Sidewalk Math: Circle Fractions, Division and Multiples

Okay, so we've walked past these two covers about a million times but today I saw them in a completely different light.

This winter/spring we've been doing a LOT with number multiples and conceptualizing multiplication and division.  Last week my mind moved toward the inevitable: fractions.  Although shivers go down my spine every time I think about fractions I'm still resolved to figure it all out for the sake of my seven year old, if not myself.  It's been sitting in the back of my mind so I guess that's why this cover caught my eye and brought me to a dead stop.

Can you see it?  Fractions and multiples!

And, a little further on, this beauty: an 8-star and some fractions!

What I really want to know is who designs these?  I want that job.  To see some other cool round things we found on a walk last spring, go read my post Channeling Tana Hoban: Juxtaposition Edition.   Now that was one amazing day for circles!

Seeing Numbers

I've been thinking on and off about ways to help my seven year old daughter come to know, appreciate, and maybe even love numbers.  Geometry is easy for both of us because it's visual.  But because my math schooling was of the memorize and drill variety I have really had to stretch myself in order to make my daughter's early experiences with numbers more meaningful than mine were.  As it turns out, I'm learning a lot in the process and, as a bonus, I am finding that numbers are quickly becoming an unexpected but welcome gang of amusing little friends. 

Since we're working on conceptualizing multiplication and division these days through various explorations, I thought it might be fun to make 'number cards' to visualize one number at a time.  Using coins (and clarifying their intent as reconciled in my recent post on units: When is a 10 Not a 10?) we looked at all the different ways to split 25 and found lots of remainders.  I suggested 60 after we read that base 60 was popular early on because it can be divided in so many ways. 

As you can see, the results are less than inspiring and it's not clear to me that the child really learned anything more from making the cards in addition to the array work with the coins.

These cards are still taped up hopefully on the wall, but I had basically abandoned the project...until I saw this fantastic post  from an international school in France that re-imagined factor trees in the most awesome way.  Isn't this the most lovely factor tree you've ever seen?  And there are more -- all as beautiful as this one.  Their project in France has inspired us here at home and has become a perfect and flexible solution for our nascent number wall idea:

I showed the girl the post that inspired me and she was entranced and eager to get started.  She is doing well with hands-on and mental multiplication and division, but factoring as an activity is new to both of us and so I decided 20 would be a nice easy number to start with. 

We took out the Cuisenaire rods and she found all the factors of 20 herself.  She double checked every rod from 1 to 10 and noticed a lot during the process, skip counting as she went.  "If I can use fours, I can use twos, too!"  "Threes...no that makes twenty one...."  

And then we wrote them out as equations and sketched a preliminary tree.  And here is where this activity became as important for me as it was for her.  I do not recall in all of my fourteen years of school mathematics (including those awful matching bumblebees in kindergarten, my tenth grade algebra D, my tenth grade geometry A, and a deliriously confusing semester of college algebra) ever learning about factors.  Ever.  So, it took me a minute, but I made it work.  (Oh, the things you do for your children!)

Then we got to work.  We drew and we drew and she whistled a happy tune (literally) for over twenty minutes.  She would not let me look at her work.  She was in heaven.  This is what she finally unveiled to me:

To me, it's glorious.  Even though it's not a factor tree, per se, it shows so much of how her 7 year old self is thinking about numbers, including an emerging understanding of factors.  All the numbers she wrote are obscured by the coloring, so I had to ask her about them.  At its base  are the numbers 10, 10 and 20.  Notice the overall symmetry of the tree.  The flowers at the base are 5+5+5+5 (do you see the 'plus' sign written on the trunk?) and, even better, the colors are inverted in each pair.  Moving up the tree, the next two longer branches on the outside are 2 and 10 (so 10x2) and the two small branches inside them are both 10 (so 10+10) and the crowning flower is 20!

This tree motif is doing so much more for her visualization of numbers than those little cards ever could!!  I'm so excited.

And, as a little bridge to future trees, after she walked me through her drawing I showed her mine.  I asked her what she noticed and what the differences were between the trees. 

She noticed a lot, but not the numbers so I pointed her attention to the fact that although each tree split in different ways (10 and 2, 4 and 5), they still had the same prime flowers -- two 2s and a 3. (Did you notice I made each prime it's own color?  And I wonder if there are other ways to bring out the number properties beside just writing the numeral?)

There is no need for us to rush into full-out factoring, but it is an interesting exploration and variation on the division we've been doing.  And, the introduction of another type of number, the primes, is an interesting development as well.  I think this is an approach that we will be able to use time and time again: create, observe, discuss, posit, repeat. 

Let's see: trees as metaphor.  I've seen fractal trees, factor trees, and trees whose branches are the positive numbers and roots the negative ones.  There is so much potential in this kind of image/metaphor for number explorations through all elementary grades.  And, you do know that part of this approach is giving kids an empty piece of paper, some awesome pens and letting them create their own trees from the ground up, right?

I know one thing for sure: our number wall is going to be gorgeous!!  [Addendum: Here's what happened the next day.]

Half A Cake

"Hey, tomorrow's your half birthday!  I know, let's make a cake to celebrate!"

Turns out the recipe called for 1 1/2 cups of flour.  Since our cake was ingredient free due to multiple food allergies, we used even parts of two gluten free flours.

"Okay so we need one and one half cups of flour.  How many halves in a whole?"

"Two!"

"Okay, so here is the 1/2 cup measuring cup.  One scoop of the buckwheat, one of the rice... Now, we need 1/2 cup more flour, using both our flours.  What's half of 1/2?"

"One-fourth!"

"How'd you know that?"

"Oh, I just knew..."
..............
Later in the morning, at the library.

"Hey look Mama!  The [really big, red tiled] square is made up of smaller squares!"

"Very cool.  How many small squares make up the big square?"

"Three, six...nine!"

"So, you wanna hear something interesting?  When you hear someone call a number 'square' that's what it means.  Look, nine squares literally make a 'square' number!  Isn't that cool?"

"Mmmm hmmm..."
.............
After lunch.

"Hey, it's time for cake!"

"YAY!!"

"But we can only eat half.  How would you cut it in half?"

[Girl gestures a line from the middle of one side to the middle of the opposite/parallel side.]

"Well, that's one way.  How else could you do it?"

[Girl pauses and then gestures diagonally from corner to corner.]

"Excellent!  But I have one more idea.  How many squares made up the big square at the library today?"

"Nine."

"So, I"m going to divide the cake into nine parts.  Here's one for you, and one for me. We'll have more later, after dinner."
...........
After dinner, and after one more piece each for the child and myself.  (Papa doesn't like chocolate, so no cake for him!)

"Mama, can I have a little more cake?"

"Hmmm, maybe.  We're only going to eat half of the cake -- let's see if there's any left....  How many pieces did we have to start?"

"Nine."

"What's half of nine?"

"Four and a half."

"So, together you and I have eaten four pieces.  There's half a piece left and you can have it!"

Ladies and gentlemen, I present to you: half a cake!

Morning Math

I'm really enjoying math in the morning.

There's something fresh and new and hopeful about mornings lately and, even though I'm not doing anything ground breaking, I'm really enjoying how connected everything seems to be these days, mathematically speaking. 

In mid-October I mapped out a basic math plan when it was clear the girl (now 7.5) needed and wanted more and different kinds of math challenges.  I decided to call the plan 'algebra' because, from what I've read, algebra combines a number of skills and concepts that we have to start learning anyhow at the primary level.  And, because the girl has often balked when I introduce new math stuff, calling it algebra motivates her to give it a try; 'algebra' is a big kid skill, and she really, really wants to be big.  

So, in the morning it's been fun to open my math folders and give my lovely child her choice of math activities.

Solve for x (conceptualizing equality and sameness, sums and differences) or math card games (3 digit mental sums and differences)?

Christmas themed beginner Sudoku puzzles or Factor Dominoes?

Growing patterns or Bean Soup (fractions, multiplication, division)?



















 "What's division, Mama?"

"You know, like when you wanted to see if you were halfway through your reader.  There were 96 pages and you figured out in your head that half of 90 was 45 and then you..."

"Oh, yeah."

Later that morning at the science museum she built this:

"Hey Mama, look!  I got the water to cover the whole area!"

"Cool!  That's an example of division, too.  The water is being distributed evenly across the table."

So, here I am, trying to connect our morning math time to the bigger math picture that I'm constructing in my brain.  Like algebra, as I've mentioned.  What is algebra???  I did some algebra when I was in high school, but failed literally and horribly (although I aced geometry).  But why let that stop me?  After some research I decided that the concepts of balance and sameness, solving for unknown quantities, and growing patterns were all pretty darn interesting and relevant whatever we called it and away we went.

We began by building and analyzing growing patterns from some pattern starters I found.  Here's one she came up with on her own:

And, when we do the occasional worksheet (the one below is skip counting/multiplication) I look for ways to extend the activity forward, even if I'm not completely sure I'm right.  The sheet below was for figuring out numbers of feathers on different numbers of hats, legs on rabbits, and petals on flowers.  She got the skip counting patterns easily, so I described the data another way by saying:

"The number of hats you have is multiplied by three feathers per hat which gives you the total number of feathers," while writing h x 3 = f.  The next two examples she talked herself through the whole thing and wrote it down.   But maybe it would have been better to say, "The total number of feathers you have is equal to the total number of hats multiplied by 3"??  Writing all this out has me thinking of another way I could have done that, but I think sometimes its okay for me to muddle through stuff like this.

And, if skip counting is coming easily, why not challenge her to fill in the chart backwards instead? Ooooh, that was cause for consternation, but she had been saying "This is easy, this is easy, " all morning.  I know for sure she needs more experience with subtraction, I know being able to invert a procedure or concept is an important skill, and I also know that if something is too challenging she'll just give up.  So, here was an opportunity for deepening her experience with a skip counting chart and providing just the right amount of challenge at the same time. 

Filling in the charts backwards really made her think.  That's what she wanted, right?

Anyhow, it's nice to be entering a stage with her where we can sit down and 'do math' outright instead of trying to leave it around the house for her to discover, although I'm keeping that strategy in my kip for now!  More than anything, it's been lovely doing math, in the morning, with the winter sunlight streaming in, with a child who is finally (finally!) allowing herself to be interested and excited about exploring this mysterious and wonderful subject.

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!   

A Game of Her Own: Discovering Division

This was one of those moments where you ignore the phone, turn off the stove, and just pay attention.

I was making a grid for a sight word bingo game, like the one above except still blank. My kid looked the blank chart over and said,

"I'm going to make a game board like that, and I'm going make it a math game."

Curious, I replied, "Okay. What's your game?"

"You divide an even number in half and then add it to another number."

It didn't actually turn out the way she had described, but she did make her own math game!  How cool is that!?  I should say that we both knew going into this that we were 'in development' with the game, that we really wouldn't know how to play it until we were done. 

The game is a bit convoluted but patterns did emerge.  She dictated the directions to me which I've put below.  I'll fill in the details later in the post.

Supplies:
really flat, stiff paper
permanent marker
ruler
pencil
pennies
drawing stuff of all kinds like crayons or something
that's about it

Step One: Make the Grid
Have the mama make the grid, 25 squares, five rows, five columns.

Step Two: How to Play the Game
Write all even numbers in the grid starting with two.

[After filling it about half way she exclaimed, "Hey, this is interesting!" and pointed out that each column included the same numeral in the one's place.]

Step Three:
Pick any number on the grid.  Use the pennies to divide that number in half.  Here's an example: One half of 28 is 14.  Color them both the same color on the grid because we know they're related.

The Kid's Final Words:
By playing this game we'll learn how to multiplicate (sic!) and we'll also learn our two times tables.

The Mama Adds:
By playing this game you'll also get a sense of 'even-ness' and 'double-ness' and 'half-ness' not to mention support your emergent abilities in division.

So, as you might imagine, this was a verrry interesting process to watch.  I should say that it was probably inspired by a game of tic tac 10 and tic tac 15 we played on Sunday.  She used the even numbers for those games and it probably reminded her that an even number can be divided into two equal parts.  That's the interesting thing -- she knew that in concept but not really in practice, until she made up this game.

The first number she chose was 18.  What's half of 18?  She couldn't do it mentally, so I suggested some Cuisenaire rods which she immediately deemed 'too confusing'.  I pulled out the change jar and she got to counting out the pennies, which turned out to be a perfect manipulative.

Counting out eighteen pennies seemed a little confusing at first, for some reason.  I casually mentioned that it helps me to put things in groups and then skip count, and that it's easy to count by fives.  One slim pause and she took the bait. 

What really fascinated me was watching her trying to figure out how to divide the pennies into two equal groups.  First she spread out the pile of eighteen pennies into something that approximated two separate groups.  From there she echoed the strategy I had suggested, above, and put the pennies into rows, some four pennies long, some five.  I can't remember how the rest of it played out, but at some point the rows were all the same amount and there were two extra pennies without a clear home.  For some reason she didn't (couldn't?) just split them up, one to each group.

I think what she was looking for was some 'even-ness' in the rows as well as the total amount in each group.  She moved the pennies around until, voila!  She had figured it out visually but we also counted just to make sure that we had even amounts in each pile.

I suggested writing the answer in the corner of the 18 box, just so she'd remember, and she continued to make notes in this manner for the rest of the game.  We looked but couldn't find '9' anywhere else on the chart.  Why?  Because there are no odd numbers on the even number chart, silly!  Time to pick another number! 

This time she chose 28.  Since we still had eighteen pennies out, I asked how many more pennies she needed to make 28.  With the answer she added the extra pennies into the mix and proceeded to divide them into two groups more easily than before.  I'd show you a picture of the process but she told me to stop taking pictures!  It was almost dinner time and she was focusing so hard, she said the camera distracted her.

Anyhow, half of 28 is fourteen, which we did find on the chart.  I suggested coloring the 14 in and she suggested coloring both numbers and using the same color so we'd know which numbers were 'related'. 

And then we were both worn out, it was time to make dinner and so we left it there for the day.  I wasn't sure if she'd want to keep going the next day, but she did.  On day two of making/playing this game there was a really interesting progression in her ability to figure out 'half' of most numbers on the chart.

The first number she chose on day two, 36, required the use of pennies to figure out and, again, she wanted the pennies to line up evenly in their rows.  While placing all 36 pennies in rows of five she said, "I think there's going to be an extra one..."  She played around with lining them up so that all the rows were the same length and, luckily, these are even numbers, right?  So it did eventually work out that way, as you can see in the picture above. 

For the next number I suggested 20 and encouraged her to figure out the half in her head.  Easy!  It was at this point that things really took off.  She was able to figure out half of 48, 24, 12 and 6 all in her head by her own strategy of dividing the tens and then the ones 'in half', and then adding the resulting ten(s) and ones together.  Pretty cool!

Then there was the 32, 16, 8, 4, 2 progression.  She had just figured out half of 16 and I wondered what would happen if she went the other way and doubled 16.  She went immediately to pencil and paper (seen above) for this which is interesting since she usually sticks to mental math and/or her fingers.  She didn't complete the sums with 'regrouping' (something she's not learned yet) and instead used the written numerals to help her thinking, something that sounded like "Okay, there are two tens and...[thinking]...thirty two!"  

All told, she used four strategies in this game to help her figure out half and/or double of the even numbers: manipulating pennies, dividing in her head by addressing tens and then ones, adding (sort of) on paper, and accessing basic addition and multiplication math facts she's internalized already.

The colors for the squares in the chart are for numbers that are 'related' as she termed it.  2, 4, 8, 16 and 32 are all pink because they are a sequence of doubling.  6, 12, 24 and 48 are all red for the same reason.  We did a final check of the squares that had not yet been colored in.  Turns out halving each created an odd number which, obviously, were not on the chart.

She was so excited and proud about making her own game.  As for me, I'm marvelling at how beautiful it is when a child literally constructs her very own understanding of math.  Neither of us really knew where the process of creating this game was going to take us, but it was obviously a worthwhile journey.  Thanks for sharing it with us!