At the very start of the school year in geometry, we started by having students make observations and write down conjectures based on their observations. We had a very fruitful paper folding activity, which students — through perseverance and a lot of conversation with each other — eventually were able to explain.
However we also gave out the following:
And students made the conjecture that you will always get a right angle, no matter where you put the point. But when they tried explaining it with what they knew (remember this was on the first or second day of class), they quickly found out they had some trouble. So we had to leave our conjecture as just that… a conjecture.
However I realized that by now, students can deductively prove that conjecture in two different ways: algebraically and geometrically.
My kids have proved* that if you have two lines with opposite reciprocal slopes, the lines must be perpendicular (conjecture, proof assignment).
My kids have derived the equation for a circle from first principles.
My kids have proved the theorem that the inscribed angle in a circle has half the measure of the central angle (if both subtend the same arc) [see Problem #10]
Two Proofs of the Conjecture:
Now to be completely honest, this isn’t exactly how I’d normally go about this. If I had my way, I’d give kids a giant whiteboard and tell ’em to prove the conjecture we made at the start of the year. The two problems with this are: (1) I doubt my kids would go to the algebraic proof (they avoid algebraic proofs!), and part of what I really want my kids to see is that we can get at this proof in multiple ways, and (2) I only have about 20-25 minutes to spare. We have so much we need to do!
With that in mind, I crafted the worksheet above. It’s going to be done in three parts.
Warm Up on Day 1: Students will spend 5 minutes refreshing their memory of the equation of a circle and how to derive it (page 1).
Warm Up on Day 2: Students will work in their groups for 8-10 minutes doing the geometric proof (page 2).
Warm Up on Day 3: Students will spend 5-8 minutes working on the algebraic proof (page 3). Once they get the slopes, we together will go through the algebra of showing the slopes as opposite reciprocals of each other as a class. It will be very guided instruction.
Possible follow-up assignment: Could we generalize the algebraic proof to a circle centered at the origin with any radius? What about radius 3? What about radius R? Work out the algebra confirming the our proof still holds.
Once we prove the Pythagorean theorem (right now we’re letting kids use it because they’ve learned it before… but we wanted to hold off on proving it) and the converse, we can use the converse to have a third proof that we have a right angle. We can show (algebraically) that the square of one side length (the diameter of the semi-circle) has the same value as the sum of the squares of the other two sides lengths of the triangle. Thus, we must have a right angle opposite the diameter!
I’m sure there are a zillion other ways to prove it. I’m just excited to have my kids see that something that was so simply observed but was impossible to explain at the start of the year can yield its mysteries based on what they know now.
The two semi-circle conjecture documents in .docx form: 2014-09-15 A Conjecture about Semicircles 2015-03-30 A Conjecture about Semicircles, Part II
*Well, okay, maybe not proved, since they worked it out for only one specific case… But this was at the start of the year, and their argument was generalizable.