New Intro to Locus and Circles

All of this actually happened today, although, well, maybe not all during the same class. So it's piece-wise true...

Part 1: Locus intro:

This was the first day of our last chapter, conics. I wanted to begin with the idea of the locus of a point. But I didn't want to actually tell them what a locus is, I wanted to show them, then get them to tell me.

I got this idea on my way into class, which by the way there has to be something to why that happens so often at that exact time. Anyway, I thought of a use for one of my geogebras that was not at all what I had intended it for. This video explains what I had intended it for, and what I ended up doing instead:




By the way, if you're interested, here's that geogebra. Next I asked my students what they thought a locus was. Here are a few samples, word for word:

• The path of a point followed by a specific function
• a locus is the path a point takes
• The path of a point of a function
• The trace of a moving point

Their words, not mine. Which, collectively, touched on all of the key points - that it's a path, that it's created by a point, that the point is moving, that as it moves, the point is following some kind of rule.

Part 2: The circle as a locus:

I then wove all of these locus ideas into this geogebra, made by the brilliant Jennifer Silverman:
How beautiful is this?

I let them play with it a bit, to draw a few circles, then identify which of these virtual things was the locus, which was the moving point P, and what rule that point was following as it moved. Here are their answers, again collectively:

What is the locus? The circle is the locus! (Just that right there was huge. All these years I've been the one saying that, and approximately no one was really seeing the circle any differently than they had always seen it - as a static thing.)

Which point traced this locus? The point at the tip of the pencil.

What rule did the point follow as it moved? It stayed the same distance from the red pin.

Then we formalized that into the locus definition of the circle, which for the first time since I've ever taught it, I didn't have to dictate or get them to fill in the blanks on pre-made notes. Okay, I did give them the word equidistant.

Part 3: The rule of the circle

Next I wanted to move onto the Cartesian coordinate system, so we reviewed that:

• the rule for the unit circle is x² + y² = 1
• where that rule came from (right triangle inside circle)
• that really the 1 in the rule was 1².

 I gave them this desmos:  and had them work on that in groups. Just like when they're using geogebra, there is no need for me to tell them if they're right or not. If it is, they'll see a circle with the right radius. Again, there were no notes, no me telling them what the rule is. It took some trial and error, but eventually everyone noticed that the radius has to be squared in the rule. After a bit we regrouped, discussed, even a few things that I hadn't expected would come up:

• Why is the radius squared in the rule? Why isn't it just x² + y² = r?
• Is it possible to get a circle that's even smaller than the unit circle?
• One student noticed that  2x² + 2y² = 1 gave a smaller circle than the unit circle.  Why would that be?

On the way out, I had another idea. I need to write this down so I'll remember it all next year. That was 9 hours ago!