Borgoña-gris-blanco Calzado de running Brooks Womens Ariel 16 Overpronation Stability Antracita / Flor de Cactus Púrpura / Primer Gris Zapatillas de running mujer HOKA ONE ONE Clifton 4 Fucsia / Caliente / Rosa ,

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 Calzado deportivo New Balance para mujer Vazee Rush V3 Blanco / Vintage Indigo , Zapatillas de running NIKE Juvenate para mujer Glacier Grey / Metallic Gold Star , Zapatillas de entrenamiento Nike Air Zoom Fearless Flyknit para mujer Blanco negro , Zapatillas de deporte adidas Vigor Bounce W Trail Runner para mujer Shock Purple / Night / Easy Coral Zapatillas ligeras para correr NIKE Womens Air Max Thea Print Trainer Blanco negro Vivien Flat de Bella Vita para mujeres Negro Prop & eacute; t Zapatillas para mujer TravelFit Hi Walking Baya Sandalia Sully de mujer caja amarilla Gris pardo Zapatillas de running Z-Pump Fusion para mujer Reebok Gris plano / grava / cerezo de neón / blanco , Sandalia para mujer Alegria Mixie Slip On Blue Harbor Zapato deportivo para acolchado 635v2 New Balance para mujer Gris negro , Calzado de running NIKE Lunarcharge Br Tobillo-High para mujer Armería blanco / claro azul-blanco , Botas de mujer Naturalizer Wanya Sage Nubuck , Crocs Flip Crocband unisex Verde (trullo tropical / blanco) Vionic Women s de Orthaheel, Shore Slide Sandal Bronce Zapato de malla Nohea de OLUKAI - Hombre Dark Shadow / Dark Shadow Zapatillas Ryka Dynamic 2.5 Cross-Trainer para mujer Blanco negro Zapato de entrenamiento WX824 para mujer New Balance Blanco , Converse Chuck Taylor All Star Core Ox Trueno / Blanco Sandalia de mujer de SCHUNA Maduna Slide Pez vela Sandalias con tirantes Rosetta-1 de mujer Pierre Dumas Blanco Bota Ecuestre Kailan de Lucky Brand para mujer Centeno Mocasines casuales Alpargata Deconstructed de TOMS para mujer Toffee Suede Cupsole Bomba Parigi de LifeStride para mujer rojo Bota corta con botones Jackie de mujer FRYE Cognac-76593 , Zapato deportivo Fiat Fusion para mujer Sangre de buey Bomba de vestido Daisie de mujer Steve Madden Ante negro , Ellie Shoes 6 Inch Sneaker Bootie Negro OluKai Ohana Sirena / Negro , Trotters Liz Loafer para mujer Light Tan Tonal

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 de Negro running ASICS hombre Gel-Cumulus Fantasma 19 para hombre Negro/ Negro/ Fantasma 6f68934 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
633400

3887156

10417587

11774874

13014391

13797312

14055686

15563004

18924105

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