Where We've Been: In this unit on logic and proof, students have been experiencing the interplay between inductive reasoning and deductive reasoning. In various contexts, we've started by examining specific cases of a geometric relationship and then making conjectures about the relationship. Finally, we've developed a formal deductive proof of the relationship, thereby establishing that the relationship is true in all cases.
Where We're Going: The proof of the triangle midsegment theorem is beyond our scope at this point in the school year, but we will be proving the theorem later on when we become more sophisticated with analytic geometry. So for now, a strong conjecture about the triangle midsegment properties will suffice.
So in this section, I activate prior knowledge with a quick overview of what we've been doing. I explain that we've been using deductive reasoning to make conjectures about geometric properties and relationships and that we've also been using deductive reasoning to prove these properties and relationships.
To make sure students are awake, I do some quick Think-Pair-Shares on some inductive reasoning basics. I start with "Inductive reasoning is reasoning based on..." Students should be able to say "...observation of patterns" or something along those lines. Next, I'll write, "After observing a sufficient number of specific cases, we then make a(n) ________________. " After that... "A conjecture is..." Finally, "We assume that a conjecture is true until we find a _________________."
Note: Although students will need to know the midpoint formula and distance formula for this lesson, I intentionally do not go over these formulas in this section. At this point in the year, my students have had enough experience using these formulas for me to expect them to memorize the formulas and know when to use them. If you have not taught these extensively, it may be good to explicitly teach them during this section of the lesson.
For this section of the lesson, the goal is for students to use coordinate geometry to study one specific case of the midsegments of a triangle. I start by distributing rulers, calculators, grid paper, and a copy of Triangle Midsegments. As noted in previous lessons, the directions on the handout are intended to be self-explanatory. Therefore, it is not necessary to give any extensive directions. The most important thing is to get students to read the directions carefully, interpret them, and do exactly what the directions are prompting them to do.
Depending on the students, this phase of the lesson can go a couple of ways:
Case 1: Students successfully complete the activity and make solid conjectures in the allotted time.
If this happens, then the next phase of the lesson (Using Geometer's Sketchpad) should be framed as a way to collect additional evidence in support of the conjectures by establishing that they hold for, not just one, but several cases.
Case 2: Students do not finish the activity, because they run out of time, or struggle with the computations.
If this happens, I frame the next phase of the lesson as an opportunity to use the power of technology to perform complex computations with greater speed and accuracy than the human mind is capable of. Don't worry about students not finishing at this point. They will return to this activity after the Sketchpad activity to verify that the coordinate geometry yields the same results as Sketchpad.
Case 3: Students complete the activity, but they do not make the intended conjectures.
It is ok if this happens. No need to steer students toward the correct conjectures. They will have an opportunity to see more cases in the next phase. After all, conjectures are usually made based on an observation of several cases, not just one.
In this section of the lesson, students have their hands on the controls of Geometer's Sketchpad. Before going over to the computer lab, I pass out the directions for the activity: Triangle Midsegment Sketchpad Activity. I want to make sure that students are clear on what they will be doing when they get to the lab so that when they get there, they can hit the ground running.
Once we're in the lab, my job is to keep everything moving forward. From the start, when I see students getting to work right away, I draw attention to these students as positive models. I also walk around with a clipboard and a copy of my class roster. When students have finished step 3 on the activity, I check them off and give them pointers and prompts to move them forward with steps 4 and 5. It goes something like this:
"So now that you have one case, how can you examine several other cases?"
"Why don't we drag the slope and length measurements next to the segments they are measuring?"
"As you look at the slopes and lengths, do you notice any consistent relationships?"
As students try their hands at making conjectures, I provide feedback to refine these conjectures. Students say things like "The slopes are always the same"...To which I respond, "Which slopes are the same?" They might then say something like "Segment AB and Segment DE". I'd then say something like, "So what about a triangle that doesn't have a segment AB or segment DE?...which segments would have the same slope then?...how can we speak in more general terms?"
In my experience, students tend to see the slope relationship before they see the length relationship, if they see the length relationship at all. To prompt students who don't seem to be seeing the length relationship, I say something like "So you found a relationship between the slopes of these two segments; is there any relationship between their lengths? Are they equal? Is there a constant sum? A constant difference? A constant ratio?
Once the time has expired, it's time to head back to class. Again, I don't worry if not all students make perfect conjectures. This is to be expected. In the next section, I'll tie it all together and make sure that everyone gets the important take-aways.
In this section, I account for the fact that students have engaged with the content at various levels. My goals are to model the inductive reasoning process and to make sure that students understand, and have a record of, the triangle midsegment properties. I also take this as an opportunity to model some of the Sketchpad functionality.
So, taking the controls of Geometer's Sketchpad, I demonstrate for the class. Eventually I arrive at the following two conjectures, which students enter into their notes:
1. The midsegment joining two sides of a triangle has the same slope as (i.e., is parallel to) the third side.
2. The midsegment joining two sides of a triangle is half the length of the third side.
Either in class, or as a take-home activity, students return to the coordinate plane to verify the conjectures about midsegments. For students who did not finish the activity in the previous section of the lesson, they will perform the calculations and explain how the calculations support both conjectures. For the students who have already finished the previous activity, they will make an arbitrary triangle with midsegments in the coordinate plane and perform calculations that support both conjectures.