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!

Math: A Secret Code?

It was free day at the library book sale.  The kid went directly for the boxes on the floor at the back of the room and started searching for old books.  She's not really impressed unless it's something that is at least 100 years old.  In the past she's found some real gems, like the full-color world atlas from the 1920's, and a 125 year speller which she used to teach herself to read.  Today she hit the jackpot when she found this, from 1918:

It even had a letter hidden in its pages.  It's from one high school (maybe college) kid to another about how he made the basketball team and has finally found time to send his friend this book.  Given that we live in Indiana (you know, Hoosier basketball) that's a pretty exciting thing to find in an old book.

But the funniest thing happened when I finished celebrating her find and went back to looking for other books. She was sitting under a table, flipping through the book, and suddenly exclaimed:

"Look, Mama!  It's a secret code!"

"Cool!  Let me look..."

If you've been following this blog for a while you'll know my seven year old is basically fluent in mental sums and differences and visual math for her age, but has only experienced the symbolic realm in a cursory way.  It's been working for us so far.  Yesterday she 'discovered' a "fourthagon" in our lunch table (short video here).  And, today she was trying to convince me to top off her savings for a set of walkie-talkies.  She's got $7.68 saved already and was imagining a set might cost around $10.   We were out for a walk when suddenly she stopped stock still and turned inward.  Then she said,

"Mama, if you could give me $2.33 I'd be able to buy those walkie-talkies."

"No," I said, "I think this is something you'll need to save for if you really want it."  Secretly I was completely surprised and impressed that she had figured that all out in her head within one penny. 

I was tickled at her reaction to that page in the Solid Geometry book too.  Without too much giggling I said:

"You're right, that is sort of a secret code.  It's called mathematical notation.  After you understand the math concepts you'll learn the way to write them down -- it's the language of math."

I'm still giggling about the whole secret code thing -- it brings to mind a secret network of mathematical spies, or some sort of secret underground club, communicating with only the most esoteric of notation, an unbreakable code that stands for all time...[cue dramatic music]. 

I think she nailed it.  Out of the mouths of babes, right?  What do you think? 

Lunch Table Math Discovery (Video Conversation)

Listen as my seven year old shares her discoveries about our lunch table, one fine October afternoon:



This moment brought to you by our Paul Salomon-inspired star inquiry.  I love the whole sequence because it captures the essence of hundreds of other math conversations and discoveries she and I have had together over the last year.  They're small moments of inspiration, identification, conversation and observation that are being layered, pieced and added together into something incredible -- comprehension. 

Fourth-agon.  I love it.

Teachers in Motion! (Video)

I was in Kansas last week working with 38 physical education teachers (both elementary and middle school).  Yes, I know it was massive, but it was a lot of fun!  The mostly hands-on, active morning was full of the core Math in Your Feet lessons: creating patterns and rhythm in our feet using my Jump Patterns tool and then combining and transforming those patterns. 

PE teachers are used to organizing moving bodies (their own and those of their students) but they are not as familiar with math curriculum, especially in relation to movement work.  It was gratifying to see them open up to the experience of both a new movement/art form and the idea of math outside the context of a textbook or a checkbook.  This program is truly mathematical 'activity'!

Here is a quick video clip of some of the the teachers' creative work.  Just like the kids do, they paired up to work through the Jump Patterns movement variables until they had created a four-beat pattern they could repeat, dance in unison (congruently) with their partner and, most importantly, be happy about.

It's a little hard to hear but the teacher in orange is saying "We combined and we reflected...alternating the reflection... [I think that's what she said -- I didn't hear her at the time either, and I guess I was expecting that the two folks on the inside would be the original pattern, and the two on the outside would reflect that pattern.  That's why I was sort of confused on tape!]  Here's what they did -- remember it's two individual four beat patterns that have been combined to make eight beats:



What was also surprising and interesting about this pattern is that the second four beats could not be reflected because it already was a reflection and congruent at the same time.  Regardless, the team wanted to make it different somehow, so that's why when the original pattern jumps forward, the 'reflection' jumps back.  After I turned off the camera we had a good laugh about how they had used the math to inform their creative work but, ultimately, it's a choreographic process and you absolutely have to stay true to the dance aesthetic as well.

That, and sometimes ignoring your students (as long as they're working on something) produces the most wonderful results!

p.s. And, if you want more  footage, it might be a while, but I have finally (!) accessed footage of students in Math in Your Feet that I've been collecting for the last couple years.  A short video summary of the program and a longer curriculum guide with an accompanying video is in the works.  It's a massive project, but I am super excited about finally being able to share the program this way.  In the meantime you can read about the program in detail by going here.

p.p.s.  We've been have a whole bunch of fun over at the Math in Your Feet Facebook page.  Why not join us?

A Small Moment of Multiplication

Last school year I made my peace with numbers, but these days I am actually starting to feel a bit of mastery, albeit at a very elementary level.  It's a long story that starts with the seeds of math anxiety that were sown as a five year old trying to count and match bumblebees on a worksheet. Over the years, it turns out that geometry is really my thing, but numbers, for the most part, give me a general and unwelcome sense of dis-ease.  That is, until I became responsible for my daughter's math education.

Over the last year both of us have increased our capacity for mental  math, which basically means we are now able to see numbers essentially as flexible, interesting objects that shrink and expand as needed.

We have developed our addition and subtraction skills mostly through games and other play and narrative approaches.  For first grade we played hours and hours of UNO and many, many rounds of Shut the Box.  We rolled miles of dice.  We counted, earned, saved, and spent mountains of coins.  My mantra was 'find your tens' and for a long time, it seemed, tens were hard for her...until one day they weren't.  Now she's a pro at double and sometimes triple digit mental addition, and subtraction is not far behind.

She and I have different approaches as we work toward an answer, but it's frankly just awe inspiring that I can add two, three or even four 2-digit numbers before reaching for the calculator.  And my kid?  I wish I had had her skills at seven -- I would have probably loved math decades sooner if I had actually been supported in developing the numeracy she already has. 

Anyhow, this fall I tried to bump things up a little with a math version of the card came War using multiple cards to find sums and differences.  After a few weeks of this it's become obvious that addition/subtraction are not a big enough challenge for her any more and so I've been searching and searching for something else.  The kid has some basic understanding of multiplication and division through a lot of conversations and play with halves and doubles but that's not quite enough any more either.  So I wondered if there was a multiplication game out there that wasn't just about memorizing times tables...

...and then Denise at Let's Play Math wrote a fabulous post about multiplication including a section on multiplication models which included a card game she created!  (It's a free pdf download and about half-way into the post.) You can play the game in the form of Concentration, Go Fish or Rummy.  We spent some time sorting out and getting familiar with the different models (sets, measurement and array) and, after we played Go Fish a few times, all of a sudden the light bulb went off in my head about how I can bring those concepts of into our daily lives.

I may be more impressed with myself than need be, but I mean, just listen to me as I walk my daughter through using the juicer this morning -- I actually sound like I understand what I'm talking about!

Me:  Do you see the 4 on the measuring cup?  That means four ounces.  You're going to juice four ounces which is also one-fourth of cup.  When the juice gets to that line, you'll be done.

Juicing commences.

Me: Okay, you're done.

Kid: That's not enough! [As I pour it into a glass that is narrower than the measuring cup.] Oh, wait.  It looks like a lot less in the measuring cup.

Me: Why do you think that is?

Kid: Well, it spreads out more in the cup.

Me: Yep.  Hey, if you're drinking four ounces of juice and that is one-fourth of a cup, how many ounces make one whole cup?

Kid drinks and thinks.

Me: One-fourth is one of four parts.  Each part is four ounces.

Kid: [Done with juice and looking at her fingers.  She pulls up four fingers and starts skip counting...] Four, eight, twelve, sixteen!!

Me: Sixteen what?

Kid: Ounces.

Me: Excellent!

I feel successful just being able to have that kind of conversation.  And really, it's these kind of small steps that ultimately made last year's math work so successful.  So, score one for a small moment of multiplication!  We're on our way -- and I can't wait to get the Multiplication Models poster I ordered off of Amazon from Maria Droujkova and Natural Math.  It'll be great to have it around us as we find more small moments just like this one.