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!

  

Adding and Subtracting Decimals (with varying degrees of success)

Working our through the unit on decimals in fourth grade, Jeff came up with a good idea to help think through place value while practicing multi-digit addition and subtraction skills.

Thursday:

Pick six cards.  Arrange them in any way you'd like to make a three-digit number.  Use the ones, tenths, and hundredths places.  Try to see how close you can get to a sum of 10.

Grid paper helps organize the computation, even if the computation is not always accurate.  We hit upon the idea of having the kids place each of their attempts on a number line, because we like number lines.  And also because of 4.NF.C.6!

And here is what we learned on Thursday:

• The kids were pretty good at adding multi-digit numbers.
• As a whole they weren't really persistent in trying out different arrangements.  Many were just "one and done".  They had to be encouraged to make multiple attempts.

Friday:

Choose 6 cards.  Arrange them to get the smallest difference possible.  No negatives!

Nice work!

Seen this before?

How about this?

Really?

This is getting bad.

One more time...

Here's what we learned on Friday:

• We've got some subtraction issues!!

     This became the lunch discussion topic.  Why, over more than two years after first being introduced to subtracting multi-digit numbers with regrouping, were so many still having so much difficulty?  Was it underdeveloped number sense?  Inability to decompose numbers?  Were there still kids not developmentally ready? Or maybe something else is happening.  Are kids overgeneralizing the commutative property?  Think about how they are taught their "basic facts". I know in our curriculum there is a heavy reliance on "turn-around" facts" (2+3 = 5, 3+2 = 5). Would it then make sense to a kid that 3-2 = 1 and 2-3=1?
    Of course when we get to subtraction in vertical columns, teachers often tell the kids, "When the number on the top is smaller than the number on the bottom we have to regroup."  Why? "Because we can't take a bigger number away from a smaller number."  Which is not true, as Jeff points out, and may make life more difficult for them when they encounter negative numbers.
    Regarding this issue, I have never been able to shake something I saw many, many years ago.  Our district was between math curricula.  We had dropped something like Houghton-Mifflin or Harcourt-Brace (can't remember which) and had not yet adopted Everyday Math.  I was among a group of teachers who piloted a program called CSMP.  I was a beginning teacher and did not know much about math education, but there was something about it that was intriguing, and I saved some of the lessons and continued to use them.  One lesson described a way to subtract multi-digit numbers that I had never seen:

The relevant piece from a 4 page lesson.

     Has anyone tried anything like this?  Of course for this method to work you'd need to  recognize that this is a problem in need of regrouping in the first place, otherwise you'd still get an answer of 456.  Any ideas would be helpful, because just today, this happened:

Not a good sign when the difference is bigger than the minuend.

Opening Pandora's Box

Because decimals can be confusing when you're nine years old, and because we like number lines and estimation180-like activities, and because we live in a world where 4.NF.C.6 exists, Jeff, Shannon, Theresa, and I decided to try something a little different with the fourth graders last week. 


 Day 1
     We decided that, instead of having the kids make their own number lines, we would give them one that was pre-made:

They were told that the amount was between $0.00 and $3.00.  We had them place their "just right" estimate, and then the actual total, on their number lines.  Instead of using a new one for each day of the series, they would use just this one.  I liked the idea of having the one number line fill up with decimals.

We made them count the money instead of simply revealing the answer.

Day 2

Day 3

Day 4

Here's a sample number line after we were finished:

     This is all very interesting to us, because as we explore more of these activities we get deeper into number sense than we ever have before and, developmentally, the kids are all over the place.

Yesterday we went back to our first activity:

How tall is Mr. Stadel?

Except this time we wanted them to estimate his height in meters.

Jeff's AM class.  

Jeff's PM class.  Over lunch it occurred to me that a vertical number line might provide a better visual for height estimates.

     When we made the decision to replace our traditional "math message" with a daily estimation-180 activity, we never imagined it would lead to places like this.  It seems as if we've opened up something of a Pandora's Box (where good things fly out?)  But it's too late now to close it, and who knows what else is in there?

Exponential Growth

You could study exponential growth by tracking the pageviews my blog has received since this happened. I suppose it's like what happens when Oprah picks your book for the next book club.  Dan was the reason I started to blog in the first place, and from there it was on to Fawn, Michael, Andrew, Christopher, and many others.  Thank you all for your inspiration.
  Dan is right about elementary school teachers.  I taught everything: reading, math, social studies, science, health, the whole megillah for over 20 years, in both second and third grade.  Now it's just math; helping struggling students, and helping teachers by supporting instruction and doing the kinds of things they don't always have the time and energy to do, like blog and find cool things to do in their classrooms on other people's blogs.  They work really hard, and I am proud to call them colleagues and friends.  Many have allowed me to turn their math classrooms into experimental laboratories, and I am grateful.  Sometimes things blow up, but when they don't it's a reminder of why we do what we do.
   Thank you all.  And in the words of #10...keep postin' and toastin'!

Clyde Abides.

A Really Good Math Book

Turns out I have a really good math book at home.

There's math inside!

     As we were finishing up our unit on multiplying and dividing fractions and mixed numbers, Rich and I decided it might be fun to do something other than grinding through more journal pages and worksheets.  I know there are loads of activities out there that use cooking and recipes to help build fraction concepts, but our little project was, at least for us, breaking new ground.  The idea was to copy some recipes (ones with nice combinations of whole numbers, fractions, and mixed numbers).  After choosing one they like, the kids would roll a die which would indicate whether they were to double, triple, quadruple, half, third, or quarter the recipe's ingredients.  They would show their calculations and results.

We added another wrinkle: after finishing the calculations, you had to visualize and draw what the finished dish would look like.  I thought the kids would have fun trying to match the recipes with the pictures.

We got two days of solid work from the kids.  They were really engaged and excited.  There was a lot of opportunity for us to work with kids still struggling with the computation, and to place the work in a meaningful context.  And there were some side benefits:

• Working with different units of measure (cups, ounces, teaspoons, tablespoons, etc.)
• Vocabulary development.  Parsley, margarine, casserole, bouillon, shortening, and rigatoni were a few of the words that groups had to google.
• Critical thinking.  This came into play when it came time to draw the dish.  Some were easy (brownies), some not so much (Chicken and Stuffing Casserole).  They had to read through the cooking directions very carefully to determine how each ingredient was used and how that would effect the look of the dish when it was finally prepared.
• Group work promoted meaningful communication.

At one point on the second day Rich and I were just standing in a corner of the room watching it all happen. "This is the way it should be," we decided.  "They're doing all the work."   We agreed it was something we need to try to recapture on a more regular basis.
   And one boy insisted I copy the Butterscotch Squares and Italian Fried Chicken recipes for him to take home.