A Useful and Beautiful Division

Me: Oh no!!  Four sets of Cuisenaire rods are all jumbled up!  I've got to get them sorted out before the next Math in Your Feet Family Night!  Do you think you could help me? 

Kid: How should we start? 

Me: I've set out four bags, one for each set of rods.  We need to divide up the rods equally so there is one full set in each bag.

Kid: (Finding all the orange rods.)

Me: "How many orange rods are there all together?"

Kid: "Twenty one."

Me: "How many orange rods go in each pile?"

Kid: (Starting with four in each group, then adding one more to each, one remains.)

Me: So, five in each group, with a remainder of one?

Kid: Look, Mama.  Four times five is twenty.  I knew that.

Yeah, I know.  I know that addition, subtraction, multiplication and division are steadily working their way into her consciousness and her skill bank, and all of this primarily by delving deeply into doubles, halves and evens.  I'm not kidding when I say that doubles/halves/even-ness comes up practically every day, one way or another.  I think it's a great way for a six year old to explore and learn about numbers, geometry, and the meaning of math.  There's actually something quite comforting in our daily discoveries of balance and symmetry whether in number play, through observations of our physical environment, or in the things we create with our own hands (the attributes matching game, for example, or our recent exploration of circles). 

I also know this kind of sorting activity is pretty basic, but having lots and lots of experience with operations in as many different contexts as possible is always useful.  In this case it was extra useful 'cause I didn't have to do this chore myself and I was able to work in some math inquiry -- sorting, classifying, measurement, division and a chance to get reacquainted with the rods -- all at the same time!   

Poem in Your Pocket Day! {Math Edition}

Yes, today is Poem in Your Pocket Day! 

"Celebrate national Poem In Your Pocket Day on Thursday, April 26, 2012!  The idea is simple: select a poem you love during National Poetry Month then carry it with you to share with co-workers, family, and friends. You can also share your poem selection on Twitter by using the hashtag #pocketpoem.

"Poems from pockets will be unfolded throughout the day with events in parks, libraries, schools, workplaces, and bookstores..."

The kid and I are going to write our favorite poems on the sidewalk with chalk!  Maybe at the park, maybe in front of our house too. 

In honor of this fabulous day, I thought I'd share one of my favorite math poems, one that really goes with the Math in Your Feet theme: squares.

School Lunch

Each food plopped by tongs or spatula
Into its own little space --
Square pizza here, square brownie there;
Milk carton cube, rectangle tray.

My snack at home after school?
Anything without corners.

From the book Sijo (Poems) by Linda Sue Park
____________________

[Edit P.S. Here's a picture of a poem my kid wrote -- I wrote it out in chalk on the basketball court down at the park and she illustrated it.]

It's a bit hard to read, so here's a transcript:

In the Night
by Isobel

The trees are whispering
in the woods
in the night.
The animals are sleeping
in the woods
in the night.
At break of dawn the animals
wake up and catch their prey.
The crystal clear waters of the stream,
you dip your paw in for rarest fish.

Circularity

For some reason, I had circles in my head.  Last night I gathered up a bunch of different sizes of circles...

...and this morning I taught the kid how to use a compass.  It took a little practice, but she got the hang of it fairly quickly.

She played around with concentric circles, circles that 'go off the page' and even discovered that three circles together can 'make a triangle!'

While she played around, I took a paper plate and a CD.  With the CD centered inside the paper plate, I drew one circle tracing the small circle in the center of the disc.  Then I played around...until, WOW!  Look at that!  I made a hexagon!  I felt like I had just discovered a dinosaur bone or something.  I decided to color it in to highlight the structure.

At that point it was time to go do Sunday things, but it took forever to get out the door.  The kid kept finding circles everywhere. 

When we got home from Sunday things, we started in on it again.  This time, the kid started experimenting with tracing the circles. A straw can make a smaller circle than the compass can.

I made another hexagon out of circles, but colored it in differently, the kid's favorite it turns out.

The kid constructed two different versions of concentric circles (mostly using the compass).  "See Mama!  They're the reverse of each other.  This one has a lot of rings on the inside, but not on the outside.  The other circle is the opposite!"  

And, then, she wanted to make a hexagon like mine so I showed her how to use the CD center circle.  I really like her design sensibility.  It's so different from mine.

And then it was time for lunch, but we still have unanswered questions waiting to be resolved.  Like, what else can you make with the compass besides circles inside circles?  Great question, kid. 

I'm fairly certain we've only just begun...

Transformation

It's amazing.  You can start with a square...

Which can turn into a triangle....

Which can turn into a square again....

And then into another triangle...

And then, after just a little bit of cutting, you get this!

It's a little bit a magic, that!

(If you're interested, these perfect little paper squares [just over 3"x3"] were 500 sheets for $2.00 on special at...um....[staples]....in six bright colors and two softer shades.  I found them in the notepad section -- looks like sticky notes, but no sticky. Also, the particular fold I used comes from a little book on Kirigami I had lying around, and the cuts were improvisational after trying out some of patterns in the book.  I'm going to use these, and many others, to decorate my summer classrooms up in the city not to mention the walls of my own house!)

Linking to Saturday's Artist at Ordinary Life Magic. :-)

Story Math: Introducing Square Numbers, Sets & Subsets

I feel comfortable doing a lot of things, but storytelling isn't one of them, especially when my kid insists I make one up on the spot. Today was no different, but luckily I got inspired.  There have been a few math-y things I've been wanting to introduce, but she's been having none of it.  Fortunately, a narrative (even a thin one) is often the way to get her hooked so I decided to go for it.  Here, then, are three of today's stories.  The first two illustrate different aspects of square numbers.  The third was actually mostly for me to start figuring out how to explain sets -- I'm not quite sure I've got subsets yet. 

As you read my modest stories, please imagine me simultaneously illustrating while I narrate.  The illustrations build as the story develops...one red cat, three orange kitties, five blue raindrop cats, etc.

Story #1: Rosie's Wish

Once upon a time there was a little red cat named Rosie [one red box colored in] who was very lonely.  She wished on a star for the company of a few new friends.  [You tell good stories mama!  Just as good as papa...!]  The star said, "I think I can do that!" and sent down three orange cats to play with.

The little red kitty loved playing with her new friends, but soon wished for even more friends.  She looked up into the sky and this time made a wish to the cloud she saw floating by.  The cloud obliged and sent her five blue raindrop kitties.  They played and frolicked, but soon wanted to share the fun with even more cats!  The little kitty made this wish to the trees around her, and the trees sent her seven little green leaf cats and, later, nine pink cherry blossom kitties. 

The whole group, all 25 of them, played and frolicked the night away.  When the sun came out in the morning, they were all tired out.  And what do kitties like to do in the sun?  Why, they curl up and take a nap!

The pink kitties lay down first, and made a soft bed of flower petals.  The green leaf cats added to the bed, then the raindrop kitties, and then the orange cats.  Rosie lay on top of the whole pile of friends, purring happily.

You probably noticed that this story ended up illustrating the growth progression of square numbers, first in a square and then in a line.  I didn't mention anything about squares or numbers in this story.  My thought was that I wanted to present the image and show the growth, which I think is cool to look at; laying the same number of squares out in lines was an in-the-moment inspiration and shows the pattern of growth more clearly. The kid loved the story and, interestingly, didn't bat an eye at the fact that the characters were represented as little colored squares!

Story #2: Lucy's Square Meals

Lucy, our cat, likes to eat.  She begs and begs us for food.  Today she said, "Give me something to eat, I'm starving.  You haven't fed me in weeks!"  So, we gave her four orange mice. 

Lucy said, "That was a great square meal but I'm still hungry -- give me more to eat!  I'd love if it could be another square meal but even bigger this time, meow!"  So we gave her nine blue chipmunks. 

Of course, that was only a tiny snack for Lucy...[I want things to fall from the sky! the kid interjects]...and she wanted another square meal, bigger than the one before.  So, we got some of the green birds that were flying around up in the sky.  As before, the meal had to be bigger than the last one, so let's count out four birds in a column, but look!  That's not a square, that's a rectangle!  If we add four more, that's still a rectangle...oh look!  Four columns of four green birds in each column make a square that is bigger than the square of blue chipmunks.  How many birds did she eat all together? 

I can't imagine Lucy is still hungry....she IS?!?  What does she want to eat this time?  Okay, a square meal out of cubes of pink cheese.  How many cubes of cheese will she get....?  I think that's enough food for Lucy, don't you?

There was actually quite a bit of buy-in from the kid on this story, especially when I was 'trying' to figure out how to make the green square bigger than the blue one.  She actually leaped in to help her apparently inept mother, ha!

Story #3: {The Family in the Fancy House}

Once there was a family who lived in a fancy house [Meaning the brackets { } used to denote a set].  The family included Lucy (a living cat, red square), Isobel (a living cat, blue square), Mama (green square) and Papa (orange square).  Sometimes they are in different rooms of the house, but no matter where they go in the fancy house they are all still part of the same family. Lucy and Isobel (red and blue squares) are included in the family who lives in the fancy house.

In addition to Lucy and Isobel, there are other kinds of cats in the fancy house as well.  There are soft cats, and china cats, and plastic cats too.  The two living cats in the house (Lucy and Isobel) were also included in a set of 33 soft (living and stuffed) cats...

...and I left it there for the day.  To you, Lucy our cat and Isobel the daughter/cat might not be in the same set, but they are to Iz.  In fact, things got a little rocky when I defined the other soft/stuffed cats as 'not living'.   The set/subsets concept is new to me and, honestly, a little challenging.  The kid and I have lots of conversations about same/similar/different, in a Venn Diagram style, but this seems different somehow.  I actually think it will make more sense when we figure out sets using numbers.  Or maybe it's time to ease in the Cuisenaire rods?

I'm not yet sure what she made of all this math-y storytelling today.  But, the beading/pattern experiment from last week is really bearing fruit this week (lots of spontaneous observation of and creation of patterns) so I'm fairly sure square numbers and sets will show themselves in her play or drawings in one form or another some time soon!

Beading Attributes: Pattern, Color, Shape, Size and...Straws!

My house is a laboratory.  My daughter is the lab rat er, cat.  I'm doing a lot of body-based rhythm and dance this summer with multiple groups of kids ages 6-12 (50-100 a week) but want to balance it out with other representations of pattern, shape and design.  I want whatever we do to have as much choice, challenge, beauty, self-expression and mathematical meaning as possible

I'm trying to figure out how to do all that on a budget.

One of my ideas is a beading project that will work well for both boys and girls in the younger and middle age groups.  I'm thinking about starting with both these books. 

I want the bead patterns to be as simple or complex as the kids require or desire.  I want there to be many possible right answers using a diverse inventory of attributes.  So far that means stiff string, pipe cleaners, spherical wooden beads with multiple colors and sizes, and...straws!!

Yes, I am making my own colorful straw beads.  They're the leftover parts of colorful bendy straws I cut to make this cube: 

 

And these.  

 

And this!

The older kids, incidentally, will be making at least the tetrahedron and the cube.  If they want to do more I plan to have enough materials on hand for that to happen.  There's a nice balance, a nice ecology, to this me thinks, what with the whole straw being used in different ways over the 6-12 age range.  Here's what I've done to make it work:

Make your first cut at the bottom of the bendy part.  The long part of the straw is about six inches, and perfect for constructing the Platonic solids using pipe cleaners as connectors.  With the remaining portion of the straw I cut the bendy part off (it's the part that expands -- in this case, I'm leaving it unexpanded, but the ridges make a nice texture.)  The top straight part, which is at the top, I've cut into half.  You could leave it longer, if you want, but I liked the shorter pieces better.  That's just me, though.

I experimented with some beautiful plastic pony beads as well but, in the end, there's only one attribute -- color.  The resulting design was really not interesting at all and I think even the youngest kid deserves more than one design element.  The wooden beads are wonderful with so many different sizes and colors and I'll keep my eye out for more sales so they can be a choice for everyone.  The straws are wonderful too because they're less than a penny per straw, offer a different/unusual bead shape, with multiple color choices AND a choice of texture! 

I'm happy with the options so far but will keep searching and experimenting.  What other kinds of (inexpensive but beautiful and varied) beads could I use?  I'd love your ideas!

Small Moments of Math

I often write about the times when mathematical inspiration hits but, most of the time, our daily lives are made up of smaller, less dramatic math moments. It's during these lovely transient events where I really get a good glimpse at how my daughter is thinking, mathematically speaking, and how she is applying her understanding in a number of different contexts.

My approach to math exploration at home has been hands-off, necessitated by a child who likes to captain her own ship.  This basically all boils down to the fact that math generally happens in bite-sized pieces around here.  It doesn't mean I am not influencing the process but it does mean I hardly ever make formal plans; instead, I am always looking for new games, thinking about what she might need or want to learn next and also how to introduce new things in a way that has the appearance of being at least 50% her idea.  I also leave stuff lying around to be 'discovered' or engage in my own pursuits, which inevitably leads to some curious inquiries from the wee bystander. 

In addition to all this stealth planning, these days I'm also preparing for some summer arts programming up in the city.  In this case, I'll have three groups of students (ages 6-7, 8-9, 10-12) an hour each day for a week or two at a time.  It'll probably be harder to orchestrate the kinds of gorgeous, in-the-moment discoveries we have at home in a windowless basement room of a run-down church building with lots of kids and no air conditioning, but I'm going to try.

Here are some examples of the small moments of math that have been happening around here lately, including some of my thinking and experimenting about the summer.  I'll start it off with half a lunch which actually started as a whole lunch, but when the kid saw the design with a whole tomato in the center she insisted in cutting it in half before eating:

A spontaneous, self-initiated design from the kid after breakfast, utilizing her rock and mineral collection.  I guess she understands reflection symmetry after all!

The kid was designing/drawing mazes while we were at the library but then got interested in a 2001 NCTM magazine on mathematics and culture I was reading, specifically an article was about a game played by the Fulbe children in Cameroon.  The game is essentially about creating designs by drawing line segments through a series of dots lined up in a grid. 

After a quick look my kid drew her own version  ("The same design equals two different things!") and then also started drawing the library ceiling (square tiles and long rectangles of lights).  "There are patterns everywhere, Mama...Oh, look a book about cats!"  And so it goes...

It's clear the kid likes functions, but the function machine she built is totally off limits to well-meaning but meddlesome adults.  The photo below is a bit blurry, but you'll see she really just likes the process of figuring out the equations using the same rule and different inputs.  The minute I said something about looking for a pattern in the answers the whole thing went south.  I've learned the hard way that, in this particular instance, there is no room for me to ask any questions about anything whatsoever.  However, I'm confident that she'll figure it out at some point like she did when made peace with the times tables last week.

She'd been multiplying on her fingers but finally realized how much time and effort that takes. "The multiplication table is like sight words," she told me, "you just have to memorize them."   Reading a book called Piaget for Teachers from the 1970's makes me realize that my approach is probably on target given her developmental age (newly operational) -- telling her that particular 'fact' myself will not help her thinking process.  The way to help her thinking process is, basically, to just let her figure it out through a variety of experiences (cleverly orchestrated in secrecy by her scheming mother, ha!).  

I made the kid a puzzle/game last week called "Make Them the Same" using elements of line, shape, design and color.  I didn't take a picture of the starting designs but if I had you would have seen that both designs in each pair were incomplete in different ways -- you had to compare the two to make them both a complete design.  I'm still playing around with this idea.  This version was too easy and my second try was too hard but I think I'll be able to make it work.  I really want to have it figured out by summer.

I spent a couple hours over the weekend looking through the math book lists at Love2Learn2Day and Living Math.  The theme for my summer program is tentatively called "What can you do with a square?" (or a pattern unit, I'm not sure which yet) and, in Three Pigs, One Wolf, Seven Shapes the answer is obviously 'tell a story with it'!  I made a sheet the kids could use while listening to the story, but when my kid discovered it she tried to put the tans on the outlines which are not to scale.  For the youngers I may have to give them each one or two real-size tan puzzle sheets instead (which I found at Mathwire), and use this sheet for the olders.

Chessboards are made up of lots of squares.  When I read this book to my daughter a few days ago to make sure it was something I wanted to use in the summer, I had the thought that I could make paper chess boards and give each kid a baggie of rice to follow along with the story. 

It might get messy with all that rice, but part of my thinking is that Math in Your Feet is about order and structure and pattern.  A chess board is so very orderly and there's a growing pattern in the story (doubling, my current favorite topic).  And, having the real rice at hand will hopefully make the story, well, more real.  Also, I'm thinking about doing more with paper quilts and tilings and the image of the chessboard would be great to refer back to when bringing up tessellations with the older kids.

Not surprisingly, more small math moments have happened during the in between times of writing and editing this post.  It seems that little jewels of math are all around us just waiting to be discovered at unexpected moments!  

As a postscript of sorts, I want to say that Maria Droujkova at Natural Math has been a huge inspiration to me over the last year and one of the main reasons I have learned to see math all around me.  I would not have grown my 'math eyes' if not for her Natural Math forum or reading about the math clubs and Math Treks she has orchestrated as well as the new Moebius Noodles project for young children. 

Real Money, Real Math

Spring cleaning unearthed a dress my mother made for me when I was seven and a bin of of forgotten pennies.

Me: You can have 'em but you'll need to count them out to see how many there are.
Kid, now wearing the dress (it fits!): How should I count them? 
Me: How about by twos?

At 100:
Kid: I need a helper!
Me: Oh, I'm too busy making dinner.  You can do it.

At the second 200:

Kid: I need an assistant!

Me: Nope, sorry.  I'm still making dinner.  Papa is busy vacuuming.

Kid: This is tiresome....Go look for a two dollar bill.

Me: Count them all and then I'll get out the paper money.

Kid: Do you know a quicker way?  By threes?

Me: How about by fives?

Kid: Okay...[counts 100 pennies out by fives]...Can I have a three dollar bill now?

At 400:
Kid: How about a four dollar bill? 
Me: Are you kidding?  There's no such thing, you goof!

Kid: I really, really want this to go faster....I know, you count out the pennies into tens and I'll count them up from there."
Me: (Nice try, kid.) Dinner's almost ready...

Done!  $4.73.  She made it an even five dollars with change from her wallet.

"I have enough to go buy ice cream!"  Well...
"It's my money, I can decide what I want to do with it!"  Well...

"I know...lipstick!"

I know it seems like this was all about the counting, but from my vantage point it was really more about strategy and predictions.  How much money am I going to end up with?  What is the quickest way to count it all up?  Can I get someone to help me?  How much do I have all together?  How much can five dollars buy? 

Real money means real life economics.  Real money means developing strategies and making real choices.  How much do I have?  Do I really need/want one more $1.00 stuffed kitty from Goodwill, or that china figurine, or do I want to wait and save my money for something else?  How much more will I need to earn/save before I can buy x, y or z?   Interacting with the realities of real money is another great example of real power.

Hexagon Poetry

Can't remember why I got out the pattern blocks; it's been a while.  'How many different ways can I make a hexagon?' I asked myself.  The kid joined in at some point and then we left it, on to other things.

Later in the day I found this. 

It says: "Each ray is different with the days."

This post is linked to Saturday's Artist at Ordinary Life Magic. :-)

Totally Territorial: Cats, Maps, Area & Multiplication

I know it doesn't look like much, but there was a battle going on.  We rolled the dice (one six sided, the other twelve sided) and staked out our territory.  We're warrior cats, you know.  You didn't?  Have you read those cat clan books?  My kid hasn't -- too many kitties getting hurt makes her crazy -- but her best friend has.  They 'play' those stories every time they meet.  You should see them in their 'warrior training'.  It's crazy.

The minute we started rolling the dice I knew I needed to give this activity a special spin.  We had just finished a 'make fifteen' game where the last person to fill in a combination of fifteen wins.  There was no real strategy involved, hence her lack of buy-in.  In fact, she had been thoroughly unimpressed with it and didn't appear too eager to try something else. 

This new activity was basically built on the same model.  The goal was to see, after lots of dice rolling, who would come out with the most area colored in after the whole page was covered.  (For the life of me I cannot remember where I got this idea, but I know it was from a recent blog post.  If it's you I apologize for my lack of recall -- and please let me know so I can credit you!) 

Since success in this game was mostly left to chance, I knew I had to think quickly. "Let's stake out our clan territories and see who gets control of the most area!" I exclaimed.  Needless to say that's all the motivation she needed.  She decided almost immediately that each new piece of territory had some kind of real-life correspondence -- campus (the scene of our recent Tana Hoban adventure), our favorite hangouts, she wanted them all.  I pointed out that she could be strategic with where she filled in her boxes and, if you look at the first photo again, you'll see that she's surrounded parts of my blue territory so that I have to cross her land to get to other parts of mine.  Nice.

Then I realized that since she was thinking of real places we could probably do the same game on a real map!  Here's what we did later in the day...

We got a free map of our fair city at the bike shop.

With her watching, I took the yard stick and marked off a 1/2" grid over the entire city.  The grid was not to scale but was consistent across the entire map.  The area unit was 'one square'.

We looked at the map and found our landmarks -- hangouts, parks, campus, lakes, our neighborhood -- and at the map's key to find the roads, schools, bike and walking trails, etc.  Then we started strategizing about which areas we'd most like to 'have control over'.

We rolled the twelve sided die first to get our length (or width, whichever, depending on where it was best used strategically).  In those boxes we placed a dot using a permanent marker.  (We wouldn't be able to read the map if we colored everything over.)  Then we drew a line to show the border of that 'territory'.  Using the six-sided die we rolled the other side and figured out the total area of our newly acquired land.

I used the words area, length, and width as well as north, south, east and west during our conversations about what parts of the city she and I wanted to acquire and why.  Some of our reasons were strategic and others were more personal -- I got a lovely lake with the hiking trails, for example, and she got the big park with two awesome playgrounds.  I also modeled a lot of multiplicative thinking.  Even though we did use skip counting to compute the total area, I would point out groupings: "How many rows of seven do you need to fill in?  Second row, two sevens are fourteen.  Third row, three sevens are 21..." 

The strength of this game is that we know our city pretty well.  We've been to a lot of the parks, we move through town frequently on foot, on bikes, on buses, and in the car.  Our personal landmarks relate to where our friends live and work and go to school.  It really is our territory which is why the map was so real to her; it made sense to her despite being an abstract 2D visual representation.  Plus, the cat clan narrative is one she knows intimately, further increasing the emotional connection to the material.  Tomorrow we might play the game again and total up the red and blue areas to see who has the most.  But then again, given the engagement level today, it almost doesn't matter.  I think we've both already won.