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?

I love the way you gave the kids the opportunity to explore their ideas on Day 1. How clever of you to back it up on Day 2 with another problem in the same set that at first glance seems quite different: little ones don’t always realise the additive nature of multiplication. You made me stop and think about the set of numbers where this ‘works.’

I’m interested to know: what’s your next step in the investigation?

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Hi Sally,

Thanks for your comment. I honestly am not sure what my next step will be. I have just a week left of classes and I need to decide whether I want to devote one more day to this investigation.

If I do end up spending one more day on this, I was thinking of having the students individually investigate different numbers and then mark off on a joint hundreds chart the numbers that worked in one color and those that didn’t in another. After a sufficient number of numbers are colored, I would have the students say their observations and conjectures.

But perhaps this will all have to wait until next year, and by then, who knows what they will come up with.

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“Little did he know that his mistake would spur a discussion that would continue over several days” – love that!

The rhythm of this investigation is just like the rhythm of this kind of thing in my class. Though I often throw in a ‘let’s all go to tables and try this out’ session too, as you do.

It’s a helpful rhythm – you don’t have to respond in a big way there and then, you can put it on the back burner, think whether it’s something you want to return to, see if the kids bring it up again…

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That’s what I was leaning towards – not bringing it up unless it comes up naturally. I’ve been amazed at how often things like this do seem to reappear. If not this year, then next year. If not this exact problem, then a similar one.

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I don’t recall ever exploring a school math problem for more than one day. Certainly not in elementary school. It wasn’t until I was an adult, that I was able to “sit with” a problem and treat it as an exploration. You are teaching your students wonderful math – and life – skills!

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Thanks! I only had the early experience of working on a problem for extended periods of time thanks to my dad, who would always give me lots of logic puzzles and rarely any answers. I definitely didn’t get it in school until much later. However, like you said, it is an essential skill and the earlier one develops it, the better!

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Wow, loved the kids involvement in the class room. Kudos!

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