Math Hombre

Jennifer Silverman made this cool motions maze the other day. (More here.) We collaborated on it a little bit, after she did all the heavy lifting. I added buttons. (I may have a button problem.) It put me in mind of these motion puzzles I used to make in Geometer's Sketchpad, and I got to thinking how much better I could make them now. So I started, with the main new feature I wanted being the ability to generate new puzzles instead of being one static dynamic puzzle.

xmin=floor(Min[{0, x(F1'), x(F2'), x(F3'), x(F4')}]) - 1
But there's a problem then - the graphics won't be in 1:1 scale, which is always nice, but especially important for motions where the two objects should look congruent!

• xm = If[(ymax - ymin) / (xmax - xmin) < r, xmax, xmin + (ymax - ymin) / r]
• ym = If[(ymax - ymin) / (xmax - xmin) < r, ymin + r (xmax - xmin), ymax]

If the sandals give a screen that's not wide enough, it uses the aspect ratio to find a suitable width. If the sandals give a screen that's not tall enough, it uses the aspect ratio to find a good height. (The If[ ] command works like If[condition, then, else], where the else is optional.

The puzzle, as it turns out, is pretty challenging. Give it a try, and let me know what you think. Or let me know an easier better way to do my GeoGebra graphics hacking.

Here's the teacher page for download or the mobile page.

EDIT: Bonus! Jennifer created an assignment to give it more structure as a lesson. (PDF in dropbox.)

That's my attempt at a Glide Reflection Frieze.

This week my K-8 students were working on motions again.  Using the Geogebra activities at Motion Sketches and More Motion Sketches.  For K-8, there's really not a need for glide reflections because they're usually not a part of the curriculum.

This doesn't fit with Euclid's vision of motions, which was strongly tied to congruence.  Any two objects are congruent if and only if there is a motion from one on to the other.  (Aka rigid motion or isometry or Euclidean transformation...)  This requires four motions, not just three.  But a glide reflection is just a slide and a flip, you may say, we don't need it.  Well all the motions can be made from just reflections, but we still teach turns and slides.

But I've been stumped as to a good way to present these glide reflections.  Students can recognize them by a process of elimination and students can make a motion that is a glide reflection.  The next level of knowing a motion is to be able to specify it.  Students are good at finding lines of reflection, and can specify direction and distance for a translation.  It is difficult for many/most to find the center of a rotation, without being told.  They can do it in a dynamic environment (cf. MotionControl, a geogebra webpage) but it is difficult for them to construct.  The first guess seems to be connecting corresponding points and trying where the lines cross.  (Which doesn't work.)  So it's really hard to get students to know how to specify a glide reflection.  Mathematicians usually describe a glide reflection with a vector and a point or position of that vector.  The vector indicates the direction and distance of the slide, and the line containing the vector is the line of reflection.

This sketch is my attempt at a glide reflection sketch - the goal was to create an environment where students might be able to notice things that would lead them to construct the idea.  I would love it, if you try it out, to get feedback about ways to make the sketch more supportive.  Thanks!



As a webpage or geogebra sketch.

In my Geometry K-8 class we've been study transformations.  Which always leads one place for me ... my love, my joy... tessellations.  Arty, playful, deep underlying structure, corner cases that require thought even still; they're perfect.  To me.  I understand how others have dabbled and grown tired, but for me they are ever fresh.

There's a probably a few too many sketches here, but let's have a look.

First - Look at a tessellation, identify the motions, and consider what properties allow it to tile that way:
Quadrilaterals:  webpage and geogebra file






Hexagons1:  webpage and geogebra file






Hexagons 2:  webpage and geogebra file










Second - Look at a tessellation, identify the motions, and then alter the tile Escher-style!

Isosceles Triangles:  webpage and geogebra file








Quadrilaterals:  webpage and geogebra file








Third - Control the properties of the tile so that it will tessellate with the given motions:

Pentagons:  webpage and geogebra file

 (A midpoint rotation and 2 side to side rotations.)






Hexagons:  webpage and geogebra file

 (Quite challenging!  3 side to side rotations.)









Bonus - Kaleidoscopes!  What's the connection between reflectional and rotational symmetry?

Control the number of Sectors:  webpage and geogebra file







Control the Angle:  webpage and geogebra file






The kaleidoscopes were to investigate an open conjecture we have.  My one disappointment with Geogebra in comparison with sketchpad is that animation isn't as easy.  It's nice to have an animate button on your kaleidoscopes. 

The Leah-Jill Conjecture:  If a shape or design has n lines of symmetry, then it will have n-fold rotational symmetry, for n > 1.  Having rotational symmetry does not imply reflectional symmetry for any n.

I don't think the sketches helped.  I can't decide if we should tackle it another way or if we should just move on.

If you have any ideas for a dynamic tessellation sketch, please let me know.  It doesn't take much of an excuse to dive right in.

These sketches are to investigate the composition of motions, starting with reflections.  For a schema activation, I asked my preservice teachers to think about compositions.

Schema Activation:  What happens when you do two of a motion?  (Same type, not necessarily the same motion.)  Please guess if you don’t know.

Motions	Result-type	Know/Guess?
Translation then a translation		K/G
Rotation then a rotation		K/G
Reflection then a reflection		K/G
Glide reflection then a glide reflection		K/G

This brings up the idea of orientation both in terms of turning and in terms of face-up/face-down.

For a focus we have the following:
Focus:  Today we’re just going to concentrate on reflections and their compositions.  We have three different sketches to consider, and will also consider the questions that could be asked about each.  A composition of motions is when you make one movement and then another.  The combination is still a motion, as the original and image are still congruent.

We're also going to consider using questions to move us forward.  The types of questions described by literacy instructors are:

1. Literal - factual answer available or quickly available by recall, or can be found directly in the text.
2. Application - answer found by applying known method or looking up with slight modification.  The method of getting the answer is known.
3. Inference - answer requires prediction or extension from known information.  Can be an outright prediction or come from reading between the lines.
4. Analysis/synthesis - answer requires combination or deduction from other known information, possibly requiring a method not currently known by the respondent.

In the three sketches, the students are asked to take more and more responsibility for the questions they are answering.

Activity:
   Two Parallel Lines:  webpage or geogebra file

We discussed these sketches together.  They asked about finding the center of rotation and saw a neat connection with the reflecting lines.  They also saw a neat connection between finding the center of a rotation and finding the center of a circle, but couldn't remember or figure out how.  They found a cool relationship between the direction and distance between parallel lines and the resulting direction and distance of the translation. 

Reflection:  What did you do during this workshop?  So what did you learn?  Now what would you want to consider next about motions or questioning?

Bonus: (or... extension)

Two Glide Reflections: webpage or geogebra file







Coming Soon:

• a tale of two lessons - why can't everything work?
• Sue's MathMama Dialogue:  what's wrong with math education?

These Geogebra sketches are meant to serve as an introduction to the four Euclidean motions.  (I am a proud supporter of the Glide Reflection.)  Probably for middle school and above.  If you try them out, I am always interested in feedback.  The Motion Intro does not actually use a lot of the dynamic nature, but generated a lot of connections with my preservice teachers.  The Motion Control led to a nice discussion of what information is needed for each motion, as well as how to find that information (like the center of rotation) for a given motion.  The webpages have the questions that I asked my students.




MotionIntro: webpage and Geogebra file.







 Motion Control: webpage and Geogebra file.

Transformations on a Graph
I've been looking for Geogebra applications for function transformations, and wanted to share a couple of the neat sketches I've found.

Michael Higdon, a math teacher at Kincaid, a college prep school in Texas, has a quadratic function in vertex form, y=a(x-h)^2+k, with a, h and k as sliders to study transformations.
Geogebra webpage: Transformation of Functions

Mike May, a Jesuit math teacher at St. Louis University, has a beautiful applet where you can input the function, and control vertical and horizontal shifts and scaling with sliders.
Geogebra webpage: Translation Compression

An overall great collection of interactive webpages appears at The Interactive Mathematics Classroom. It has a nice search feature and a good breakdown by area of mathematics.

My first attempt at a transformations sketch is with a cubic as the starting function. Although you can change the function.

As a webpage, and as the geogebra file.

Slope in Linear Equations
A nice collection of middle school or Algebra I activities for linear equations from mathcasts.org: Slope Explorations
Mathcasts are screencasts of writing with voice-overs. They have mathcasts for K-12, and a nice collection of interactive math activities.

Here's my first slope sketch. It tries to get at the idea of the slope being constant on a line regardless of what points are selected.

As a webpage or a geogebra file.