Katie’s Triangle

Those familiar with Pascal’s Triangle would probably immediately recognize its beginnings in the following:

ImageHowever, as I was giving this to Katie, I realized that from this much information the pattern is far from obvious.  For example, a 4 in the first blank spot seems like a very natural choice to me.  Still, I decided to see what Katie would make of this without any ‘hints’.  All I told her was that she should find a pattern and continue it.

She pretty quickly filled in the following, staring at the last two blanks and saying that she didn’t know what should go there.

ImageI couldn’t help her since I didn’t know why she had placed the 1 between the 4’s and what one can do with it.  Then she looked at me as if realizing something, and produced this final result.

ImageI asked her why she chose this particular way of continuing, but couldn’t quite follow the explanation.  I do recall her saying something like “there were 1’s before and so there are 1’s here now.”  I said something along the lines of “interesting pattern” and the conversation ended there.

About a week later I stumbled on this piece of paper again and an interesting continuation of Katie’s pattern jumped out at me.  Here it is.

ImageIn this pattern the triangle contains itself as a sub-triangle.  I was pretty excited.  I don’t know whether this is what Katie was thinking – probably not quite – but perhaps it was something along these lines.  I am now glad that I didn’t give Katie the hint of trying to figure out the blank from the two numbers directly above it.  This puzzle turned out to be much more open-ended than I originally thought, and Katie discovered something unexpected.  I think that I learned a lesson here :-).

About aofradkin

I enjoy thinking about presenting mathematical concepts to young children in exciting and engaging ways.
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2 Responses to Katie’s Triangle

  1. toomai says:

    I love it! I think I’ll do this with my kids sometime.


  2. sannichka says:

    this was a thoroughly enjoyable read!


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