To Prove or Disprove

Today in class we were analyzing the pattern of ninths when converted to decimals.

Students had to write the decimal equivalents in the table and notice patterns.

It went according to plan.  Students write 9/9 as 1 but noticed the pattern of the numerator repeating in decimal form, which begs the question: is 9/9 or 1 equivalent to 0.9 repeating?

It's always a fun debate because the thought that it could be just blows their minds.  Thus setting the tone for 7th grade math where we take everything they think they know (rules and overgeneralization) and turn it upside down.  

Today's math debate however just blew my mind.  For those who have not been following my blog and tweets, I have been working with the ideas in Becoming the Math Teacher You Wish You'd Had by Tracy Zager.  Everything we do I put through the mathematician filter.  With that filter on, the debate was really a chance for us to question what we were seeing in this pattern and prove our conjecture (whichever side you were on).  

As usual, most kids were on the side of 9/9 is not equivalent to 0.9 repeating.  Student on both sides were giving their reasons for why they stood on that side of the room.  The patterns show they are equivalent on one side.  0.9 repeating is close to 1 but there is a number that we cannot name that would have to be added to get to 1 on the other side.  

Then one boy crossed the room, very nonchalantly while someone was talking.  The conversation went something like this:

Me: Tyler, I noticed you switched sides.  Do you want to talk to us about that?

Tyler: As I listened I realized that we didn't really have an argument that proved the other side wrong.

And that, my friends, is the difference between teaching students to explain/justify and having them prove.  In order to prove you are on the correct side of the room, you need to disprove the other side's thinking.


If you are interested in my other  Becoming the Math Teacher You Wish You'd Had blog posts you can find a complete list here.