All right class, let's get going. Jason, sit. Thank you. Congratulations to Summer and Marta. You did awesome in last night's basketball game. Also, if you haven't seen "Much Ado About Nothing," go see it--Sarah is hillarious and it will probably take you a while to recognize Matt--it's great.
Rashad, could you man the computer and open median23.gsp? Thanks.
Recall that yesterday you proved that the medians of a triangle are concurrent and we called that point the centroid. Although we didn't formally write out the proof, we experimented on GSP, and I sketched out for you how we would prove that the centroid ends up being 2/3 of the way down the median. Ah, here is the picture from yesterday:
So, given the fact that we know how to bisect segment AG, we actually have AD cut into three congruent pieces...what do we call that? Amy? Yes, trisected.
Well, here's the plan for today. Rashad, please open median23a.gsp. Thanks.
A conversation ensues in which the students in the class infer where the new ray, circle, and segments came from. Having verified the correct inferences, the class is divided into 8 groups of three and 1 group of 2 (since Page is absent)
Now, I would like you to discuss in your groups the new segments and figures that are created by this new point H...Yes, Chris? No problem. For instance, are the new segments we created similar at all to any segments we allready had? What can you say about the new triangles that have been constructed? Allright, you have 4 minutes.
As the information is discovered, the diagram is relabeled as follows:
Great--I am impressed with how well you remembered those theorems and postulates. Now, the reason that I wanted to do this exercise today was to do a construction...please hold your applause until the end of class.
Here is the setup--We want to construct a triangle given only the lengths of the three medians. How do you think we should start? Anyone, anyone, Buehler? Yes Terry, I agree..let's start by drawing the three medians. Amy, will you do the honors? Thanks. Why don't we make them different sizes just to make sure we are generalizing. Oh, and just for the heck of it let's make them three different colors. Any logical suggestions for those colors? Good..green, blue, and red to match the medians we had in our last exercise.
Now for the big question---how do we construct the triangle given these three medians? Hint: study the diagram we drew before--in particular, is there a triangle which involves all three of these medians?
(Pause of 45 seconds)
Yes, Paula?....I agree, there does appear to be a triangle which involves part of each of these medians. How much of them? Tellula?...Yes, 2/3.
Let me suggest this: If we could construct triangle AGH, we could then construct our triangle ABC. It's OK if you don't see that right now, it will come--for now let's concentrate on constructing triangle AGH, whose sides are each 2/3 the length of the medians of the triangle. If only we could get 2/3 of our segments above. Does anyone know a good way of getting that?
Yes Darren? We do have a script which trisects a segment!--and yes, that would "rock" if we used that. Amy, do you know where to find that script? Good, let's run it on the three segments.
Good job, Amy. Let me have the hot seat for a minute and let's see if we can finish this up before we go to the Trampoline Assembly.
Mr Leatham constructs the triangle AGH using the strategy they had used previously in class.
Now that we have triangle AGH, we must get triangle ABC. Let's take another look at our diagram from the beginning of class:
After several minutes, and a few leading questions, the class decides to get point B by drawing the line GH and then drawing a circle centered at G through H. The intersection of the line and the circle will be B. They decide to find C by finding the midpoint F of GH and then do the same.
The following diagram is constructed.
Okay, we have 2 more minutes. Good job folks. We'll talk about this problem a little bit more at the beginning of class tomorrow.
Now, we have been assigned to sit in rows D and E in the auditorium. Let's get down there.
To acces a GSP document of the above diagram, in which the lengths of these medians can be altered, click here.