Wednesday, July 16, 2014

Super Mathio

Hedge wrote this fun post on Mario Brothers and math - specifically parabolas - that got me to tweet: I also now want a video game where you jump by clicking the vertex to make a precise parabola based on where you are. She asked the good question: Can you make that?

Should have thought that one through.

Fortunately, though, it's Twitter, so brilliant people to the rescue.

Desmos tweeted:

Pretty sweet! Find it at Desmos:

Beyond my current Desmos levels, but pretty amazing. Great image use. 

Then Andrew Knauft tweeted:
Find it at OpenProcessing.

You click to do the jump. Beyond my current Processing skills, but excellent.

So I did -of course- have to try it in GeoGebra. I'm pretty happy with the result. The math wasn't too hard, though the scripting the buttons and resetting graphics is always tricky for me.

It's at GeoGebraTube for you to play with. There's a data mode that let's you try to calculate the best high point first - which is where I would want to go with it. It's an insufficient data situation as I purposely left off the coordinates of the bonus box.

There are some things to notice about parabolas as you play it, and for deeper work it would be interesting to think about how to add scoring.

GeoGebra Note: Andrew Knauft also helped with this! You define piecewise functions in GeoGebra using the If command, which also has an else variant. I.e. either If[<condition>,<then>] or If[<condition>,<then>,<else>]. I was getting errors with the inequalities to make the towers, and Andrew figured out that GGB didn't like an inequality with the else form. Redefined them as Ifs alone, and I was good to go.

This was very fun. I'm curious to know if or how you would use it, or what features you might add. As always, if you have an idea for dynamicizing, let me know!

Friday, July 11, 2014

Complex Instruction

Complex instruction isn't real.

OK. I think that's enough of that. (Complex instruction is an actual instructional approach for differentiation, check it out.) This post is part my general thinking about complex numbers and, in other part, a very specific visualization.

I. In General
This summer teaching the math-history-themed capstone class for our seniors, one of the themes that came out was how complex numbers are the natural setting for much of mathematics. In my own experience, I didn't really appreciate that until grad school, and our majors are graduating without enough experience to feel like they are really numbers.

My perspective: I think counting numbers feel like numbers to us because:
  • we have an intuition for them as quantities, 
  • we can represent them in multiple ways, we can compare them fluidly, 
  • we are comfortable with what operations mean, 
  • and we have multiple ways to compute with them. 
The more of those aspects are broken down, the less they feel like numbers to us. We see that with students at large whole numbers, fractions, decimals, radicals, etc. I've surveyed about 50 math major seniors and inservice math teachers now, and they are significantly less comfortable complex numbers than radicals and transcendentals. When I shared some of those results on Twitter, several mathematicians wrote back that this means the students have not developed an appreciation for how weird the real numbers are. Part joke, all truth.

Part of the power of being a mathematician is being able to make this transition for wide classes of objects: functions, matrices, operators, manifolds, algebras, etc.

Complex numbers came up in math history in the solving of quadratics with the great Islamic mathematicians; the solving of cubics with Cardano, Bombelli and Tartaglia; the great acceptance of them under Euler; and the deep understanding of them that followed Gauss. The students saw the connections amongst the lack of acceptance for irrationals, negatives and complex, but - in general - saw their discomfort with complex as justifiable, while rejecting irrationals or negatives was just a bit silly. The exception to this was the few students who had taken a second semester of abstract algebra and saw C as a beautifully complete field.

I think the problem might be that algebra is before we see our students, and the focus on calculus and linear algebra in our program has to support the engineering students as well as serve math majors. Algebra and linear algebra seem like the natural places to start seeing complex numbers as natural and necessary.

It was probably Michael Pershan who really got me thinking about this, as he started sharing his usual deep reflections on the topic a couple of years ago.  Bonus: Mike Lawler has a couple extensions (one and two) which my students this summer found interesting and valuable.

II. The Fundamental Theorem of ... Algebra

One of the morals of the capstone class was that if mathematicians labeled a theorem as Fundamental, it's worth your focus and understanding.

If you have time, please watch at least the beginning of this terrific Numberphile video with David Eisenbud, director of MSRI at Berkeley.

Marvelous. One of the things that so infatuated me with my advisor, Nigel Higson, was his ability to motivate and moralize the mathematics. The dog on the leash is the nice metaphor here. I think the power of this representation is not in the static image he draws, but his ability to manipulate and use this imagery to connect ideas. For most of history, this has been a significant barrier in mathematics.

III. The Visual
Of course this has to end up in GeoGebra...

The ability of GeoGebra to have two graphics windows allows us to dynamicize Eisenbud's picture. Here's the GeoGebraTube page for this sketch. In works online (the student page), but is better if you can download and open it. GeoGebra does not compute in complex numbers natively - yet - so this sketch has a lot of workarounds to treat points as complex numbers. This post isn't about how to make the ssketch, though, it's about using it.

GeoGebra tip: when a point or slider is selected, you can move it with the arrow keys. Hold down shift while clicking an arrow for smaller changes.

The sketch has one window for input to a polynomial, and the second window for the output. For example:
 To get that feel for what Eisenbud was doing, I turned on the trace for the points. Complex valued functions and analysis is very much about paths and what happens to them under transformation.

We can use this sketch to find real and complex roots. When you click circular path, it plots all the points in the input plane with the same magnitude as A, the blue point. The output plane shows the result. So when the orange transform of that circle goes through the origin, there's a root with that magnitude. The green slider rotates around the circle, allowing you to findthe exact value. Above is a real root for that function, \(f(x)=x^3+2x^2+x+1\), near \(x=-1.76\). Where are the complex roots?

One more example: Find the first root.

So, since this is a real coefficient function, the conjugate must be a root. I made this cubic to have the root \( x=2 \), so I show that root also. The transformation of the rectilinear path by the function is quite fascinating, no?

Studying different kind of roots can be pretty interesting. What do you think is going on here?

We can investigate the winding number phenomena as well:

There are so many things to notice once we have a visual. It supports more people in developing the kind of intuition and acceptance that might some day see accepted status for our complex brothers and sisters. After all, the reals are complex, too.

Seriously, I'd like to think that we're coming into an age where technology makes these ideas more accessible to a wider number of people. Can you think of other features or visualizations that we should try to make? What are the key concepts of complex numbers that need the support of a dynamic visualization?

  • GeoGebra's gif export only does the active graphics window, so these are done with the RecordIt app for Mac - highly recommended.
  • It's a little weird that the function has real coefficients but complex constant, but it seemed worth it to get at some of the points that Dr. Eisenbud raises in Numberphile video.
  • The work arounds consist of making separate real and imaginary functions from the input function. It looks like this:

Friday, June 20, 2014

Playing with Math

Today's the day! The crowdfunding for Sue Van Hattum's book Playing with Math opens up. I'm excited about the book, proud to be part of it in a little way and so happy for her.

If there's one phrase that captures my approach to mathematics learning and teaching, it's 'playing with math.'  So I'm really cheesed that Sue has stolen this title for memoirs... wait. That's not where I was going with this. Besides, the full title is Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers

I didn't meet Sue until after she had moved away from here (West Michigan), but got to know her via what is now the Math-Twitter-Blogosphere, and then in real life on one of her return visits. In this book she has gathered together many of my favorite aspects of the math community and culture, plus more that I have yet to know.  I got to be a realatively early reader of the manuscript (found a v1 file on my computer!), and have seen it start good and get better from there. This is going to be an amazing resource. Her philosophy in building the book was very much about building a community, sharing the people and math of which she is so fond with you. I think you'll find it intriguing, entertaining and helpful. Bloggers, math circles, living math forum... Sue is great at connecting people.

I'm a great believer that teachers get better through conversation, and every piece in this comes across as powerful teacher or learner sharing. It's a rare anthology where you feel like you wouldn't cut a thing, but this is one of those. The pieces I have returned to more than once already include Bob and Ellen Kaplan's reflection on a prison math circle, Maria Droujkova's rejoicing in confusion, Malke Rosenfeld's mapping the territory, and Allisson Cuttler's putting herself in her students' shoes. And... it could easily become the table of contents. In editing, Sue worked hard to preserve the author's voice, make the book very inclusive of student and teacher diversity, and to represent each of her three communities.

And each teacher story finishes with a puzzle or game. Tanton, Halabi, Gaskins, Salomon... Van Hattum. In addition to editing, Sue is a great and reflective teacher, and her own writing and games are an important part of the book. It is very much like a teacher weaving a lesson together from student work and responses, the way she tells her vision of mathematics learning from such a wide variety of different authors.

Nix the Tricks and Moebius Noodles are both great examples of books that are from and for the math community, and this is a great next step. Please consider supporting it; I think you'll be glad you did.

Some other resources, reviews and comments:
A family favorite to which we were introduced by Sue. Our semi-annual gaming get togethers are now pretty highly anticipated!

Tuesday, June 17, 2014

Capstone Book Club - Summer 14

Time for another installment of Read 'Em and Weep... no, that doesn't sound right. See a previous edition here.  Students grouped with others who read the same book, plus a group of readers with unique books. After they have a chance to discuss for a good bit, I ask the groups to think of what they want to say to the whole class about it. In a whole class circle, we discuss each book in turn. These notes are from the whole class discussion.

Accessible Mathematics:
shifts emphasize how to adjust your classroom… eg. multiple representations. Liked it b/c it wasn’t just changes. Like minimizing what is no longer important; calculators for example. A lot of specific examples… real life. $1.89 v easy numbers…
Love and Math:
Good for non-math majors so they can understand what we’re talking about. Example of symmetry: which is more symmetric a circular table or a round table? If you left the room and I turned. More of a story and Frenkel’s search for knowledge.
e: the Story of a Number
We agree that the beginning - the history of the mathematicians who worked on it - was very accessible. Could not follow the shifting representations of problems in the last few chapters. The end was significantly harder to understand. The part about the personal life of mathematicians was cool, and who winds up getting credit for achievements. Publish or perish keeps some people who deserve credit from getting it. Qualified recommendation. Music and logarithms. Some review that was helpful. Interesting to see how they discovered what we study, and how hard it was to find some of what we take for granted.
Euler: the Master of us All
Is Euler human? Today you have to be very specific, but Euler did so much in so many areas. He goes by topic, what was known before Euler, hen what Euler did. Proofs, but open to people who can handle logs and series. Highly, highly recommended. Popularized complex numbers. His proof writing was like Romeo and Juliet; the balcony just works. Blind for a lot of his life…
Review: Kyle Ferguson: plus some extenisve book notes.
    Gödel, Escher, Bach
    Really about Gödel and the incompleteness theorem. He’s trying to explain to a general audience. Tortoise and Achilles, a fable, then ties it into a serious discussion of the mathematics. He also ties in art, music, zen philosophy. Not a clear path to the Incompleteness Theorem, but about what is interesting along the way.

    A Journey through Genius
    Paint a picture of mathematics that is logically sound and aesthetically beautiful. He picks the best proofs, a bit of history about the man, then an explanation. You could read this in sections to understand something specific and the culture it came from. Not just about the applications, but appreciating it for what it is. Logic still holds true centuries later.
    Review: Jason Lohman

    The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Prob-
    lems, and Personalities by William Dunham
    Also by Dunham. Proofs and problems of mathematics, arranged alphabetically. Which is confusing vs historical order. Chose some really interesting proofs. Any natural number as distinct integers and odd integers.
    Review: Nate De Maagd. Includes some of his proofy highlights. (click through to Gdoc)
      Vision of Elementary Mathematics
      Good for non-math majors teaching elementary school. Good for visual learners, lots of diagrams. Also for getting kids to discover math on their own. Gives alternate perspectives that could help in talking to students. It’s a little repetitive. But we’d really recommend it.
      Mathematician’s Lament
      Problems with math teaching as it is right now in schools. The differences between teaching art and teaching math. The creativity is taken out of the math. There’s nothing to explore or discover. Like teaching art with only paint by numbers. Couple issues with some of the arguments: like teachers try to make it interesting but it already is interesting. Or you learn when it’s relevant to you, but he had argued against relevance. Also doesn’t describe how to make it better. Says math should be a free for all - against all structure. Everything he’s telling us is pointless. He’s against drill but favors real world problems. We teach definitions for no reason. (Quadrilaterals, for example. But then how do we communicate math?) Has a mathematician’s perspective, but he didn’t have a teacher’s perspective. What he offers doesn’t seem appropriate to most schools. The conversations at the end of each chapter were confusing. “What to do with elementary students?” “Just have them play games.” Strong opinions but no evidence. You should read it but it will make you frustrated.

      I feel like how we’ve been taught to teach does help handle a lot of these issues. It would work if everybody loved math like he does.
      The Joy of X
      Insightful and helps make math more accessible. Analogies from Sesame Street. “Fish, fish, fish, fish, fish, fish…” all the way up through complex numbers. Split up into 30 short chapters. From numbers to infinity. I liked the way he uses personal stories. Very readable. Not for diving into math but there’s a lot of stuff that helps clarify. Even for someone who struggled with math. But it won’t change your mind. Great book for college freshmen who have to start making some sense of math. Also good for middle school and high school teachers.
      The Math Book
      Looks intimidating but it’s an easy to read. A page and a picture. Goes through history including lots of things that you would not think of as math, like tic tac toe or mancala. Also see a lot of mathematicians come up multiple times, which is neat to see

      Students choose their own books and that really pays off. Very positive feedback this time around. We follow it up with a book swap, so you get to read a second book that is of interest to you, or at least to skim it. (That's why some people have more than one review linked.)

      Some of the reviews are fabulous; if you're interested by the blurb here I'd really recommend following up. And of course, if you have a chance to comment on a student blog, that's excellent.

      Friday, May 16, 2014

      I See Number Theory

      We had a really interesting week in my number theory class. We are really a seminar, seven teachers and I investigating elementary number theory together.  I hope they're learning half as much as I am.

      This week we were exploring primes and modular arithmetic. The first day we were thinking about the \(4x \pm 1\) and \(6x \pm 1\) structures, and the results that there are an infinite number of each type of prime.

      To gain modular arithmetic practice, we played Modular Skirmish. (Cf. this post on Gauss.)

      Then we started looking at this GeoGebra sketch:

      The numbers increase from the bottom left corner up the column. My first attempt was a growing square, but that let you see asymptotic distribution of primes more than the modular structure.

      We put this sketch up on the front screen, and advanced n. Teachers noticed the empty top rows (multiples of the modulus) and how some values separated the primes into rows: 2, 4, 6, 8, 10, 12... while others seemed to form diagonals: 3, 5, 7, 11, 13... We wondered about which were the most consecutive primes or gaps in a row, and whether that would change as m increased. (Personally I got wondering about where are the largest square gaps.) Teachers connected many of the patterns to the rows in 6. For example, in 7:
      The diagonal really means the next prime is +7 -1 or 6 apart.So it goes back to that 6 structure.

      Modulo 10 is really just looking at the last digits. We noticed that no digit seemed more or less common out of the four possible. Also, no consecutive dots more than 2. Is that always true?

      The two coolest structure theorems are with respect to four and six. I think these helped in understanding why primes are of the form \(4x \pm 1\) or \(6x \pm 1\). Which may have also helped with the proof that there are an infinite number of primes of the form \(4x - 1\) (or \(6x - 1\) ).

      We did find a modulus where there was a row of 8 consecutive primes, but I can't rediscover it!

      Understanding the six structure also helped us understand a diagram that we were looking at the previous week, from a designer who was really impressed with a 12 structure. (Source in reddit/r/mathpics. The picture isn't super precise, but did offer a lot of making sense opportunities. And colorful!)

      Rather than make the course a tour through the great theorems of number theory, my hope is that it can be an opportunity to do math ourselves. So instead of necessarily illustrating a theorem, I'd rather find a way to notice things that might lead to the theorem. Since we're interested in K-12 applications, divisibility tests and primality tests are of interest; that means exploring the ideas in Fermat's Little Theorem.

      So the idea came - given the success of the modality/primes visualization - to visualize exponential patterns in the modular context. This sketch is what I came up with.
      Oh! The patterns they found!

      Definitely a lot of things that I had not noticed. Not, interestingly, Fermat's Little Theorem, but there were many observations that will lead there.

      A lot of our discussion was about pairs of cycles. The visualization made it clear when two different bases created the same path, up to direction. Eg. \( 2^m \mod 5\) and \( 3^m \mod 5\).

      Furthermore, they noticed this awesome pairing within the cycles. Here's the nicest mod 13 pair.

      Look back at the other data... there's a lot to notice. And it definitely has me wondering. (Copyright, trademark and kudos to Max and Annie from the Math Forum.)

      It's hard to imagine that introducing a theorem and sharing a proof would have resulted in building any more understanding, and there's no way it would have led to doing any more math. And this will make the theorem so much more meaningful when we get there. If we do, with such a fine boatload of conjectures to explore.

      Friday, May 9, 2014

      Safe or Sorry

      Penrose Impossible Triangle
      (best source I can find)
      There was no specific math goal for today, so that always puts me in mind for number sense, mental math and operational fluency.

      I've been interested in push your luck games (Farkle, Zombie Dice. Pig from the Interactive Mathematics Poject..) but those mechanics can be lame for a whole class math game. Students get bored waiting for the turn to come around. Not enough decisions to make. So I thought of a variation that offered a little better opportunity for mental math, and more activity for the players. The game was a moderate success, but the game design discussion at the end was even better.

      Safe or Sorry
      Dice game for two or more players.

      All the players roll a die. Add them up for a total.
      If the total is a multiple of 5 –  turn is over, zero points.
      You can stay and take that many points, or keep rolling.
      Whoever is still in, rolls again and adds the points to your first total.
      If the new total is a multiple of 5 –  turn is over, zero points.
      After each roll you have to decide:  stay safe and take the points or keep rolling. If the total is a multiple of 5, though, sorry, the turn is over and you get zero points.

      Winner is the first player to 150. If more than one player is over 150, they all win.

      Example: 3 players: Ann, Bill and CeCe.
      Ann rolls 3, Bill a 4 and CeCe a 5. Total 3+4+5=12
      Bill stays and scores 12. Ann rolls a 2, CeCe rolls a 6. 12+2+6 =… 20! That’s a multiple of 5.
      Score: Ann =0, Bill = 12, CeCe = 0.

      Next turn: Ann, 4; Bill, 4, CeCe, 3: 4+4+3 = 11. Ann stays and scores 11. Bill and CeCe roll: 3 and 5. 11+3+5=19. Bill stays and scores 19. CeCe rolls again: 5. 19+5=24. She rolls again: 5. 24+5=29. She rolls again: 6. 29+6 =35! She loses all the points.
      Score: Ann = 11, Bill = 12+19 = 31, CeCe = 0.

      You have to know when to stop, CeCe!

      Pretty bare bones, but it's the end of the year so I want them doing more of the game design work.

      I played a game against the class (Golden vs guys vs girls) to introduce it. The main confusion was whether the multiple of 5 was the total on the roll or the total for the turn. Maybe I need better terminology?  It wasn't a persistent confusion, though. The original game was to 100, but they wanted to 150, and that worked pretty well for the playtest.

      The mental math level was appropriate and pretty diverse: some adding the single digit dice for some and writing down the two digit sums, mentally keeping the running total, or considering their score + running total.  Having multiple dice to add gives options for summing, the group nature had people doing it in different ways. I saw everything from counting on to efficient fact use (doubles or sum to ten).

      The game was mostly engaging. One group couldn't get into it, two groups played a full game then stopped, and four groups played until awwww, time was called. But, interestingly, the discussion afterward was full engagement, even the group that tried it the least. They all agreed that the game was a keeper, even though it needed some work. There was a fair amount of discussion about the zero condition. Some people wanted it to come up more often, others felt it was okay as is. One student suggested, "what about on multiples of 5s and 10s?" which led to a quick but strong class discussion about that. Some students wanted as hard a condition as even numbers. They recognized that the more common the condition, the more risky continuing to roll, the more often you should stop. The strategy discussion was strong; I was surprised that they recognized they were not stopping enough, but the fun of rolling was worth it. Most of the stories they were telling were of the "I got to 147 on one turn! Then 155 for a zero, of course."

      One interesting rules discussion was about whether players who were out be able to come back in. At first the majority thought yes, but then someone pointed out that this made the decision to stay or keep rolling less important. That changed public opinion, but there was an interesting suggestion: when someone else opts out you can choose to come back in.

      There does need to be a catch up mechanic, because when you're behind, the risky behavior is not enough to get back in. The opt in might be one way to do it. Other suggestions were that the risk should elevate. A really interesting idea was that there should be extra zero conditions the farther they go. One cool idea was that the zero condition should be a multiple of the number of players!

      The game could use a context. The best idea the class had might not be suitable for school! "What about, you're a burglar, and you keep doing jobs, but you might get caught and go to jail. That's the zero." I do like that as a pushing your luck context; would it bother teachers to have their students playing criminals? Do you have a better idea for a context?

      I think this is how I'd try it for the next iteration, adding in the extra zero condition:
      Safe or Sorry
      Dice game for two or more players.

      All the players roll a die. Add them up.
      If the your total for the turn is ever a multiple of 5 – the turn is over, everyone who rolled gets zero points. Now you can take your points and be safe, or keep rolling.
      Whoever is still going rolls again and adds the points to their turn total. If the new total is a multiple of 5, the turn is over, zero points.
      After each roll, you have to decide:  stay safe and take the points or keep rolling. If the total is a multiple of 5, though, sorry, the turn is over and you get zero points.
      But the longer you stay in, the riskier it gets! If the total is over 50, multiples of 3 also give you zero.

      Winner is the first player to 150. If more than one player is over 150, they all win.

      Friday, May 2, 2014

      Four Corners

      5th grade games are delayed for a week, so that gives me a chance to write up our last game that was a pretty good success.

      The commission from Mr. Schiller was a game to introduce graphing in the first quadrant.

      Yes! Graphing! So game like already, thanks to Battleship and Connect Four. A lot of ideas came to me - too many, to be honest. And too complicated. My first thought was to put points in a line. The choosing dice mechanic has been so good, I was thinking about that...
      On your turn, roll four dice. Use those numbers to make up the two coordinates of a point, but you don’t have to use them all. For example, if you roll 2, 2, 3, 5, you could make (2+2,3+5)=(4,8), (2+3,2+5)=(5,7), (2, 2+3+5)=(2,10), or even just (2,2+5)=(2,7) (don’t use the 3) or (2,2).

      On the graph put your point using your initial or symbol.

      Each player makes 5 points.

      Then draw 1 line through as many points as you can. The winner is the player with the most points on their line.
      original image source

      Great game for beginners, right? I thought there'd be lots of vertical and horizontal lines. Filed that away for a future algebra game, though. Maybe.

      What about a chase? I went through lots of variations of trying to get to an escape point and the other team hunting you down... like a Battleship where you could move your ship. I couldn't figure out the hunt mechanic, though, and it didn't emphasize the placement of the points. Felt more vectory. That's a word. File that away for linear algebra.

      If they're going to make vertical and horizontals anyway, why not go with that? Maybe the game could be about making rectangles. An early attempt:
      On your turn, roll four dice. Use those numbers to make up the two coordinates of a point, but you don’t have to use them all. For example, if you roll 2, 2, 3, 5, you could make (2+2,3+5)=(4,8), (2+3,2+5)=(5,7), (2, 2+3+5)=(2,10), or even just (2,2+5)=(2,7) (don’t use the 3) or (2,2).

      On the graph put your point using your initial or symbol.

      The first team makes one point and then each turn after you make two points. A game is 12 points for each team.
      Team 1 – one point; Team 2 – two points; Team 1 – two points; Team 2 – two points;
      Team 1 – two points; Team 2 – two points; Team 1 – two points; Team 2 – two points;
      Team 1 – two points; Team 2 – two points; Team 1 – two points; Team 2 – two points;
      Team 1 – one point.

      The goal is to make rectangles up to 12 squares in area. When your points make the corners of a rectangle, draw in the rectangle, add the area to your score.
      Still the dice choice. You can see me struggling with the advantage of going first. It still felt too complicated. Variations on capturing opponents' points were even more complex. But as I was testing this, I started to notice. It was pretty hard to make a rectangle. I didn't have to worry about multiples... it was hard to make one! That was the key. I also stopped worrying about getting rid of the going first advantage. The randomness of the rolling really reduced the impact. So...

      And of course you'll need some graph paper. (Here's the 12x12 labeled axes I used, 2 to a page.)

      I launched the game by playing vs the whole class. Students made connections to other games, especially Minecraft. (Engagement +2 immediately. "This stuff is in Minecraft?")

      We've settled into a routine for the whole class play. Pick someone who is ready to contribute to make the class move, then they pick someone of the other gender to make the next move. I intentionally picked some hard rectangles to complete by picking x or y coordinates to be low or high, and they won by actually making two rectangles at once. The whole class play provided opportunity for the instruction on placing points, and they got to be self-correcting pretty quickly. As well as developing some strategy. We debated whether you could use zero or not, and after resolving how you'd even plot those points, everyone agreed that would make a better game.

      As usual, they played mostly in teams of two, though this game was direct enough that some felt comfortable playing one on one.

      At the close, they had lots of things to suggest for strategy, some conflicting. The blocking aspect was brought up by one team and there were a lot of "Oooooh"s. Some conflict over whether you were better off towards the middle. Some agreement on putting them in lines if you can. Engagement was very high, and some people felt this was the best game all year.  There were variation ideas from the students, some of which they had already tried. Three people competing on one board, for example.

      Overall, Mr. Schiller and I were both very happy with the game, for engagement and mathematically. This offers lots of advantages over Battleship. The making rectangles aspect helped a lot with getting the idea of the coordinates, and almost all students knew about common x or y-coordinates making a line, and several were finding new points from old instead of plotting from the axes. More strategy and thinking in terms of the coordinates you want and how to make the best use of your dice roll. Deeper strategies available for students ready to think about it. Lots of practice placing points with another team to monitor you. (I was impressed how few students were doing the x-y reversal by the end of the game.) It just has the good feel of simple rules and reasonable depth.

      Of course, if you have ideas or get to try it, I'd love to hear back.