A number talk that turned into an investigation

It all started out with

12+12+12

a problem that was not particularly difficult for any of my 1st/2nd grade students.  The students took turns suggesting strategies for doing this computation and I recorded them on the board (this was an example of a “number talk”, something that we do on a regular basis at the beginning of class).

However, one of the younger student’s in the class, made a typical mistake for his age of getting the digits reversed in his head.  So instead, he solved the problem

21+21+21,

correctly, I must add.  Little did he know that his mistake would spur a discussion that would continue over several days (he was very proud of this afterwards).

Here is what the board looked like at the end of this first number talk.

Except that it wasn’t the end at all!  After looking at the board for a moment, one student raised a hand and voiced an observation – he noticed that 63=21+21+21 is the reverse of 36=12+12+12, just like 21 is the reverse of 12.  “Does that always happen?” he wanted to know.

Some students immediately said “yes”, while others said “no”.  And so the investigation began.

I asked them what other number they wanted to try it out on first.  Someone suggested 15.  We tried it, and it didn’t work!  Then someone suggested that the number had to be smaller than 15, and we tried 14.  That didn’t work either.

Now that it was clear that the property doesn’t always hold, the students started hypothesizing when it does hold.

“It works only for numbers smaller than 14,” one student suggested.

“But it worked for 21,” another one countered.

They tried a few more numbers, and the property did not hold.  At this point, most of the students had the sense that in order for the property to hold, the numbers couldn’t be too large.  But it was time to move on to a different topic.  I promised that we’d come back to this problem soon.

The next day, we had the following number talk

It wasn’t long before someone commented that the number 23 satisfied the property from the day before.  So the students did a bit more investigating.

And now the story should end with them figuring out exactly for which 2-digit numbers the reverse of three times the number equals three time the reverse of the number.  But they didn’t – not quite.

I thought I heard someone say something about the digits at some point, but when I asked them to repeat it, they weren’t quite sure what they had just said.

I told them that we would come back to this problem again soon.  Some students wanted to know the answer, while others really wanted to figure it out for themselves.

To be continued…perhaps.

What is your earliest memory of working on a problem for more than one day?