math

#notabookstudy: Constructing Fractions

OK you guys!  I am so excited about my math class on Friday.  Perhaps too excited??  IMPOSSIBLE!

First, I’ve been thinking a lot more about the learning that takes place when students develop a model on their own.
Second, last year I worked a lot on the open number line with my grade 3/4 class.  I thought we were doing so well!  Then we started working on fractions and they didn’t know where 1/2 fit on the number line.  They could identify 1/2 of a set, or half a whole, but 1/2 of a distance was too much. I was going to say “at first”, but I never really felt like they owned that piece of information.  They just sort of borrowed it from me for a while and then silently put it back on the shelf, I think.  Though this is giving them some credit for putting things back where they found them, and that might also be misplaced. So…this year I wanted to add fraction of a distance to our study of fractions.
Today I wanted to start talking about fractions.  Yes, this is a bit last minute.  (I’m not going to tell you the whole story of why this happened.)
So how to develop the context and ask everyone to develop their own model?
Typically, I would gather the class, show them a half, and them let them go make some halves. Then we’d do fourths and thirds, and then we’d try out other things that make kids happy like 5ths, 6ths, etc.   I would do fractions of a set one day, then fractions of a whole another day, then practice them a variety of ways – sometimes alone and sometimes side by side. Then we’d make some connections to half of dollar, or half of an hour, and then we’d be done with fractions. We are a grade 2/3, so equivalent fractions would show up in there, and so would mixed numbers, but that is not our focus.
But new-me wanted them to develop more of this on their own.  All of it, I hope.
Students in my class sit at tables.  There are 4 of tables, so 4 groups.  I put colour tiles on one table, pattern blocks on another, LEGO on the third, and snap cubes on the last. I told them to go play with the blocks for 5 minutes, and then we’d start.  It’s May, and we’ve used the math tools a lot, but they seem to still need to just play with them for a few minutes every time.  If you can’t beat them, meet them where they are, I always say.  Or at least I say that sometimes.
After 5 minutes, I had them all join me on the carpet.  I wrote 1/2 on the board. (I had 1 on top of 2) “What is that?” I asked.  I got a variety of answers.  The 4th or 5th student called it “half” but others called it 1 divided by 2, 1 out of 2, a 1 and a 2, and 1 = 2. I decided to give it to them.  I wrote “half” under the fraction.  “Oh, half, right,” a few responded.  “Now, go to your table and use the tools I put out for you to show me half.”  Away they went.
Here are some responses:
I was surprised at the number of students who weren’t sure what to do.
One student had this:
img_3432
I asked, “Is this half?”
She said, “I’m not sure.”
I turned to her neighbour who had this:
img_3431
I asked, “Is this half?”
He said, “Yes.”
I asked, “Can you explain to (her) how you know it is half?”
This was really interesting.  He kept saying, “Well, it’s half. See?  It’s half.”  Then after a few repetitions for this, he used his hands to separate the two halves and said, “See?  Half are here and half are here.”  He’d separated them by colour.  He wasn’t sure how to explain it, even though he knew he had half.
I asked her if she could see it, and she could.  So I challenged her to make her own.
At the pattern block table, I took this picture:
img_3434
“No,” the student said.  “You have to take the picture from above.  Then people can see that half is  yellow and half is red.”
“What about this one?”  I pointed to 2 red trapezoids.
“It’s cut in half by that line.”
img_3435
Here was a very interesting one:
img_3430
When I asked him to explain to the class, he said, “Well I have 4 blocks.  2 are over here, 2 are over here, so half are in each place.”  I was really surprised by this because I thought he was showing me two models: one half orange, half red, and another that was 2 halves making a whole. I explained this to the class and they got it both ways.
One of the people at the connector cubes table had this example:
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I use the Show Me app and the Apple TV to project pictures during an activity like this so we can congress what has taken place.  Sometimes I write within the app, and sometimes, like in this picture, I write directly on the whiteboard.
He had set it up two ways. First he said, “Half are here and half are there.” Then he changed it and said, “I mean half are yellow and half are something else.”  I think he was surprised when I said that each was a different way to show half and he’d been write two times with two different answers.  They are used to me saying this kind of thing, but I think he thought he was wrong the first time.
After we congressed this, I sent them back to their tables.  “Now that you have seen some other models, go show me 5 ways to make half.”  And they did.  I mean, AND THEY DID!!!!!
We had to skip math on Thursday because of an assembly, so I decided to do double math today. I told them that after recess I’d have different blocks on their tables and they could use a new tool to show me half.  They came in and got right to work.  That’s a lie.  They got to work eventually.  But they all showed me half in a few ways with a new tool.  Photographic evidence:
One child had this, which she explained like this, “When my mom says, ‘Eat half.’ this is how much I would eat.”  I was happy to see that connection to fractions in her real life, even though she didn’t quite understand that each of her piles should have an equal amount in them.  In reality, who counts every exact pea?  But I think that might start happening, and I should probably call her mom and apologize in advance.
img_3444
I then read them a great book called, “The Cookie Fiasco” by Dan Santant.  It’s part of the “Elephant and Piggie Read” series.  It’s funny!  Four friends, 3 cookies, how can this be solved? I read up to the part where the friends are completely befuddled about how to solve this problem.  Then I gave them 3 square cookies made of paper, and asked them to figure out what the friends could do.  In the book, the nervous hippo had already broken the cookies in half, but there still weren’t enough cookies for everyone.  So, all of my people cut their “cookies” in half. Most of them cut them again, and within a few minutes everyone had figure this out.
Congress:  “What did you find?”
Everyone agreed that everyone got 3 pieces of cookie.  I asked, “Ok, so how much cookie did they get.”  This took some discussion.  I had to reassemble them into wholes, then move them around until someone said, “Well, they are quarters.” and then we counted by quarters, “1/4 + 1/4 + 1/4= what?”  Finally someone said, “Well, it’s 3 quarters.”  and the lightbulbs that flashed on above all the heads nearly blinded me! It was awesome!  They got it!  I mean THEY GOT IT!!!
Then I showed them how some of our classmates had cut the “cookies” into squares and some into triangles and some into rectangles. It took a bit more discussion, but we all saw how they all ended up with 3/4 as the answer to how much cookie each friend got. I told them they could put the cookies in the trash, and about half of them put the in their backpacks instead because sometimes trash = more important to take home than our actual work or, you know, report cards.  You’re welcome, Moms!
Finally, fraction of a distance  Last week in gym I asked everyone to run halfway across the floor and stop.  It was trickier than you might think, especially given the fact that there is a giant black line halfway across the gym. But then I said, “From there run halfway to the wall.” and this time there was no line to signal halfway.  Some of them did it, and some of them didn’t.  We ended our super day of math by going outside into the glorious sunshine.  I had two hula  hoops and asked kids to stop halfway between the two.  More of them did than didn’t.  Of those that didn’t, most overshot the mark by a good amount. We’ve obviously got some work to do around fractions of a distance.  I’m thinking they need to do it in a place where they can’t really run.  I think they are going too fast and 1 meter into the full out sprint they have forgotten they are supposed to stop halfway, then one classmate stops, or they see the end goal, and something deep inside them says, “Wait? Was I supposed to stop somewhere?” and by then they’ve gone more than half way.
Tuesday we’ll do some work with writing down fractions, looking at the models and pictures and writing the number.  Then on Wednesday we start EQAO with the math section and I am super paranoid about being accused of cheating so I don’t teach math during the 2 days when we are doing the math sections (which is where we begin this year.)  That gives me some time to think about a few more ways to connect their  fraction learning to their past learning.

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