Math Hombre

It was time for my last game with the fifth graders, and the content was multiplication and division of fractions.  Having just been critical of a game that was very computation focused (see my Math Evolve review) I was very wary of doing the same thing.

I'm at the point now where I've designed more games than I can remember easily, so one of my first steps in game design is to search my own stuff. Failing finding a game to use, maybe there's one to revise. Failing that, maybe one to revise. I did stumble across my post on multiplying fractions. Oh.

I have many lessons that get at the meaning of the operations, can be used to start discovering, exploring and justifying the rules... but only the Product Game (adapted for fractions) for practice. But that post closes with an Ant Man and Wasp cartoon, and we'd been talking a lot about them because of the Avengers movie.  My son is a comic book fan (sounds better than monomaniacally obsessed, right?) so this was quite a debate. He loved the movie maximally, but would have loved it even more with the original comic book Avengers crew. (Joss Whedon wanted characters to whom non-super powered audiences could relate as well as the super powerful ones.)

By Xavier Golden, Super Hero Squad style. "The hexagons are Pym Particles."

So I was struck with the idea of a size-changing game. But why would our heroes have to constantly change size?  To get past obstacles! Sometimes they'd have to grow, sometimes to shrink. I tried to think of a battle game because there were several boys always interested in that, but for battle it seemed like you'd almost always want to be giant-sized instead of ant-sized. So a kind of maze... so it could be a race game.

I tried to think of a way to turn dice rolling into fractions for multiplying or dividing, or to roll three and choose two, but that didn't feel appropriate for such new content. I wanted actual fractions to see and think about. 






If you have transparent spinners, this is a good place to use them; I just used bobby pins, which make excellent spinner needles.  I experimented with the spinner entries and maze heights to find settings that were not too immediate but not too difficult either. Thinking about the framework I've been using...

1. Goal(s) - solid. I wanted students to get the understanding of the effect of multiplying and dividing by fractions, so contrary to their expectations. I wanted to get some sense of estimation, and some experience with calculations that would lead to support the symbolic rule they'll learn later. I'd also noticed that they were very interested in calculators, but had little experience with using them. This put everyone on an equal footing, as the numbers were messy and required a calculator.
2. Structure - like the stretching/shrinking as a context for multiplication/division. The spinners allowed a lot of flexibility in getting values to be used. Makes the game highly adaptable. And the intention of having to choose a multiplying or dividing spinner helped get across the stretching or shrinking effect.
3. Strategy - weak as it was.
4. Interaction - typically weak in race games, though .
5. Surprise - spinners help here and with...
6. Catch-Up .
7. Inertia - not meant to be a game that requires a lot of replay.
8. Rules - basic premise, spin and change your height. Move forward when you fit.
9. Context - thought this was strong, plus pop culture tie ins to a heavily advertised movie. Kids were interested and engaged, though I sold it a bit explaining about Ant Man and the Avengers. There's a little suspension of disbelief, as Wasp could just shrink and fly through all the obstacles, and it's rare that she grows in the book.

I added the Spin Again option to help with catch-up, strategy and interaction. But most students were so immersed in their own spins that they rarely used them! The other idea that I like quite a bit was the customizable board. Most of the fifth graders were happy to use the board as printed, but a few experimented with rearranging the board.  Maybe with middle school students, more would be interested in giving it a go. Designing a board for your opponents is a great opportunity for some open-ended problem solving.  I picked up a couple packs of mini post-its, and they were perfect for keeping track of the players' heights.

It was a good last game of the year. In the debrief, they definitely got the point that multiplying and dividing by fractions did not just have the same effect as multiplying by whole numbers, and a few kids were noticing that dividing by unit fractions was like multiplying by the denominator. I also saw considerably increased skill with the calculators, and some sensible rounding of the decimals involved.  (Parentheses were almost entirely new to them.) They asked me to leave the supplies so they could play later, and it got almost 100% thumbs up for keep or dump - both good signs.

Hopefully you can get a chance to give this one a try. It has some interesting features, and I think the choice of spinners and rearrangeable board will show up again - good game mechanic features. I'm always interested in your feedback, if you have any ideas or get a chance to use it with students. One dramatic need: the name is a terrible joke, and of absolutely no use with middle school.  Ideas?

This is a pretty focused game and it is undone.  Even the fifth graders couldn't help me finish it... maybe a reader can help?  We were playing Manga High games with my preservice middle school teachers this week, and they noticed something.  The context can be completely silly.  They liked "the penguin game" in particular.  They thought it was mathematically worthwhile, teaching estimation.  They found the problems worth doing in the game and felt like it could help students improve their estimation and computation.  But we never really have to chuck penguins to safety.


There's something appropriate to the context.  We're approximating, acting quickly, there's the idea of what fraction of the way across are we... but it doesn't bear much scrutiny.  It's just silly and a bit of fun.  I think that's an element missing from some of my practice games.

The Game
The game premise is complete simplicity: race from one end of the paper to the other (25 cm) by drawing rectangles with area = 10 sq.cm.  Each turn your rectangle is determined by rolling the width with 2 dice.

That's it.

The Lesson
I shared that we were playing a game that had no name and no context.  I needed their help.  Immediately they began shouting names and ideas.  If I were a game designer, I would do this.  Just get a bunch of kids and let them throw ideas at you.  Holy cow were they enthused and specific.  Sadly, I had to tell them I wanted ideas for this game, and we'd have to play it first.  Me versus the entire class, like we usually do.

I showed the graph paper and explained that the goal was to get to the top by drawing rectangles with an area of 10 sq. cm.  I drew a 1x10 rectangle.  But that wouldn't be very interesting. So instead we're going to roll the width of the rectangle.  I rolled a 2 and asked, "how tall should it be?"  A couple students quickly jumped to 5, so I drew it and we checked - there were 10 squares.  We rolled for them.

Aside: that worked well enough that I would fake a 5 if doing this again.  (Faking a 2 is greedy.)

Their roll - a 7!  That's how wide it should be, but how tall? "3!" kids are quick to say. Hmm - that makes for an area of 21.  "1!"  That's only 7.  "1 and a half!" Let's check.  They talk me through 1.5x7. (Not a few couldn't recall.) Hmm, 10.5, too much.  "1.4!" They walk me through that... so close! 9.8. "1.45?" We try it and get 10.15.  Wait, I say, aren't we really doing division?  What times 7 equals 10? We started an saw that 7 goes into ten once.  "It doesn't work," someone said.  Another student said, "add a zero!" I used the notation my son has been using, which I like.  We got to 1.428, and then realized that was close enough for drawing the rectangle.  "Is it point-4-2-8 repeating?" "Will it ever repeat?"

We finished our practice game, and recorded our calculations on the white board.  When they played for themselves (mostly 2 on 2, with a few 1 on 1's), they referred to the ones we had calculated, but had also several others that needed to figured out.

In the summary, they thought the game was fun enough to play, and had good math.  25 cm made for about the right length of game.  (Hah!) They really got into trying to come up with a context. REALLY.  "Monkeys climbing!" "Monkeys escaping from sharks!" "Tiger sharks!" "People escaping from sharks!"  "You're escaping evil aliens."  "It's an alien trying to climb to the moon!" "It's ..."

OK... it should connect somehow to what we're doing.  I see why the climbing.  But why sharks?  Where is the shark chasing?  Could it connect to what we're stacking? What would you stack that has different sizes? "Students escaping an evil teacher by stacking homework!" "Books!" 

OK... maybe the game remains undone.  Maybe it should be unfinished?

Reflection
The game made a good context for modeling the division on which they needed practice.  The representation of the rectangles was supportive.  We talked about the connection between 10/2 and 10/4, for example.  They got the indirect variation aspect of it, and were rooting for small numbers for themselves and large numbers for me.  The familiarity with common division computations was good.  For such a vanilla game, it might be a nice aspect to keep it vanilla, and keep the game design aspect to the lesson.

If I was developing this into a video game, the subsequent levels would introduce some variability of the dividend also.  I also think you could have kids estimate the division.  If they overestimate, no block.  Increased decimal places would get you farther faster.  One variation that I ruled out was to have the game be a race to 2.5, so that the kids were dividing 1 instead of 10.  But then the area is .1 sq. units - not nearly as intuitive.  My next lesson would be how you can use these results to get at other computations, and to strengthen that multiplication connection.  I'm convinced that 99.9% of kids do not see the partial product connection with the division algorithm.

My 1cm graphpaper (pdf) is at Scribd if you'd like it.

I've been working a lot on division lately.  Long division with my son in 4th grade, with his classmates in small groups and with my preservice teachers.

Xavier's teacher sent home a note before the unit, showing the four ways they would be approaching it.  From the traditional algorithm to the "forgiving algorithm" and a couple in between steps.  My wife couldn't make too much of the note, despite being bright, living with more math than anyone would think is reasonable, and being comfortable with her own computation.  Don't know what it was like in other homes.

Xavier was making pretty good sense of it.  The curriculum sensibly starts out with dividing by 5 only, which was a nice touch.  I think I might even start with 10s, then 5s, having seen how well this worked.  He transferred this pretty well to working with other single digit numbers.  For example, 61/4.  When working, he used appropriate language and responded to questions like "how many fours go in 61?"

I did the soda machine problem (from Contexts for Learning, my favorite curriculum, Exploring Soda Machines: a context for division) with a small group the week before, and they did amazing work.  In the problem, you describe a pop machine (in Michigan, soda=pop) which has 6 flavors.  When full, it has 156 cans of pop.  How many cans of each flavor, if there's the same of each?  But wait, that's a lot of pop.  When I buy pop it's usually in a six pack.  How many six packs will that be?

Students do the most amazing work wth this problem.  Most choose to tackle the 6 flavors question first, and drew their own pop machines.  Six columns, and fill in pop cans 1 per column until they have 156.  156 is an inspired amount - not so much as to be overwhelming, but enough so that they usually have to start some kind of record keeping to keep track, often multiples of 6 as they add cans.  Sometimes drawing a line, with the total amount under that line.  Some students draw neat stacks of cans in a row, and some draw crazy piles of circles of many sizes.

Not very many students saw the connection with the two problems.  Only one realized that both were 156/6.  Others started to think about it when they realized the answer to both questions was 26, and then saw a connection. Many used their pictures of the flavors and started circling groups of 6.  Which was interesting, becuase then there are 2 left over of each flavor.  Some had 24 six packs with 12 left over, and some had 26 six packs.  (Talking they agreed on 26.)  The student who made the division connection shared, but the other students didn't really seem to hear her.

The formal language for what's going on here is that there are two different division actions.  When students can solve some division stoies but not others, sometimes this is the underlying cause.  (Quotative and partitive division, from learner.org, with kid video, too.)  When explaining this to our preservice teachers, we often use the terms fair share (how many of each flavor) and measure (how many six packs) for the different actions.

The next week I wanted to bring another context.  I brought a bunch of play money that I wanted to sort into 6 bags for my preservice teachers.  We counted up the 74 quarters together by making stacks of 10. I wanted to have enough that they were counting up in 10s, so that when we were dividing we'd see the benefit of the 10s.

Do we have enough for 10 in each bag? "Yes."  How much left? "14."  (I kept the notes on the left.  10x6=60)  How much more can we put in each bag? 10? "No!!"  "Two," one of the kids suggested.  People agreed, so in went 2 each. 2x6=12.  2 left, not enough for even one more in each bag.  "That's the remainder."  Excellent!

We then divvied up the rest of the coins.  We had the most pennies, so I asked for a volunteer team.  Then the next team chose dimes over nickels.  96 nickels, it turned out.  But then they lumped all of their neat stacks of 10 together again!  149 dimes.  And 491 pennies.  (One of the kids even noticed the anagram.)  The nickel team was done pretty quickly, 10 in each, then 5 in each and 6 left over.  "Oh, that's enough for one more in each bag."  The dime team did 10 each, then 10 again.  5 each... not quite.  Get one out of each of the five bags with 5.  Remainder 5.  The 491 team did multiple rounds of 10 each.  Then just kind of scooped that last 11 into a bag.  All the bags had lots of pennies by that point.  Then each team made a number record like on the left, and nobody saw the connection with the division algorithm they have been doing.

Wow!

Okay, that's immediately a preservice teacher activity.  At GVSU we're blessed with a goodly pile of manipulatives.  So each table got 6 tubs of blocks (2 each with unifix cubes, wooden cubes, and snap cubes.)  5 (or so) minutes to play with them, because play is important.  Of course they made many mathematical designs and structures.

Then it was time for the task:

1)    How many of the object did you get?
2)    Physically divide them up into the 6 tubs evenly.  How did you do it?  How many in each tub?
3)    Show with a number record what you did.
4)    Use a sense-making method to do the associated division problem.  How would what you did make sense as physically dividing the objects?  Why does your method work?
_________________________________________

1)    ___________ blocks in each tub.


2)    Description of method:


3)    Number record:


4)    Sensible division problem:

And then to make a poster of the connections between their number record and their division work.  Here's what they did! (Click on the images for full scale.)




























And the piece of least resistance:



Those are some beautiful connections!