The other day I made up a triangle centers lab for my 8th graders.

Here is how it went:

**Day 0**: Homework for tonight is to make a triangle larger than your hand out of some material heavier than paper. Cardstock or cardboard are ideal.

**Day 1**: Open GeoGebra and get to work. Kids got after it. Some slowed themselves down by not reading directions very well. The nice thing about GeoGebra is that it’s easy to erase.

**Day 2**: Most finished the lab and went onto extension activity. Those who didn’t finish had a difficult time managing time. They could do the work, but staying focused was the issue.

**Extension**: Now that you know how to find the circumcenter and incenter, construct an inscribed and circumscribed circle using only compass and straight edge. These students haven’t done anything with a compass, so I offered a 6th point (assignment was worth 5) for those who could figure out how to do the constructions on their own. If they chose to look up the “how to” of constructions, they then would have to *prove* that the method of angle bisecting works.

David came up with his own extension. He asked, “why does the centroid allow you to balance the triangle?”

“Nice question. Now go away and come back with an answer.” He’s figured out that the three medians divide the triangles into six smaller triangles with equal area and that would account for equal weight distribution from the centroid. He can see it in GeoGebra, but is working on a formal proof. Brandon tried to backdoor me with a proof by contradiction, “well the six triangles have to have equivalent areas because if they weren’t, the large triangle wouldn’t balance.”

Don’t try to beat me at my own game, son.

Chris’ reflection: “Hey Mr. Cox, you just kinda gave us a test without giving us instructions.”

“Yeeeeaah kinda, huh?

**For Next Time**: Stamp each page after students have demonstrated the correct constructions. Then allow them to go to the next page. Take a little more time discussing the difference between “drawing” and “constructing.”

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“Take a little more time discussing the difference between “drawing” and “constructing.””

There’s something I should’ve thought of a couple months ago. I am now adding that to my to-do list for the next time I do a geometry unit.

The question of balance is a very advanced investigation – what an impressive question to ask! Careful about the generalization of balance: triangles of equal area won’t necessarily balance even though they are of the same weight, since a long skinny triangle will create more torque than one more compact.

@joshg

I had done a lab re: draw v. construct, but it was in the first quarter. Needed to refresh.

@Riley

I had the same conversation about torque with a colleague yesterday. Does that apply when dealing with median? To be honest, if he can just prove that all six triangles have The same areas, I’d be happy.