My students did this activity yesterday, in which they unwittingly drew a parabola using geometry instead of algebra. So they "knew" about the focus and directrix of a parabola, and that all the points on a parabola are the same distance from the focus as they are from the directrix. And they "knew" the formula c = 1/(4a).
But I had a sneaking feeling they didn't really KNOW, you know?
So today I showed them this:
...and asked "Which point is the focus of this parabola?" There were guesses for each of the colours. Some said they all could be the focus. Bingo. They know what it is but not what it isn't. That's not KNOWing.
"If the pink one is the focus, then which line has to be the directrix?"
Everyone went with the pink line of course, but their reasons were varied: because the colour matches, because it's the farthest away, because it's the same distance from the vertex as the pink dot.
Right. We've just established that as soon as you have a focus, you also have a directrix - they work as a team. That's a slightly bigger picture. Also I need to knock it off with the colour coding.
"Is it possible that any one of these could be the focus, as long as we pair it with the correct directrix?" Some said yes, some said no.
Time to test out their hypotheses. I had everyone make a dot somewhere on the parabola with their initials.
"Draw L1 and L2 for your point using the green focus/directrix."
The board looked something like this:
It was very fortunate that K picked the vertex for her point!
"Does anyone's point have L1 = L2?"
K's did, no one else's though.
"Well does that mean the green point is the focus or isn't the focus?"
Great discussion on why it isn't - it's not good enough to have one point on the parabola with L1 = L2, they ALL have to have it. They knew that yesterday, but this was knowing on a different level. I think the fact that each person "owned" a different point reinforces the all or nothing idea here.
Then each person picked a new points, and we tested out the blue pair. As soon as ONE person's point didn't work, the reaction was immediate - it can't be the right pair.
Now they knew that there is only ONE possible location for the focus and directrix of a parabola. Move anything and it doesn't have the L1 = L2 property.
We finally tested the pink pair and found it to be the actual focus and directrix. Now they knew where it was and where it wasn't, and there's only one possibility for the former.
Way to drop the ball McSquared:
The final point I wanted to make was the connection between what they just did and the formula c = 1/4a. Not the algebraic connection, but the bigger, deeper one:
The numbers are connected like this: Change the value of a, and you'll change the value of c, and vice versa.
The things are connected like this: Change the parabola, and you change the focus and vice versa.
Unfortunately, that part was just me talking, which I'll replace next time with them doing something, not sure what. Something involving matching parabolas with c values and focus/directrix pairs.....it was time to zoom out here and I dropped the ball, but I'll pick it up next year.
But I do think I helped them to KNOW, you know?