Introducing "How Many of Each?" Problems

 

"I have 6 toy ducks hidden under this box.  Some are green and some are red.  How many of each could I have?"

"How many of each?" problems were some of the first open-ended problems that I used in the classroom, prompting me to feel that it is possible to differentiate math instruction without driving myself crazy teaching multiple lessons or juggling small groups.  I believe that they are a great way to challenge students at a variety of levels within the same context, and I have posted more about my experiences with "How many of each?" Problems here.) 

"How many of each?" problems are also some of the first open-ended problems that I introduce to students each school year.  I expect, of course, that students will bring a wide range of understanding to the table.  First graders who are only a few months out of kindergarten have limited experiences recording their thinking on paper.  Second graders who were in my class last year understand a lot more about what I expect in terms of paper work and their number sense is usually much more advanced.  In order to help all of my students find success on our first day working with these problems, I begin with a very concrete example.  (More abstract problems can come over the next few days.) 

In the school basement a few years back I discovered a stash of toy rubber ducks in a variety of colors-- I find them to be the perfect concrete context for this lesson.  Are rubber ducks required?  No, of course not.  Teddy bear counters, bottle tops, crayons... if it comes in a variety of colors, it would probably work well. 

I begin by showing students the ducks and we do a little counting.  Then students hide their eyes and I put some ducks under a box.  They guess how many are under the box and we check and count.  Students hide their eyes again and this time I hide 5 ducks-- 2 red, 3 green.  Now I ask the first "How many of each?" problem-- "There are five ducks under this box.  Some are red and some are green.  How many of each could I have?"  As students suggest combinations we use the extra ducks to show possibilities, and to stay organized we keep each combination on its own piece of construction paper.  Once we have all the possible combinations covered, we have a big reveal complete with drum roll.  

And just as there is a range in the group of mathematical capability, there is a range of social/emotional maturity.  It is seemingly inevitable that one of my youngest will crow that he was "right" about what was under the box and another will tear up a bit because he suggested a combination other than the one I had hidden.    We go on to have a conversation about how there were lots of great ideas about what could have been under the box and how getting the "right" answer was really just lucky. 

After all of this opener, we finally get to that open-ended workshop question.  The kids close their eyes, I hide 7 ducks, some red and some green, and they are sent off with blank paper to show how many of each there could be. 

At first as they work, I circulate, but as students begin to finish, I ask them to check in with me and show me their work.  My policy is to say something positive and encouraging... "I like how you drew all the ducks."  "I like how you labeled your picture."  "I like how you wrote down the numbers."  ... and also push them to just a little further... "Could you label your picture?"  "Could you add those numbers?"  "Could you think of another possibility?"  Some students take another piece of paper or two or three or more and staple their pages into a booklet of possibilities. 

When the work period is over, we gather once more on the rug.  We share some different strategies and answers.  We have a final big reveal.