On the first two days of our Tools of Geometry unit, we talked about perpendicular bisectors and angle bisectors in the context of a task from Illustrative Mathematics. On the third day, we talked about medians and altitudes, absent of a context, except to call the centroid of a triangle its balancing point.
We did emphasize two Mathematical Practices.
1. Attend to precision.
I showed students the definition of a median and asked them to sketch one on paper:
A median of a triangle is a segment that goes from a vertex to the midpoint of the opposite side.
Do you know how many students would prefer that we just show them a median instead of having them make sense of the definition? We are getting good at waiting.
Then I had them construct a median using dynamic geometry software. Of course several students went on and constructed all three, because they predicted that the medians are concurrent at a point.
2. Look for regularity in repeated reasoning.
Is the location of the centroid significant?
I sent a Quick Poll to see what students predicted about the location (the diagram is on the left side of the picture).
16 of the 30 were correct – and while I have the correct answer showing now, I didn’t show the students the correct answer.
We used our dynamic geometry software to explore the measurements. Students observed the measurements and predicted the relationship.
Then without confirming our conjectures, I sent the poll again. This time 21 (22 if you count the last submission) were correct. I still didn’t show them the correct answer.
The document was set up for automatic data collection. While students were moving the points of the triangle, the spreadsheet on the next page was filling up with data. Someone stated a conjecture. We tested the conjecture in the spreadsheet using a formula.
Then I sent a different poll to assess student understanding. We were up to 24 correct. What question did the students who got 12 answer? An opportunity to both construct a viable argument and critique the reasoning of others. An opportunity to learn from our mistakes.
We have yet to actually prove that the medians meet at a point. One proof of this assumes information about midsegments (which we will prove later). Another proof of this uses coordinate geometry (which we will get to later). So for now, we have observed that the medians meet at a point, and we know how that point partitions each median.
After medians, we spent some time on altitudes.
I gave them the definition and had them draw one on paper.
Then we constructed an altitude using technology.
Then we figured out whether the location of the orthocenter could give us information about the type of triangle.
Then we worked a problem involving an altitude.
Then we used technology to explore a triangle with all four special segments drawn from one vertex. Can you deduce which segment is which? How?
For what kind of triangle are all four special segments the same triangle? Equilateral. How do you know?
We used the angle measurement tool. Is the triangle equilateral? No. How do you know?
What would happen if the triangle is equilateral? How do you know?
Note: I used the Geometry Nspired Special Segments in Triangles as a guide for this lesson.
Was this lesson important?
I haven’t figured that out yet. I do know that I don’t want to waste my students’ time.
And so the journey continues …