Chapter 0

Software and Transformations

Ford “spheres” and their reflections - Chapter 0 icon

Ford “spheres” and their reflections

 

Before we begin it’s important to understand what the four icons below represent. They will be used throughout the book to make directions clearer.

The This computer shaped icon indicates the directions which follow, are part of this 8th grade course. Learners are expected to fulfill the requirements of those directions, and to sustain a determined effort over a period of time when necessary. icon indicates the directions which follow, are part of this 8th grade course.  Learners are expected to fulfill the requirements of those directions, and to sustain a determined effort over a period of time when necessary.

The A book shaped icon that indicates the following directions are opportunities for further information, exploration, or learning. These opportunities are optional, unless the learner finds them helpful, interesting, and/or necessary for algebra credit. icon indicates the following directions are opportunities for further information, exploration, or learning. These opportunities are optional, unless the learner finds them helpful, interesting, and/or necessary for algebra credit.

The The pencil shaped icon indicates the following directions involve recording and defending work and discoveries on paper, in order to fulfill the requirements of the previous directions (templates for doing this will be provided by the teacher). icon indicates the following directions involve recording and defending work and discoveries on paper, in order to fulfill the requirements of the previous directions (templates for doing this will be provided by the teacher).

The The smiley face icon is a friendly stop sign, to remind the learner to defend their work, participate in mathematical discourse with at least one other student, keep the applet/program open, and then check with their math teacher before continuing icon is a friendly stop sign, to remind the learner to defend their work, participate in mathematical discourse with at least one other student, keep the applet/program open, and then check with their math teacher before continuing.

On the next page you will be asked to use an applet to discover and explore four transformations of the plane. These transformations will be used throughout the book to geometrically describe algebraic ideas, and to help deepen our understanding of those algebraic ideas.

Below are snapshots of an applet, which will allow you to discover and explore four transformations of the plane.

Below are snapshots of an applet, which will allow you to discover and explore four transformations of the plane.computer icon Run the applet “transformations“, and explore until you can describe what the 4 transformations do to objects.

pencil icon Use a copy of the .pdf appletnotes to record the descriptions in your own words, along with sketches.

book icon To see how to access and use this Geogebra applet, watch the movie “transformations.mov“.

smiley face icon After recording the descriptions of all 4 transformations in your own words, and defending them with sketches, review the icon chart again to make sure you understand what the icons mean, before going on to the next page.

Using the transformations applet was soley a geometric way, of concretely and informally discovering what those four transformations do to mathematical objects (triangles) on a plane (flat surface).

  • It was soley geometric because all you had to explore with, were triangles (there were no numbers, and the plane wasn’t even marked with a grid).
  • It was concrete in the sense that you were directly changing these objects with the cursor.
  • It was informal because there were no right or wrong answers, just what the four transformations (shown again below) meant to you.

Below are snapshots of a program containing 10 challenges, which will allow you to further explore transformations.

computer icon Run  program 001  until you can solve at least 8 of the 10 challenges.

pencil icon Use a copy of the .pdf  programnotes1g  to record modified descriptions of any of the 4 transformations, if necessary, to make calculations or sketches in order to solve the challenges, and to record any further discoveries.

book icon To see how to access and use this program, watch the movie  “program001.mov”.

smiley face iconAfter modifying descriptions, solving at least 8 of the challenges, and reviewing the icon chart again, go on to the next step.

Having an x-axis and a y-axis in the background, allowed for a slightly more algebraic exploration of the four transformations. By controlling the transformations with numbers, and seeing the resulting animation,we were able to deepen our understanding.

Combining geometry with algebra to better understand a mathematical idea is a helpful strategy,and will be used throughout this mathematics book.

For example, using the applet, we discovered we can scale 2-D objects to make them bigger or smaller. Using the program, we discovered we can scale horizontally and/or vertically. Understanding this will be important later on in the book.

Below are snapshots of a program, which will randomly generate 24 transformation problems for you to solve. 

computer icon Run  program 002  until you can solve at least 20 of the 24 transformation problems given.

pencil icon Use a copy of the .pdf  programnotes2g for calculating and/or sketching, and to record any further discoveries.

book icon To see how to access and use this program, watch the movie  “program002.mov”.

book icon For information about coordinates, open the pdf  xycoordinates.

smiley face icon After solving at least 20 problems, recording discoveries, and reading page 2 one last time, go on to the next page.

As we discovered, when the mathematical objects are points, we can further understand transformations, by looking at their coordinates.

For instance, a vertical translation is when we add to, or subtract from, the y coordinate.   In the example to the right, the red triangle was translated down 5 units.  Notice, the y-coordinate went from a 6, down to a 1.

A reflection about the x-axis is when we take the opposite of the y-coordinate. In the example to the right, the red triangle was reflected about the x-axis. Notice, the y-coordinate went from a 3, to a -3.

Earlier, we described scaling as making objects bigger or smaller. Points however, don’t have size.  Scaling actually makes objects farther or closer, which in turn makes 2-D shapes bigger or smaller. A vertical translation is when the y-coordinate is multiplied by a scale factor.  In the example to the right, the red triangle was vertically scaled by 2.  Notice, the y-coordinate 3, was doubled, and became a 6 (the point was moved twice as far from the x-axis).

A rotation of 90º is when we take the opposite of the y-coordinate, and switch the order of the ordered pair.  In the example to the right, the red triangle was rotated 90º clockwise.  Notice, the 6 became a -6, and was then switched with the 7 (the point was reflected about the x-axis and then reflected about a diagonal).

In the last program, geometry and algebra were fully merged, by focusing on the x and y coordinates of vertices.

Some people may have viewed the program as solving algebraic problems geometrically, and others may have viewed it as solving geometric problems algebraically (or a combination of both).

Usually, the geometric aspect focusses on something specific and makes ideas more concrete, and the algebraic aspect focusses on things in general and allows for a more abstract, but deeper understanding of ideas.  This mathematics book will use this dual nature of mathematical objects to help us navigate, as we travel deeper into the beautiful world of mathematics.

As the “ambigrams” below symbolize, the use of technology along with the language of transformations, will allow us to better describe the symmetry of geometry, and give us more than one way to look at algebra.

The above “ambigrams” are based on John Langdon’s artwork.

book icon To try making your own ambigrams, run the applet  “ambigrams.html”  and/or  “ambigrams2.html”  .

book icon To get more practice using applets, try determining which letters of the alphabet have reflective/rotational symmetry.

Below is a picture of a sculpture by Robert Indiana, located on the grounds of the Farnsworth Art Museum, in Rockport Maine.

How would you describe the picture of the sculpture, using transformations of the plane?

book icon If you thought the picture at the beginning of this introduction was interesting (a copy of it is shown below), take the opportunity to practice using GeoGebra applets and Chipmunk Basic programs to explore Ford circles.

Ford “spheres” and their reflections - Chapter 0 icon

Ford “spheres” and their reflections

 

 

 

 

 

 

Below is a snapshot of the chapter 0 programs menu program.

book icon Run the program  “0menu.bas”   , or its alias to access every program used in chapter 0.

 

 

“…the feeling of mathematical beauty, of the harmony of numbers and of forms, of geometric elegance.  It is a genuinely aesthetic feeling, which all mathematicians know” — Henri Poincare —

“Geometry enlightlens the intellect and sets one’s mind right” — Ibn Khaldun —

“…for no human inquiry can be called science unless is pursues its path through mathematics exposition and demonstration” — Leonardo da Vinci —

“The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.” — I.N. Herstein —

“I must study politics and war that my sons may have liberty to study mathematics and philosophy.”  — John Adams —

 

 

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