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.

Hip, Hip, Array!

I was mourning our math walks.  For almost six months, every time we went out our door we'd find math.  Mostly geometry, though, but it was everywhere.

Then, this fall, the kid (age seven) went back to numbers.  We hit them hard -- a review of addition and subtraction which, twelve weeks into our school year, has her easily adding three digit numbers together. On her own volition.  In her head.  Without her fingers. 

It was clear I had to do something else with numbers.  Obviously multiplication was next but I had no clue how to proceed since I knew the whole rote-memorization thing was not going to fly.  She's not that into charts either.  Luckily, a great post over at Let's Play Math saved me, and ultimately inspired us down a fantastic new path of inquiry.  There's more to this story (having to do with the magic beans) but for now let me just say that, indeed, if one starts to think about what multiplication really is, multiplication turns out to be EVERYWHERE.

Especially in the form of arrays.  In the last week or so we've been finding and analyzing the arrays we find out in the world.  At the park:

 

Many leaves, not an array, but very pretty!

 One column of long rectangles.  1 times many.

Is it an array or is it a gradient of trapezoids?
 

Once again, our city's sewer covers provide a fabulous opportunity for math.  We saw enough of these on the walk home from the library one day that the 4x4 fact is permanently etched in my daughter's head.

Arrays are everywhere, even in a truck grill.  This one is conveniently divided into four parts, so if you really wanted to you could find the number of holes in one section and then figure out the total number of holes in the grate from there, and then, because there are two sides, double that number.  Fun times!

These we find at almost every crosswalk.  They all seem to have either eleven rows or eleven columns or both.

Is this wall an array?  Why or why not?

Simple and sweet.  (An upturned recycling bin!)

At the post office!

In front of the library! (How good are your eight times-tables?  Or, perhaps you could figure out the number of dots a different way?)

Three weeks or so ago, when I was trying to wrap my head around teaching multiplication without focusing too much on the facts, it was really quite difficult for my seven year old to analyze and compute an array over 3 x 3.

This morning we were working on growing patterns with some square pattern blocks, but while we were cleaning up I played around to see if the number of blocks I had in my pile would make an even array.  They did!

I thought to ask my daughter if she could figure out how many there were.  It was like magic!  Unprompted from there she immediately saw three in each row, counted six rows, noticed a top half and a bottom half in the design, and started skip counting.  Even as recently as three weeks ago this kind of structure was not evident to her eyes, but today it took only a couple seconds to thoroughly analyze what she had before her.

"Eighteen, Mama, eighteen!  Three times six is eighteen!"

I never knew multiplication was so interesting and so ubiquitous.  Thank goodness I get to do elementary math all over again.  It is SO much fun!

p.s. It's also fun over at the Math in Your Feet Facebook page!  Check it out -- today I put up a link to a video of fun, family friendly percussive body music!

Critter Combinations: Grids, Arrays, Multiplication & More!

When I was in third grade I learned multiplication.  Well, really, it's more like I memorized the facts.  I learned about fractions too, but I never really understood them...at all.  To this day, I have only a cursory understanding of ratios and percentages.  I have higher hopes for my seven year old daughter, though, especially now that I know all of these subjects are related. 

My daughter and I have explored a lot of different aspects of math over the last year (mental math, sums and differences, lots and lots of geometry, fractals in various forms, mathematical stars, flexagons, functions...).  It's been great and we're still going to keep exploring the beauty and structure of patterns wherever and whenever they come up.  But, I recently came to the decision that though it might be a challenge for both of us, multiplication was next.  And I knew that my focus, for now, was not going to be about facts or memorization, it was going to be about comprehension. 

My daughter has basically understood the groupings concept of multiplication for a while now, well before she mastered addition and subtraction.  It's of the 'cookies on the plate' variety, except for one of my favorite lessons using multiples of threes to make stars.   Fortunately, I recently found a great card game from Let's Play Math (a free download, which I wrote about here) that helped me introduce the idea of multiplication as a number sentence, arrays, groupings, and measurement.  We've also started playing with this Primitives Application (which is really more about factoring, but for the elementary mathematician the visual groupings can't be beat).  I first found out about the primitives app from Maria Droujkova of Natural Math.  And, I recently purchased and hung Natural Math's Multiplication Models poster in a prominent place just so I can ponder all the wonderful information in it. If my kid gravitates toward it, all the better.

As I expected, soon after I hung the poster, I got inspired.  This little activity was completely influenced by the combinations portion of the poster pictured below:



































I was pointing it out to my daughter -- "Look, a cat-dog!  Let's call it a cog!"  I was having fun coming up with silly names for each combination and that's when it hit me.  I was actually understanding the process of combining something other than dance steps.

Then I thought: Grids. Arrays. Multiplication. Perfect. Let's make our own!

I made a quick grid that night and upped the combinations to 4x4.  I asked my daughter what animals she'd like and found some clip art that worked.

The next morning I ran into a roadblock.  The kid was not happy about cutting up all the animals in to heads and bodies.  Not.  At.  All.  I quickly changed that to 'tops' and 'bottoms' hoping for a less gruesome association to only partial success (she covered her eyes every time I cut an animal in two).

To insure that this activity didn't drag on, I did most of the cutting and pasting.  I suppose it would make a good fine motor activity, but I wanted her to focus on the combining rather than the preparation.

We started with the pig 'top' and bear 'bottom' (hee hee) and called that one a 'Pear' and continued haphazardly from there.  In retrospect, I wish we had moved left to right and top to bottom on the grid, pasting parts and making up the names from there.  Giving them combination names was a good challenge but if I had another chance to do this again I would formalize the naming process by using the animal top to blend in with the animal bottom or vice versa. You can't name a pig top/horse bottom the same as a horse top/pig bottom, right?  They're different creatures altogether.  So, for example, Pig/Bear would be "Pear" and Bear/Pig might be "Big".

Although I wish we had filled in the grid in a more organized manner, it also did work out the way we did it, too.  I'd ask, what combination do you want to do now?  She'd choose and then I'd ask her to find the square where that combo animal had to go on the grid.  She didn't quite get it at first, but after a few combinations she had the hang of it. 

At the end I guided her through counting the rows and columns and finding the total number of combinations we had made.  I also focused on writing the answer in two different number sentences, as multiplication and again as repeated addition.  By the end, I realized just how dense this kind of activity is.  If I do it again, I'd plan for a large grid (maybe 6x6?) but I'd have us combine three animal tops and bottoms to start with.  Depending on the kid, you can add one or two more animals to the mix or even just (aha!) predict how many more combinations you would have with the addition of each new animal.  And, then you can make a connection to square numbers.  Cool!

Okay, I'm revising the list of math activity/topics in this post: growing patterns (algebra), square numbers, grids, arrays, multiplication, and combinations.  When your students are done with this deceptively simple activity give them a big pat on the back and then take them out for ice cream.  I also owe my daughter a big thank you for being a fairly willing guinea pig on what turned out to be a very tired day for her.

If you want another similar and super fun grid/combinations activity check out this great post from Yelena at Moebius Noodles called Mr. Potato Head is Good at Math.  It's fantastic.

And, don't forget to check out all the fun we're having over at the Math in Your Feet Facebook page!  Today we found the math in an incredible flower from Argentina!