Stake Your Claim

My daughter's 3rd/4th grade class has been working on arrays over the last few weeks. (I recently found out they are using the TERC Investigations curriculum, which seems really strong. Arrays are the first investigation for the 4th grade sequence.)  One of their homework assignments was to find, draw and quantify examples of arrays in their environment at home and at school.  Of course, I love that! 

It reminded me of a game I created for my daughter and I to play in the spring of 2012.  The post about the activity is called Totally Territorial: Cats, Maps, Area & Multiplication.

Essentially, I drew a grid over a map of our fair city and, armed with dice and some pens, we staked out our claim.  It was totally fun and my almost-seven-year-old was completely into it.

Since they are in the middle of a world geography unit (meaning maps!) in addition to their array inquiry I thought it might be fun to formalize the game for the class. I searched for 'maps of imaginary places' and found a bunch at The Explorer's Notebook, created some grids and a game structure and....voila!

The game right now has five maps.  The rules are simple, but I think the narrative context provided by the maps is enough to make this fun:

Purpose of the Game:To capture as much area on your map as possible.

Rules of the Game: 

2 players

Roll one dice to determine who starts the game. 

Roll one dice. First roll determines the number of horizontal squares/length. Second roll determines number of vertical squares/width (include the first horizontal row in your count). 

Outline or highlight your full array in your chosen color. Compute total area (total number of squares) for each turn. 

Game is over when all territory is captured or both players have both had two unsuccessful turns (meaning not being able to find enough area to capture based on their roll). Player with the most territory/area wins although you may want to consider the value of land vs. water in the final value of area claimed. 

They played it today (while I was working with small groups working on half-ness, midpoints, congruence and three-ness with this Sierpinski Triangle activity from The Fractal Foundation).  Their teacher said they were all completely into it for the entire math class.  From what he overheard, it was unclear if the kids had developed any over-arching strategy at this point.  More likely, he thinks, they were simply enjoying rolling the dice, computing the area, and capturing territory. 

I know I'm not on the cutting edge of game design, but the fact that 27 kids were happily engaged in mathematical activity for almost an hour probably says a lot. The teacher and I did discuss upping the difficulty by adding a rule about only being able to add territory attached to previously won territory.  I love the idea, but think you'd need a bigger playing area for that.

Wanna' play? Here are the files! Please do let me know how it goes and what modifications/improvements you made to the execution, or any design suggestions you might have.

New Math Game: Change Your Rule

This post is in two parts: The first part outlines the basic flow and structure of a new game idea which, I think, has the potential to support the conceptualization of math concepts such as combinations, permutations, patterns, variables/attributes and reflection symmetry.  

I know that's a tall order so in the second part of this post I briefly discuss my reasoning, pose a few lingering questions and then ask for feedback.

Part One: Basic Flow of the Game
The idea for this new game started with a picture:

Photo via: http://www.fotomat.es/arte-combinatorio/

I immediately thought how cool it would be to have this kind of mathematical art/visual inspriation in a living or learning environment, something to hang out with that evokes noticing and wondering.  In this case, a real-life shelf with spheres and cubes just begs for interaction, something to play around with and ponder. I want one! In the process of thinking about how to get this kind of installation in my own house, a new game was born.  Here's the basic idea:

Make/build a rule.  Find a way to change it.

These are the game pieces. Set 1 includes two shapes, four colors.  Set 2 includes six shapes and six colors.

Rule Change #1: Make a rule/pattern. Using the same shapes/colors how can you make the next iteration different from the first.  Does order matter?  How many different combinations can you make?

Rule Change #2: Make a rule/pattern. Change one element of each shape to make a new pattern.  For example, orange turns into red, or square turns into circle.  For each new design change only one variable for each piece.  How many times can you change the rule? How many new rules/patterns can you make from your starter rule?  Can you ever get back to the original rule/pattern?

Rule Change #3: Build your pattern then reflect the design. What do you have to do differently when you build the reflected pattern?

But what happens if you only use one color and shape?

How many combinations can you make using three of one shape and one other shape?  Would it change things if each of those circles were different colors?

On another line of questioning, what larger design can you make by building your design line by line using the same shapes/colors in each line?

Part Two: Thoughts and Questions

Even though Part One seems like a lesson flow, it's really meant to be more of a general framework for exploration.  My intent was to provide a little structure, some basic 'rules' and a lot of room for inquiry.

I'm wondering if it really feels like a game, or is more of an activity?  If it's going to feel more game-like, does it need more structure?  A timer? Some 'change your rule' cards?  Or, maybe, some cards that say how many different attributes to use in the pattern?  How many different rules would be enough to create a sense of chance? What other 'change rules' would you include?

Based on trying this out with my own kid this morning, if you were to do this more as an activity/lesson perhaps it would be helpful to have some really easy starter examples, like all one color, or two different colors to get the ball rolling in a positive direction.  I asked my daughter to build a pattern with three pieces but then gave her the 'change rule'.  She didn't really like that and we left it there.  The game pieces and board are still out, though, hanging around.  I'll see if she wanders back over.

Questions I still have include: When combining or creating permutations, how much does the second attribute matter?  For example, this first design (two colors, one shape) looks like it combines in exactly the same way as the second design (two colors, two shapes):

Another one of my goals for this game/activity was to also explore the ideas of attributes/variables in design -- how well do you think it does that?  I intentionally did not use pattern blocks -- only one attribute/thing to change -- and instead made my own (Set 1). What would pull the attribute/variable idea out a little further?

I hope it's clear that I am hoping to dig into the brain trust that is my modest but wonderful readership.  That means you by the way, so please if you have any thoughts about all this, I'd love to hear your ideas.  

Things I would love feedback on: key ideas about combinations/permutations at the elementary level, thoughts about the game structure, ideas for other 'rule changers' to add into the mix, and any other thoughts you may have.  Thank you!!

p.s. If you want to leave a comment but don't see the comment section below, please consider refreshing this post or closing the post altogether and come back.  This template has some bugs. If you don't see a green-lettered header and/or the page menu at the top, then chances are the template did not fully load.  Sorry for the hassle. 

Thrifted Math

Here's something you probably don't know about me -- I've got something of a golden touch when I'm out thrifting.  Not always, but I do have a track record that is (to me) quite impressive!  See what you think about these highlights from the last year:

A down jacket, perfect fit, perfect color, just needed a wash and it was brand new again; I paid $4 and am on my second season with it still in great condition

A Leap Frog talking globe, like-new, retailing for $350; I paid $4 and the kid learned her continents and is onto countries, plus she makes up her own dances to the 'music of the world' option.

A sturdy elementary microscope with three lenses and swivel eye piece, like-new, retailing for $160; I paid $5 and, although we haven't really used it much, I just know it'll come in handy at some point.

We also get a lot of great science, social studies, history and art resource books at our library's resale bookstore.  And, just a week or so ago I found two old math games at Goodwill!  The first one is Scan, a "split second matching game" from the 1970's.  I found a comparable version it online offered for $50.  I got it for $1.99 and we've already had a ton of fun with it!!!

When I saw it on the shelf I immediately recognized it as mathematical in some way. There are four categories on each card: color (square, circle, square, circle), position (four black dots on a 4x4 grid), pattern (different combinations of x's and o's) and shape (various irregular purple polygons!). 

Our game didn't come with directions and at first I thought it the point was to match all four categories at once.  I even went so far as to try and analyze and sort the cards so that I could understand how it might work.  Here's one attempt at sorting by grid pattern: 

Ultimately, no pattern emerged and I also determined there were no 4-way matches to be made.  I'm not very good with combinatorics but even I can figure out (albeit after a bit of struggle) that you need more than 26 cards to have a match for every possible combination and permutation.  (Anyone want to figure it out?!  Just kidding.) 

I went online and found the directions and was relieved to find that you only have to match one category on the center card to a card on the table to win the round.  The box said it was for ages 9 to adult, but I decided to try with my 7.5 year old.  Initially I thought maybe I'd have to half-size the deck for the first few games so she'd have a chance to get the hang of it but for some reason I put out the full deck on our first game and, what do you know, she beat the pants off of me! 
 

I found the next game about a week after I found Scan.  It is not branded so I can only describe it with this picture:

Well, actually, as I was editing this post I noticed the company name Garlic Press in the bottom left corner.  Apparently they're still in business and have other similar two-sided 'self-check' math fact puzzles.  Essentially, you piece together this round puzzle by doing your multiplication facts...

...which, incidentally, are not in any particular order around the ring, which is a great feature.  When you've placed all the pieces you flip it over to see if you got them right.  If you did, you have a picture that makes sense.  If you didn't you have a good giggle about the parrot's beak and wing being in the wrong places and try again. 

I'm not that big on drilling math facts, but I thought it'd be interesting to try and a nice addition to our other work even at the sky high thrifting price of $4.99.  What do you know -- the puzzles are fun!  My daughter has only done the 0, 1, 2, 3, 4 and 10 times table puzzles so far but she's motivated to do about one per day.  I mean, who wouldn't want to solve your 8 times tables if you got a cute little tiger cub at the end of it?!

You use the same procedure to solve each puzzle, which makes me optimistic that she'll start developing some reasoning strategies around the skip counting she doesn't already know backward and forward -- like 11 times something is similar to 1 times something.  Or, if I do the 0, 1, 2, 3, 5 and 10 times first in any puzzle, the rest are not hard to figure out. 

The multiplication puzzles might be a little hard to make yourself, but if I knew about Scan and couldn't find the game itself I'd totally make a version myself with my kid or other students!  The Scan cards are just four quadrants with a basic concept and pattern rule for each quadrant.  Also, you need two copies of each card. to play the game.  It's essentially a quick-paced matching game for big kids;  I think upper elementary kids might really enjoy the challenge of making their own versions and variations.

New Math Game: Factor Dominoes!

Lately I've been looking for different ways for my seven year old and I to conceptualize multiplication. As has happened many times before on our math journey, this graphic showed up at just the right time (albeit somewhat circuitously through the excellent influence of the Math Munch blog).

 

 

My favorite thing about it is that it's not about numerals; when I look at factoring trees I can make some surface sense of them, but my mind goes numb pretty quickly. In this visualization, however, there is an incredible connection to shapes and grouping. I find this visual especially well-suited for kids in general and at least this adult specifically.

Last night I printed out the graphic and left it advantageously on the kitchen counter. I thought maybe my kid might be interested but was truly surprised by her reaction when she found it this morning.  It is probably the first piece of math my daughter has ever admitted she was excited to know more about, which is saying a lot.

She wondered what it was about so we looked it over together.  At first it was basically 'count the dots' and notice that each configuration was one more dot than the one before. Then, in the same way we tackled the 100's chart last winter, we started looking around and noticing things: The ring of seven dots on the far right column has multiples of seven underneath it.  The 6 shape shows up two more times on a descending diagonal. It's fun just to look and talk about what you see.

It's the geometry of the design that really shows the relationships between numbers. And, even though this was not meant to be a multiplication chart, it's probably the best one I've ever seen.

All our talking and looking got my mind spinning. What if...what if I made little playing cards out of each factorized number? What kind of game would it be? 

I was about halfway through constructing the cards when my big AHA! moment hit. As I made and sorted them one by one it became completely clear to me that the integers 1 through 7 formed shapes that were echoed in the other factorizations.  As an attempt to organize my growing pile of cards I laid out a top row of 1 through 7.  But where to put the other cards? For example, 5 is a pentagon made out of single dots and 10 is a pentagon group of two dot groupings. Where does it belong?  The 2's column or the 5's column? This kind of question is at the heart of the new game.

Here's how my daughter decided to sort them in a 'get acquainted' activity before we started playing:

As we went along I refined the language she needed to help her make her choices. Was she going to place a particular card based on its large grouping (outer shape) or the smaller groups? As you can see above, there's a 5 shape of 3s in the 3 column, because the smaller group is a match to that number. But, every other 5 shape is in the 5's column. She's also got a 7 shape in the 3's column for the same reason -- the smaller grouping matched and, ultimately, the whole 3's column is consistent on that criterion.

For some comparison, here is how I sorted the cards, earlier in the day. I was trying to match to the category of 'outer shape':

I'm not sure I got it the way I wanted it, but no worries.  There is probably no one right way to sort these cards and the activity in itself makes for some really interesting thinking and conversation.

After she familiarized herself with the cards we started in on the new game which I'm calling Factor Dominoes (with a side of Scrabble). The title alone should give you clues as to the game's aesthetic and procedure, but here's how to play:

Split the deck equally between two players. Player 1 puts down the opening card. Player 2 tries to find a match. If Player 2 has no match the card is put aside face up for future use and play returns to Player 1. You can find a match either by outer grouping/shape (triangle, square, pentagon, weird six shape and seven ring) or by similarity between the small dot groupings. In our game we also matched 'echoes' -- small groupings that are the same shape as another number's outer shape.

For example, in the picture below the first card is a 5 shape with small groups of 2.  The 6 shape next to it works because even though it's a different shape it also is comprised of 2s. And, the card directly below the first card also works because the smaller groupings of 3 match the 5 shape of the larger grouping. Make sense? 

Here's another example: The top line of matches have the 3 shape in common. The bottom row connects to the top with small groupings of 4.

And, here's a picture of a couple more interesting matches.  See if you can figure out our reasoning on this section of the game:

Play the game until there are no more cards. This is a cooperative/conversational game but feel free to give it a point structure if you like. You can also make the game bigger and more complex for older students -- just cut out more factors and make more cards! That's what I'm going to do for our next round of play.

Here is our completed first game:

 

Based the exponential growth of my personal understanding of primes and factors, gained in just one short day, I am firmly convinced that a wide range of ages, experiences and abilities can get something of value out of this game. 

My seven year old was perfectly challenged as we focused on groupings, but what if you added the prime numbers beyond 7 into the mix? How would that deepen or change things? What about adding exponents as a match category? What if you figured the value of each card and matched them in sequences (like {25, 26, 27, 28...} or {4, 8, 12, 16...} or even a sequence of primes, in order)?

If you do play this game PLEASE let me know how it went and what other ideas you have for it.  And, please do consider joining us on the Math in Your Feet Facebook page. We're having a good time over there!

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Malke Rosenfeld delights in creating rich environments in which children and their adults can explore, make, play, and talk math based on their own questions and inclinations. Her upcoming book, Math on the Move: Engaging Students in Whole Body Learning, will be published by Heinemann in Fall 2016.