37 lbs of Garbage

So while we were out walking a mile around the school, I noticed the kindergartners picking up trash on the school grounds.  We waved to them, they waved back, we continued our walk, and they continued to collect garbage and put it in the large trash bags their teachers had brought outside. I ran into one of their teachers at the end of the day.
  "Great job!  It looks a lot better out there," I told her.
  "We collected 37 pounds of trash!" she told me excitedly.

They used the scale in the nurse's office.  But first they had the kids estimate how much it would all weigh!  Not only that, they found out who had the closest estimate and how far off they were.  And to top it all off, they put the bags in order from heaviest to lightest.  

Our First Real 3-Act

We tried our first real 3-Act on Monday with Jeff's fourth graders.  Motivated by issues of perimeter and scale (which I blogged about here), and a great post by Matt Jones over at The Math Lab @ Rm 27, we started out with this photo:

Lots of excitement!  First we asked the kids if they knew what they were looking at.  We were surprised that so many of them knew right away that it was a photo of our school.  In order to orient them, we identified some features together and marked up the photo on the smart board:

Those scale-related issues had led us to Andrew's estimation180 series on flight distances, which in turn led us to wonder: Do the kids even know what a mile looks like?  Could we relate that to the perimeter of the school grounds?  Our school does a Halloween parade every year so...

ACT ONE: 

How many Halloween parade routes would you need to walk to go a mile? 

The red line indicates the parade route and also defines the perimeter of the grounds (not including the parking lot and the large field.)

After discussing individual too lows, too highs, and just rights, the class agreed that the answer would lie somewhere between 1/2 times around and 9 times around.

Although each student is responsible for their own personal number line in their math notebooks, we also like to have a class number line as well.

Then it came time for...

ACT TWO: 

What do you need in order to figure this out?

Some representative responses:

• "How long is a mile?"
• "How long is half the parade route in kilometers?"
• "I want a long tape measure that I could stretch around the school."
• "I would need a meter stick."

But we did find enough to build on:

• "What is the perimeter of the school?"
• "How long are the sides?"
• "How many feet/inches/yards in a mile?"

We told them that we would give them some of the information they requested:

The measurements are approximate.

They liked this, but soon realized that they needed to know how many feet were in a mile.  No one had this answer, though there were some guesses, including 12 and 100.  We had them find the answer in the measurement table in the back of their journal.

There's quite a bit of information back there, and it wasn't easy for them to locate.

After we established that there were 5,280 feet = 1 mile, we set them off to work.

ACT 3

Figure it out!

Everyone's first instinct was to add up the side lengths.  Good idea!  Only problem was there was a disagreement about the total, due to both computation errors and the fact that some kids left out a side length.  But everyone went back to check their work and we settled on 1,880 feet for once around.

The kids realized that they would have to go around again...

We liked that the computation was just the means to an end, not an end in itself.  And it turns out that twice around is still not a mile, but three times is more than a mile.  Hmmm.

The kids needed a bit of help with that...

We needed to break the side where the parade starts into two separate segments and add on lengths from there until we got close to 5,280 feet.

     This is how the class determined that they would need to walk two Halloween parade routes plus most of a third.  We approximated the stopping point along the driveway entrance to the school.
 And that would have been the end of that, except...

ACT 4

How long does it take to walk a mile?

We couldn't resist.  Here was a perfect opportunity to build a personal referent for 1 mile, based on something that each child knew well: the route of the Halloween parade.  And since we are now programmed to think of most of our tasks as estimation180-type activities, we had to make number lines for our too lows, too highs, and just rights. 

This student had too low of 5 minutes and a too high of 60 minutes.  Her "just right" is 10 minutes.  Does she think that 10 is about halfway between 5 and 60, as its placement on her number line would indicate?  Jeff and I have talked about this issue, and what our expectations should be.   

Ultimately the class was satisfied that it would take us somewhere between 1 minute and 2 hours to walk a mile.  And then we all went outside and walked a mile, almost three Halloween parade routes, as Jeff and one of his students kept track of the time.  It took about 20 minutes.
  We collected the kids on the grass under a tree.  "What does a mile feel like?" I asked.

• "A mile doesn't feel like a lot."
• "A mile feels like exercise."
• "It was easy."
• "It wasn't as far as I thought."
• "A mile is tiring."
• "A mile makes me feel hungry."

Jeff and I discussed what had happened and what we might have accomplished.  Did the kids now have an idea about what a mile was?  Were they clearer about the relationship between feet and miles?  Did the activity strengthen their ideas about perimeter?  Elapsed time?
 As the discussion wrapped up, one student gazed over at the large soccer field across the entrance driveway.
 "I wonder how long it would be around that big field?"
  Funny you should ask...

Pie-Eating Contest

     Sandwiched between a unit on long division and a unit on perimeter and area, our fourth grade curriculum called for a very quick unit on angles.  As it does quite often, inspiration came from Andrew Stadel's estimation180 site; in particular the activities for days 112, 113, and 114.   Relating angles to pieces of pie led me to adapt the Everyday Math game Angle Race into a "pie-eating contest" for our fourth graders.

The game consists of two circles (the pies) and a set of Angle Race cards.  Partners take turns drawing a card and then using a protractor to measure off the right sized piece.  Keep going until you've eaten your entire pie.  Whoever finishes the pie first is the winner!   I liked this better than the original game because it added the skill of drawing angles with a protractor.

Jeff and I started a demo game under the document camera.  We wanted the kids to keep a running total of their angle measurements from 0 to 360. Later we decided to change this by asking the kids to start at 360 and subtract down to 0.  We felt they could use the subtraction practice more than the addition practice.

An example of each.

     After looking at their completed pies, I had another idea.  Cut out the slices and classify them as acute, obtuse, right, or straight.

Jeff and I thought it would be a good idea to let them combine angles.  Here two 15 degree angles and a 150 degree angle have been combined to make 180 degree angle.

Three 30 degree angles make one right angle.

All done!

I had some other ideas that I did not get to try out:

• Have partners cut out their slices, mix them up, and then reassemble the pies like a puzzle.

I tried this out.  This was not the original configuration.

• Play a game where the pieces get all mixed up.  Players take turns  pulling out a piece at random.  Larger piece (or smaller piece) wins, and player gets points equal to difference in size.

I'm sure there are lots of others. Feel free to comment and add your own ideas.

The Gift That Keeps Giving

Andrew Stadel's estimation180 continues to inspire us.  The seemingly simple idea continues to amaze as it takes us into uncharted waters of number sense, measurement, computation, and now, quite unintentionally, scale.
  This last go-round started with something that occurred to me at home, watching my daughter get a snack.

My idea was to just throw it up there, with no review of perimeter or mention of what units to use.  I was curious to see what would happen.  Within a few seconds several hands shot up:
    Students: What's perimeter again?
    So a quick review.  Then:
    Students: What units should we use?
    Me: What do you guys think would be good?
    Students: Feet!  Inches!  Centimeters!  Meters!
    I was relieved that at least they were all linear measures, but clearly we needed work on choosing appropriate units.  After looking at a meter stick, and a 12 inch ruler, we narrowed it down to centimeters or inches, and I told them we were going to use inches as our unit.
   They made their number lines, with their too low and too high boundaries, and just right estimates.  Then I showed them the reveal:

They were going to have to work for the answer.

There were cheers, groans, sighs, and audible expressions of puzzlement.  It was clear we had some work to do, and here's what arose from the class discussion:

• We would need to use the cracker's shape to infer the other side lengths.
• The cracker was rectangular, which meant that opposite sides were equal in length.
• The two little marks after the numbers meant "inches".
• The lengths of each side would have to be added to find the perimeter.
• It was a chocolate graham cracker.

After various number models, from 5 + 5 + 2 + 2, to 10 + 4, to (2 x 5) +  (2 x 2), to 7 + 7 were vetted, and the perimeter was established at 14 inches, I showed the kids this demonstration under the document camera:

The graham cracker framed by anglelegs.  They now represent the cracker's sides.

 

The four sides laid end to end.  I hoped that this visual would reinforce the idea of perimeter as the total length of the sides.

Day 2:

After discussing some too lows and too highs, a girl in the back volunteered her "just right" at 4 inches.  When asked why, she explained that she drew the wheat thin in her notebook, and figured each side to be 1 inch long.  She was close:

By providing the measure of only one side, I wanted them to infer that it was a square.  I also wanted them to practice adding fractions.

I loved that the girl had drawn a picture of the wheat thin in her notebook, and we all took a look.

Notice that her too high is 14 in, which was the perimeter of the graham cracker.

Day 3:

What's the perimeter of the Club Cracker?

Working with the idea from the previous day, I asked the kids to draw out a diagram of the Club Cracker and estimate the side lengths.  After a few minutes, I asked for some perimeter estimates, and was surprised to hear a response of 5 inches, which was the answer from the previous day.  I wondered how, given the visual, a student could come up with the same perimeter for both crackers.  A quick look at her book revealed how: She had drawn a picture of the cracker and measured the sides of the picture!
  Clearly there was some confusion.  Is it possible that in the student's mind there are actually three crackers: the actual cracker, the picture of the cracker on the smartboard, and the picture of the cracker drawn in their notebooks? Hmmm.  So this is what led us to a discussion about scale.

Another opportunity to add mixed numbers.

Day 4:

What's the perimeter of the hamantash?

I liked this one because it was a triangle, and provided another opportunity to add mixed numbers.

This also led to a discussion about how to classify the triangle based on its side lengths.

Here's an example of  student's notebook.  This was an excellent estimate, only 1/2 inch off!

Day 5 was another hamantash, this one a little larger.  Which led us into...

Day 6:

What is the perimeter of the matzo?

Again, we talked about scale (on the smartboard, the side lengths measured around 22 inches), and the importance of using the wheat thin (the side lengths of which had already been established) as a "ruler".

Another example of a student's work.

After the perimeter was established at 24", a student piped up, "That's 2 feet!"  I built a frame with 6" straws connected with twist ties around the matzo, which I then removed and stretched out against a tape measure so the kids could see the sides laid end to end, like I had done with the graham cracker.

     We then proceeded to finish off the perimeter series with three whiteboards: a 12" x 9" student whiteboard, a larger one attached to an easel, and finally one mounted on a classroom wall.

    So in addition to building number sense, we had also touched on perimeter, adding mixed numbers, selecting appropriate units, measurement equivalencies, scale, and classifying two dimensional shapes.  What I loved about it all was that the concepts arose naturally from the activity.  As we move into a unit on perimeter (and area) in a few weeks, I am curious to know what effect this will have on their understanding.  I am becoming more convinced that we can pre-teach, preview, re-teach, and reinforce many skills and concepts in very meaningful ways by embedding them within these estimation activities.

Trying to Make Some Sense Out of Long Division

Raise your hand if you look forward to teaching long division.  For me, anticipating long division time in grade 4 has always left me with a sinking feeling.  Jeff, Shannon, and I have tried many different strategies, sequences, and procedures, and though most kids ultimately learn how to divide using the traditional algorithm, too many seem to struggle.  So this year we decided to try something new.
  I was inspired by  Nicora Placa's post on tape diagrams.  It seemed like a natural fit, because the kids were already comfortable using the diagrams to solve "fraction of"problems, which I've blogged about here.  I envisioned the move from a partial quotients algorithm to a traditional algorithm unfolding in phases.  So we start with a division problem:

Phase 1: Model with a horizontal tape diagram:

The decision I made here was to continue to strike out the number in the total box (dividend) until only a remainder was left.  The subtraction work was kept on the side.  The only difference between this and the "fraction of" diagram we used was that the fraction 1/4 did not appear in each box.  I think this nicely illustrates how the 157 is being gradually and equally distributed.

We did some guided practice,then it was on to...

Phase 2: Model with a vertical tape diagram:

Why rotate it?  Ostensibly because we now have the ability to keep all the subtraction work inside the box (this is what the kids were told).  But really it's because I wanted a way to position the dividend to the right of the divisor.  And by the way, let's now record the amount in each box (quotient) on the top.

More guided practice:

Phase 3: Let's get more symbolic:

The 4 now stands in for the four boxes.  We need to imagine those.  We don't have to write the partial quotients 30 and 9 over and over again.  They can go on top.  I wanted to reinforce the placement of the divisor to the left of the dividend and the quotient on top of the dividend.

More guided practice:

We told the kids to "draw the rest of the box".  Now the tableau makes sense.  The division bracket is just what's left of the vertical box that houses the dividend.

Phase 4: Traditional algorithm

Here's what's changed: We remove most of the horizontal box.  Instead of writing the quotient as 30 + 9, we record it as a 3 in the tens place and a 9 in the ones place.

Jeff did a great job keeping track of which students were successful and comfortable with which method and assigned some differentiated homework.  We grouped them up all week, moving kids around depending on which method they were comfortable working with.

This was the page you received if you still liked to draw the tape diagram...

...and this one was for the kids who had moved to the next stage.

This was for the kids comfortable with the traditional long division algorithm.

Friday was assessment day.  I was curious to see what the kids would do.  Here's what I found:

11 kids used the horizontal tape diagram with the divisor boxes and partial quotients (phase 2)
4 kids used the horizontal tape diagram without the divisor boxes & with partial quotients (phase 3)
18 kids used the traditional long division algorithm
4 kids used alternative methods from home

Not all of it was completely accurate, but Shannon, Jeff, and I all agree that we are much farther along with getting them all comfortable with the traditional division algorithm than we have ever been before.
Here's what I liked about using tape diagrams:

• The traditional long division tableau now makes sense because it comes from somewhere.
• Beginning with a tape diagram reinforces the notion of division as equal sharing.
• Not necessary to remember an acronym "DMBS" (divide, multiply, subtract, bring down).

Here are some issues:

• The tape diagram does not reinforce the notion of division as equal grouping.  We thought quite a bit about that.  But we felt that since the diagram was going to lead into the traditional algorithm, and that was what would be used to solve division problems, we would need to set  that aside.
• The diagram gets more problematic for larger dividends, but that is what drives the need for the algorithm.

For our first time through, I think I'm pretty happy.  Thoughts and suggestions are encouraged and appreciated!

Factors, Primes, Composites, and Some NBA Action

     Love him or hate him, most agree that Carmelo Anthony is the MVP of the Knicks.  Unless, that is, you're a fifth grader in Rich Whalen's class, in which case...

Are you kidding me? 

     Yes it's true, turns out Andrea Bargnani is the Knicks MVP!  How can this be?
     The kids were getting a bit burned out on all the fraction work, and we still had a ways to go in the unit, so we decided to take a time out and circle back to something we explored in September: factors, primes, composites, and the game "Factor Captor".   At the time, Rich and I had played around with a project centered around football rosters.

2013 New York Giants. A disappointing season.

Questions.  A disappointing lesson.

 It wasn't so great.  There were too many numbers, too many questions, and not everyone was secure with the requisite skills.  We got a little mileage out of it (mostly with the high achievers), but did not follow up.
  I liked the idea of using basketball rosters.  Fewer numbers, and easier to work with.  My idea was to play off the rules for "Factor Captor" and have the kids take each player's jersey number, find its factors, identify the number as prime or composite, and then add the factors together to come up with a "score" for that player.  The player with the highest score would be the team MVP.  Then all the scores would be added, giving the team one final, total team score.  Then we'd set up a tournament where you would match up against another team, and the team with the highest total score would win and advance.

First, we gave the kids the rosters and asked them to take a look and talk about what they noticed.   The rosters contained quite a bit of information and we made sure everyone understood what they were looking at.

Making progress with the Bobcats roster.  Rich's student teacher, Shannon, helped design and create the record sheet.  There was still some confusion about which numbers were prime, especially with 0 and 1.

Calculators were allowed.

Why is it necessary to check your work for accuracy?  Because if you miss any factors or make a calculation error, the opposing team will "steal" those points!

Shannon had a great idea about how to change the game: instead of simply matching total score vs. total score, why not match player vs. player?  Team 1 would identify a player, Team 2 would match him with a player from their roster, and the team with the highest scoring player would receive the point differential.  Any missing factors (on either side) could be "stolen" and added to the team's aggregate score.  Then Team 2 would select a player, and so on.  The differentials would be added, and the team with the highest total would be the winner.  This added an element of strategy that was missing in my original conception, and had the added benefit of echoing the rules of "Factor Captor".

Opening Day finally arrived:

Schedule for the AM class.

Knicks vs. Heat.  Garbage time already?  Cole Aldrich makes an early appearance for the Knicks.

Bargnani posterizes Chris Bosh!

Knicks lose, 258-231.

     Of course I had to get into the action.  I was partnering with one of my students (we were the Knicks of course!)  I knew beforehand that the Heat had a higher aggregate total than the Knicks, but was hopeful that by implementing some type of match-up strategy we might pull it out, and I openly discussed this during the game.   One of the boys running the Heat roster said that he didn't think it mattered, in any order the Knicks would lose.  I asked him to explain why, but he wasn't able to articulate a reason.
  This got me thinking, and I pulled Rich aside.  We began to talk it through.  Did order matter?  If it didn't, then the outcome of every game was predetermined, and playing served no purpose other than having kids practice finding the difference between two numbers and doing some column addition.   Quickly Rich and I tried different combinations of only three players per side, and here's what we found:  The scores of each game would vary depending on the order the players were matched up, but the difference in the scores would always remain the same.
   I called my supervisor and explained the problem.  That afternoon he showed up at my door with this:

Using only three players per side, this is how he explained to me that order did not matter.  Here Team B will always win by 39 points.  That's why he's my supervisor!

I had to try this for myself.  I could understand why order didn't matter, but I was fascinated by the fact that the score would change.  I had to play it out on my whiteboard.

Two hypothetical teams.  Team A has a higher aggregate score (15 point difference).

Scenario 1.  Team A wins 31-16.

Scenario 2.  Team A wins 43-28.

Scenario 3.  Team A wins 18-3.

This was really cool.  In each instance Team A won by 15 points...but the score of each game was different.  So we were faced with a decision about how to proceed with the project: we wanted the kids to discover this for themselves, but having them play each other multiple times with a 15 man roster would take too long, and the inevitable computation errors would confound the discovery process.  After some thought, we decided to have everybody play the same 5 on 5 game with Eastern Conference vs. Western Conference All-Stars.

We gave everyone a roster with the individual scores already calculated.  We asked each kid to find the aggregate score for each all-star team.  Then we put the question to the class: Given this information, will the West all-stars always win?  There was widespread disagreement.  Many felt, as I had, that the East could win given the right match-up circumstances.  

We set them off to work.  They were given the option to work alone or to choose someone to partner with.

They were anxious to get started.  The 5 on 5 game was easier to play and took much less time than the 15 on 15 game we had originally conceived.

They used the sheets in different ways.   Rich copied them back to back so they could be used to play multiple games.

As the scores came in the kids posted them on the chalkboard.  After a while, a pattern emerged and we took a time-out to discuss what was happening.  Seemed the West always won by 3 points.  Coincidence?  Could the East ever win?

Confounding data got them all excited, and there was a mad rush to check and see if the results were accurate...

...which of course they weren't.  The most common error was using a jersey number instead of the player's total factor score.  So the scores had to be fixed.

They kept working like this for around 40 minutes.

The board filled up...

...and we had to use a whiteboard for additional results.

Finally we had to call it quits.  Some were now convinced that the West would always win by 3 points, but there were many who just wanted to keep trying.  And that's where we left it.  Unresolved.
Here's what I learned:

• When you experiment, you never know what you might discover.  The original lesson morphed into something totally unexpected but really amazing.

 Here's why I think it was successful:

• It wasn't really complicated.  They could all do it, and they were all in it working together.  No one person could have played out all the possible combinations, but with everybody trying we could gather some good data.
• The fact that it was connected to pro hoops made it alluring to some, but there were many who had no idea who any of the players were.  I think they got excited because I was excited.  So, yeah, enthusiasm is contagious!

Here's what I'd do differently next time:

• Start the same way, with the kids finding the scores for all the players on their rosters, but have them select the top 5 to play with.  Calculate the aggregate score for just those 5.
• Play another team multiple times.  This should go quickly.  It should not take long for the kids to get suspicious about what's happening.  Kids get a chance to hypothesize: can the lower scoring team ever win?  Is the outcome predetermined?
• Then do the all-star game experiment.  This time keep track of all possible scores on chart paper.  This way they can be cut out and then organized (lowest to highest?) to look for patterns and repeats.

And we're not done with this project.  So stay tuned for some more NBA action!