Flip Flop

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.

The user moves points A and B to try to find the line of reflection between the two flip flops. When you hit the Check button, it shows you the reflection over the line you're trying. 

What follows is my GeoGebra geek out over trying to make it look right. Here's the puzzle if you want to skip that: Flip Flop.

One thing I love about Jennifer's sketches are her excellent images. So I tried to step it up with some nice flip flops from openclipart.org.

It turns out the trickiest part was getting both sandals to always show up. That's why I'm writing this post. (A lot of my individual sketches I post at the tumblr.) The key to being able to do this is that in the graphics window you can put variables in for the window dimensions. Define those from the objects in the sketch, and,  voilà, you can see both the sandals. So I defined xmin, xmax, ymin and ymax from the two sandals. E.g.,

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!

The Corner[ ] command is my new best friend. Corner[n] for 1, 2, 3 & 4 return the coordinates of those corners. Corner[5] returns a point with (width, height). Corner[image name, number] returns the corners of an image. This was handy for finding the corners of the reflection, F1' etc. in the command above. 

So using Corner[5] I could find out the aspect ratio of the graphics window. It took me a few minutes, but I hit on the idea of making the bottom left corner steady, and then altering the top right corner based on the aspect ratio. I defined r = y(Corner[5])/x(Corner[5]) (so, height:width) and:

• 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.)

Puzzle

Try Yourself

How many squares, of any size, can you find on this chess board that do not contain a Rook?

 

Answer: There are 116 squares without a rook.

1 x 1 : 61 squares.

2 x 2 : 37 squares.
3 x 3 : 15 squares.
4 x 4 :   3 squares.

 

Too Puzzling

The content objective: adding and subtracting fractions with unlike denominators, for 5th grade students.

What game? On short notice... I have a couple race games and Mr. Schiller has pattern blocks - but those use the fraction cards which I like, but were also at my office.I was thinking about a game where students start at one, and one team tries to race to 2, while the other team takes away and races to zero.  That would be good for building intuition, and learning to record fraction number sentences, but this is their last day with fraction operations before moving on to something else. What could serve as a review?

I thought about a game you could play with a small number of cards (one sheet), or a game where they would be doing review problems and checking each other as they play. Somehow I was reminded of a tarsia puzzle. (Here's a bunch at tes, among other places. Not sure who tes is, really.) Mostly those are triangular, but I decided on a square pattern, more familiar to the students. Here's what I wound up with. Made the grid in GeoGebra, of course.


A lot of thought went into the puzzle. I repeated a couple values to make it so that a match did not conclusively mean that two pieces went together (harder). I made all of them doable with 24ths (easier).  I decided on a 3x3 vs a 4x4 (easier). I made almost patterns on borders (harder and easier). Made sure some of those were nonmatches (easier). Related problems, like 1/4+1/2 and 3/4-1/2 (easier). Some operations where the common denominator is one of the present denominators, 2/3-1/6. And one mistake. (Crazy harder. Fixed version above.) I tried to put in some common characteristics on adjacent tiles. (Hmmm?) If they've been doing these problems in general, I thought that this would be enough support for everybody, especially working in teams.

To differentiate, then, I was mostly thinking upward. On the original puzzle they could make 3 more squares to make it a 12 piece puzzle. I made one with some triangles blank for the students to work out sums, differences or make up a problem, and then an entirely blank one for them to make up their own Tarsia.

I launched the puzle by showing cut out pieces, telling them it was a puzzle and asking how it might go together. Through whole group discussion they figured out that the sums and differences matched some of the fractions. I compared it to the puzzles that are all squares that divide up pictures, which are pretty tough. After finding a few of the matches together, they formed pairs and came up to pick up their choice of puzzle. Only one group took an option on the partially filled in puzzle.

They did not like it. Found it too hard, or didn't know how to start. Mr. Schiller and I circulated and helped people get started. They found lots of matches, but before our time was up were moving on to other pursuits. Not interested in making their own.

To wrap up, we came back together and did some together to get a firmer idea of how to do it. They confirmed that it was beyond them. When I asked for words of wisdom, one student volunteered: "You might want to try it yourself, first. If it's too easy, add stuff to make it tougher. If it's too hard, make it easier."

Wise words, indeed.

To use this in fifth grade again, I think I might concentrate on first getting a square of four made, and then try to grow it. Mr. Schiller recommended either a fraction equivalence puzzle or a fraction-decimal equivalence puzzle to get it launched. I still love these kinds of puzzles, but fee like I learned something about introducing and using them with younger learners.

Post Script:
Jeff says: The equivalent fraction version of that worked a lot better.  I sort of sneakily encouraged them to also incorporate equivalent decimals too.  In the version we played in the afternoon,  they created their game board in pairs and then they matched up with another team and exchanged puzzles and it became a race to finish the puzzle first.  I sort of mentioned in an offhanded way - oh yeah, and if you wanted to make it a little more challenging for your opponent, you might include some equivalent decimals, too.  (That did the trick)

Rigor and Relevance in Parallel

(The math is at the end of this one.)

Last week I had one of those teaching collisions where it felt like every idea was dovetailing.  First some twitterer retweeted Terie Engelbrecht's post on rigor and relevance in the context of motivating students with respect not points. Her post was a riff on this International Center for Leadership in Education chart. My preservice high school teachers had asked for parallel lines, circles and proof for our geometry topic.  Elissa Miller tweeted about parallel lines in a way that also brought up relevance.

Rigor, Relevance, Relationships

Fig 1.1 from this ASCD article

Some comments from my colleague Dave Coffey connected the Relevance framework to the Levels of Transfer, a professional development framework from Joyce and Showers. Before I switched to a communication framework, I tried to grade using that framework! Very ingenuous, as most students are not at an executive level of transfer, and it is not fair as an expectation for a large quantity of work. I did like the sense of ownership that it promoted, and the encouragement for students to show that they had made the learning their own. When it came time to convert to grades, Integrated Use was an A, and Executive an A+. Is that fair?

Terie's central question was: 'So why not change the "doing the work for points" idea into a "doing the work to improve my learning?" idea?' This has been an issue all semester with this particular group of students for me. Because this is a teaching class, we've even explicitly discussed from whence comes thier lack of responsibility. (As in the Condition of Learning, Responsibility; not as in a guilt trip from your parents. I hope.)

So I brought the framework into class and asked them to classify the kind of lessons they saw in observation... 10 to 15 lessons over the course of the semester.  I was seriously surprised how well distributed the X's were on the relevance framework, but not surprised by the low range of rigor. The students discussed how the ABCD was confusing, as D was the "best." They had quite an interesting discussion about whether math should be 'real world' or not. You all know the discussion: why do they ask us (math teachers)? Do they ask their history teacher? What about learning the math for math's sake? Sometimes the applications are so artificial.  The students hate the applications so much, why make them do them?

I think, from their discussions as well as their observation journals, I would have put most of their X's in A.  The application in the rigor framework led to a lot of the marks being along that line, as we haven't studied Bloom's, nor have they in other classes.  Personally, I do wonder about what the classroom should look like in terms of this scale. I like the idea of flexible access, and I want students to transition to analysis, synthesis and evaluation. The application framework seems like you do want a diversity of those problem types. Regardless, I'm really interested in what other teachers think about the framework.

To support the students in thinking about implementing this, I brought what I had thought of after Elissa's prompt. Their task:

Our question is: how can we plan a lesson or lessons that will support our students in moving towards being able to make proofs, understanding the required angle content, and engaging them all the while?

Consider the following and please add your own:
a) a puzzle made from parallels and transversals
b) a map of city streets
c) a classic Japanese problem using these ideas

Where do these ideas fall on the rigor/relevance chart? How should we sequence them or structure class to get to our objective?

Explore the activities in your groups, or design your own, and then we’ll come back together to discuss the ideas.

The puzzle: (Actually revised a bit from their use, based on play; changes were made to make it more accessible and focus on the mathematical properties. Click on the image for full size.)

The map: (Grand Rapids)

Tasks suggested for this were identifying parallel and perpendicular lines, vertical and transversal pairs of angles, combining measurement of street intersections, make informal arguments for congruent pairs, etc.

The Japanese angle problem:

(The top angle is 50 degrees, the bottom 30 degrees.)


Find the angle x. “Please solve the problem and if you can make an explanation that is amazing.” From the TIMSS video at http://timssvideo.com/66 (Love this problem; thanks to Rebecca Walker who first found it.)

Students tried all three, and they typically wanted a combination of them in their lessons for their hypothetical students.  The discussion of rigor in these problems gave us a lot of material (by which I guess I mean gave me a lot of informal assessment data) for a discussion of answer, solution, argument, proof in a later class.

I think these problems are pretty good samples of A, B and C lessons; I confess to being a bit stumped as to what a D lesson/problem/activity might be for this topic. The nature of abstraction in mathematics will lead to a lot of good lessons proceeding from B to C. Opportunities to venture onto D might be more rare. I'm very curious to hear what other people think.

Triangle Puzzle

Have you ever had a nice problem that you just thought about at odd moments?  Boring meeting, stuck waiting somewhere, few surprise extra minutes in a day?

For a while now, my favorite problem like that has been finding a nice way to divide up a square into the seven triangle types.  I love tangrams, and I like Pierre Van Hiele's mosaic puzzle even better.  If you do too, stop reading right now and try this problem.  It's fun and worth a surprising amount of thought.  (For me, anyway.)  Then suddenly this week, one of my little thumbnail sketches worked out.  I don't know whether to be happy or sad.  Being a geogebra nerd, I wanted to make a sketch of it, and that led to making a puzzle out of it.

You can print this picture of the pieces to try in real life, or try it with the Geogebra file or as a webpage.    (A solution is an option on the file or webpage.)

But... now I'm left wondering what to think about in those rare extra moments.  Then on Twitter, Justin Lanier (@j_lanier) tweets:

Had an insight in the shower this morning. Example: .717171... = .717171.../1 = .717171.../.999999... = 71/99 (!)

 Hmmm.  Really?  Maybe it's a coincidence, because 100 times .717171... minus the original leaves you 99... hmm.  Would it work for .717171.../.6666... ?  It does.  Tweet back:

@j_lanier cool. So is .a_1 a_2...a_n repeating / .xxx... =a_1...a_n/xx...x (n times) for any x? Or divided by .b_1 b_2... b_m repeating ...

Which connects to another problem (from Dave Coffey) I like thinking about: how many digits does it take 1/17 to repeat and how can you tell?  In general?

OK.  Deep breath.  There's always more problems.