math, Patterning & Algebra

Patterning

So – Patterning.  I’m thinking a lot about this skill and how to make it meaningful for my mathematicians. I’m thinking a lot about its connection to algebra and how to set my grade 2’s & 3’s up for success and start them on the road to algebraic thinking.

I put them to work on Monday.  I put baskets of math manipulatives out and told them to go make patterns.  As predicted, they made a bunch of repeating patterns.  They were quite proud of them in fact.  On Tuesday, we talked about growing patterns.  They weren’t really showing an understanding of reading the pattern left to right, so we had a bit of a chat about that on Wednesday when we talked about shrinking patterns and about how the direction matters. As seems to happen often this year, they were amazed by this knowledge.  I think it will stick!  Here is one of the examples I built to show them that direction matters:

Today, Thursday, I asked everyone to actually put their pattern on a number line.  We have done a lot of work with number lines this year, and with the 100 chart.  I feel like it is really paying off!  I started with some guided inquiry.  What, I asked, would my pattern look like on a number line?

Together we constructed a few:

img_1022
Really glad I put this on top of the gross old tape line that children have been slowly picking off for 2 years!

Then I sent them to make some patterns of their own, and map them on number lines.  I didn’t hand them the paper until they had their patterns made and could talk to me about how the pattern was growing and shrinking (by ones, by 3’s, etc.  Actually, no “etcetera” because everyone either did ones or threes, like our example.  I’m not worried though because tomorrow I can tell them they are too good to stick with ones and threes and they need to choose something else!)

img_1039
This one led to a conversation about how they actually have 2 different patterns: add 3 tiles to each term, OR add one column to each term. They hadn’t noticed the second and were trying to figure out why I was so excited about it! I feel like there is a double number line opportunity here, but we aren’t ready for that!

I know it might not be right to have favourites, but this is my favourite conversation:

First, there was this:

img_1031.jpg

The child who made this pattern was insistent that it was a growing and shrinking pattern.  His partners were not convinced.  In fact, they were downright mad because he was so sure and they couldn’t see it. I couldn’t see it either, to be honest.  I wanted so badly to tell him that this was not going to work!  But Cathy Fosnot’s voice echoed in my head, “Productive struggle…productive struggle…”  so I handed him the strip of paper and a marker, and walked away.  A few minutes later, I returned to this:

img_1037

He’d figured out on his own that to make a number line his “special stones” needed to be laid out in a straight line.  He was also able to finally show us that the green stones aren’t actually part of the pattern.  They just mark the end/beginning of each set of clear stones.  As soon as it was straight, he could help his partners see his thinking – he could explain it so much more easily.  He’d made it through the struggle and came out successful on the other side.  (He did write in the numbers and finish the number line – I didn’t get a picture though.)

Two others made this pattern. (I’ll add the picture later!)

When we chatted about it, they told me that they knew 22 should come next, but didn’t have enough special stones. This was a huge piece of info for me! I thought they’d just been rote counting, but are perhaps ready to make a line without having to build the concrete pattern first.

I am, however, left with one question:  How does one put a repeating pattern on a number line?  ABABABA patterns, or ABCABCABC patterns – can they be put on a number line?

*update* Today I challenged everyone to try something besides 1 & 3.

They tried 2, 4, 5 & 10.

img_1974
…backwards…. We still have work to do!
img_1973
This one went on and on, but there are kids in the picture near the end.
img_1976
They made a “counting by 1” pattern and had to be convinced to add the second row and then to understand the second row. I had to start the number line for them, but it felt like an appropriate scaffold.
img_1966.jpg
I can’t show the “100” at the end. But I can say there was a great conversation about why 10 and 50 were so close, but 50 and 100 were so far away! They decided they needed a “refresh” and drew a new underline with perfectly iterated lengths between the numbers. Also, they realized they didn’t need to build every term in the pattern to draw the number line.
math, Number Sense & Numeration

#WODB

I wrote back in August about a great book I wanted to use during the first two weeks of school.  It’s called “Which One Doesn’t Belong” and is written by Christopher Danielson.  (You can read about that here.)

Today was an inclement weather day, meaning the busses were all cancelled.  It was our second in a row, and we have actually had a lot this winter now that I am thinking about it.  I needed to spend about an hour doing an activity with a bunch of kids (grades 1, 2 and 3), over half of whom are not in my class on a regular day. Actually, probably 3/4 of them aren’t in my regular class. I decided to pull this book back off the shelf.

I explained the concept and read the first few pages.  I made sure that every child knew that on every page there would be 4 things, and they could think of at least one reason why three of those things would go to together, but one wouldn’t belong. I explained that there is not one right answer for each page, but what matters is justifying your own thinking so others can at least see what you mean, if not actually change their own mind.  It’s great math!  And it is also so interesting to see how children think.  I have been through this book a few times now, and I am always amazed at how they come up with answers and justifications that I haven’t noticed.

After, I challenged them to create their own using LEGO, two colour counters, attribute blocks, colour tiles, and poker chips (I got them cheap – over 1000 in lots of different sizes and colours at Value Village.  Best investment ever!)  Here are a few:

I took photos and we projected them on the whiteboard so we could share our thinking.  They LOVED it.   This one with the dominos really intrigued me.  I immediately saw kids doing some counting, but nobody used the counting in their answers.  I decided to take that one a bit further.  I wrote the totals on the board beneath each domino.

 

Several thought the 13 does not belong because they are all in descending order, but it is out of place.  Some thought the 12’s do not belong because they each have a twin and none of the others do.  Finally, several thought 9 does not belong because it is a single digit number (they actually said because it is less than 10 and the others are over, so I pointed out the single/double digit difference.)

It was a fun activity, and I think all of the students learned something!

 

Guided Math, math, Measurement, Number Sense & Numeration, Number Strings

Use the 5’s and 10’s, PLEASE! I’m Begging You!

This week I started a new Context for Learning unit with my grade 2/3 class.  Prior to this unit, we have completed the “Collecting and Organizing” unit, which encourages the use of the 5 and 10 structure to organize and then count large groups of items.  We counted books in our classroom because that was a meaningful thing for my class.  The parent council had recently offered up money to buy more books, so I tied that all together. After that, we completed the “Double Decker Bus” unit, again using 5’s and 10’s and thinking about adding and subtracting.  Simultaneously, my grade 3’s – who were already doing well with the models and strategies taught in the bus unit – were working on “The T-Shirt Factory” unit.

Measuring for the Art show comes next on the recommended order list.  I should be starting “Grocery Stores, Stamps, and Measuring Strips” with the grade 3’s.  However, I really want to solidify this number line business, so I am not going to go forward with that unit for another week…maybe two. I am going to extend the numbers well past 100 in this unit so the grade 3’s are still challenged. Picking the numbers is my job this weekend.

So…here we are, measuring for a fictitious art show, and also thinking that we will run this year’s school art show.

I gave groups of children baskets of cubes in 2 colours and set them the task of using the blocks to measure the papers.

As you can see, there was some great measuring going on!  We even agreed on the measurements!

Despite all the work we have done with counting things in groups of 5’s and 10’s, some of my little friends really can’t stop counting by ones.  I asked myself, “WWCFD?” (What Would Cathy Fosnot Do?) I finally had a serious talk with them about it.  “WHY?????”  I screamed. But out-loud I said, “I know you guys can count by 5’s and 10’s, but you keep counting by 1’s even when we have a lot of things to count.  What’s up with that?”  They gave me the blank stare.  “Here’s what I think,”  I continued.  “I think you know how to count by 2’s, 5’s and 10’s, but you’re not sure you are getting the right answer so you always count by 1’s because you are sure that will give you the right answer. Am I right?”  There was a lot of vigorous nodding.  “What I want you to do is keep counting by 1’s.  But do it after you count by 5’s or 10’s. Do it to double check your work.  But challenge yourself to grow your brain and do it the harder way.  I know this is going to help you feel more confident!” So now we are doing that, except a lot of them quickly realized they were getting the right answers the first time, and it was a lot more efficient to skip count.

After 2 days of this, including a congress when we had the above conversation, I asked them to help me make a number line, organizing the cubes into groups of 5.  Believe it or not, there was magic!  As soon as I had a long string of cubes up on the board, out of everyone’s reach, 15 out of 18 immediately saw the value of using the 5s and 10s.  We worked on related Number Strings for 2 days, and then I asked them to make a number line like I had been making using their own cubes and a piece of adding machine tape.

The group pictured on the left kept counting  by 5s, but when they got to the mis-matched groups of 5, they realized that maybe I am a genius after-all and they should have listened when I said, “Make all 5 the same colour!”

So everyone make beautiful number lines, with mostly iterated units.  We put the cubes away and I didn’t get them back out. When I asked them, the following day, to figure out where numbers like 13, 23, and 33, should go, they did a great job of reasoning their way through the problem.  I can look at these and see some immediate needs I need to address on Monday or Tuesday.  But I feel like we are on our way!

img_9218

In a VoicEd.ca radio broadcast (You can listen here!) , Cathy Fosnot said she hoped that teachers who were listening would stay curious and keep wondering about the things their students are doing.   For me this is some of her most valuable advice.  Being curious about why my students are doing something, especially if it is something that makes no sense to me, has paid off so many times.

So…there you go, Cathy Fosnot.  You were right again.

math

Not Done Yet

Yes, I am in fact still teaching math.  We only have 4 more school days, and 2 of those are booked solid.  This might be my last real math class of the year, but there’s still hope for Thursday!

One day last week, I saw this on Twitter:

img_4827
This is not my picture! I clipped it from Twitter one day last week and now can’t find the original source!

I really wish I had not just saved the photo and could tell you where I found it.  But honestly, it was probably around 10:00 at night and I’m surprised I even remember I copied the picture.  I appear not to have retweeted it because I have really searched and can’t find the original source.  If you know who it is, please tell me!

Today I had a long math class (a full 90 minutes, which happens once a week. We don’t always use the whole time, but I am always glad for the extra!) so I pulled this picture up on the Apple TV.

We talked about what we saw.  Then I asked, “How can this child say that 3 hexagons is equal to 11 other shapes?”  I was met with blank stares, which is exactly as I had hoped it would be.  I sent them back to their tables with pattern blocks to see if they could figure out what this random stranger was talking about, and if s/he was actually right.

They played around for a few minutes, and then one child showed me this:

img_4837

I asked him to explain.  “Well, I saw that 2 of the red were the same as the yellow, so I just tried out the others and it worked.”  Good old “Guess & Check”.  As soon as he explained, other people at his table gave it a try.  In the meantime, on the other side of the room, another child had discovered the same thing.  When I talked to him, he added, “And now (X) is copying me!”  The child next to him had made the same representation.  “It might seem like copying, but I think X was just learning from you.  And look what (X) is doing now.”  I asked her to explain, “Well, I wondered if we could make more hexagons with other shapes, not just the red, triangles and the blues.”  That inspired the other children in her group to start exploring more options.  Could anything be done with the square, for example.

We all gathered on the carpet and I put up a few examples of the student’s work.  “Now,” I told them, “I’d like to see if you can come up with balanced equations of your own.  Maybe you’ll come up with something that doesn’t even use hexagons!” and away they went. Here are some of the things they came up with:

I have to say I was pretty happy with the results!  There was a lot of deductive reasoning going on as students built one shape and then reasoned that they could build other similar shapes.  It was interesting and I am sorry I don’t have more pictures to share!  My favourite was a long row of rhombus’ on one side of the balance, and a long row of triangles on the other, all neatly stacked in a row that equalled the length of the rhombus row.

This is my other favourite:

img_4848

She built her own balance!  At first I thought she had too many blocks for no reason and was very close to questioning her when I realized it was a model of the balance.

I figured out, a few weeks too late, that we need more practice with shape names. But I was happy with the spatial reasoning I saw!

So, #SorryNotSorry kids!  You may have thought you were finished with the year, but, alas, you were not!  And stay tuned to find out what we are doing on Thursday!

 

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:
img_3449
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.