Pretend I'm Not Here

Yesterday we worked on this pattern. 

By the end of the period, we had two different rules.

n + n + 5     or      (n - 2) + (n - 2) + 9

Today we had to decide whether or not these two rules were equivalent.  We had a brief discussion about the different ways students could make their argument:  numerically, visually, symbolically or verbally.  I asked each student to choose a method they preferred and spend a few minutes constructing an argument.  The plan was to then have them pass their journal around the group and have their partners help them make their arguments more convincing.  

As I circled around the classroom, I noticed the work of a particular student who doesn't yet have the confidence I believe will eventually show up.  I stopped and asked him about his work. 

Me: So, tell me about what you have going on here?

Student:  ...

Me:  What type of argument are you trying to make here?

Student: Numbers. 

Me:  Ok, so what numbers are you choosing?

Student:  I chose 55.

Me:  Does it work for both rules?

Student:  Yes. 

Me:  Now that I'm sitting here with you and hear you explain, I can totally understand what you're trying to say.  

Me:  Let me ask you something:  Do you think that if you ripped this page out of your journal and left it for me to read after class, I'd be able to understand your argument?

Student:  No, I don't think so. 

Me:  Can you treat this as a rough draft and try to convince me as if I wasn't here?

Student:  Yes. 

Me:  Ok, great.  I'll come back and check in a bit. 

After a second pass around the class, I come back to this:

I asked if I could have his permission to take a picture of both and show it to the class.  We'd keep it secret if he wanted, I assured him.  When I projected the first iteration, other students tried to explain his thinking.  When I showed the work of the "second student", we all agreed it was much easier to follow the thinking.  Then I said, "This is the same kid."

Class:  "Wait, WHAT?!  

The coolest part of this was that when I wouldn't say the name of the student, many of his classmates said, "It's obvious Mr. Cox.  Look at him."

He was beaming. 

I Love This Problem

I love simple problems with simple proofs.  And I love GeoGebra even more for making it possible for my students to use their induction to help the proof along.

Problem:

Find the shortest path from point C to point D that touches line AB.₁

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¹ This problem is one of the many reasons I'm glad I finally got around to reading this.

If Only It Was Always Like This

Instead of working through the problems in the geometry book, we decided it would be fun to try to prove all the theorems as we come to them.  A couple of weeks ago he proved the midsegment theorem.  Now we are on to trying to prove that the centroid is 2/3 of the way down the median.  So we sit around the white board easel and discuss how we might go about this.  At this point, I don't even know how we're gonna prove this thing.  We learned from the midsegment theorem that defining the vertices using (x₁, y₁),  (x₂, y₂), (x₃, y₃) helped out greatly.  So, what the heck, let's try it again. 

We know we can define the midpoints generally as well, so that's what we do.  Then it hits! We can define the vector from the vertex to the midtpoint of the opposite side, use a scalar of 2/3 to determine the vector from the vertex to what's supposed to be the centroid and then translate the vertex to said point. 

Ta-freakin'-da

 Turns out that the centroid is 1/3(x₁+ x₂+x₃, y₁+ y₂+y₃). 

I didn't know that.  If this is what it's like to become a co-learner with your students, then sign me the heck up.