I banished the textbook slope, distance, and midpoint formulas from Geometry class this year. I feel like such a rebel.
First of all, kids have trouble with subscripts. They write them like exponents and then madness ensues. Second, that big long radical sign freaks them out. Third, if they learn and remember it conceptually, they won't try to memorize a formula they don't understand and dink it up later.
So here's what they did learn, and even though I'd like a more organic way for them to get there, the outcome was not bad at all.
Really Just the Pythagorean Theorem. Distance2 = (subtract the x's)2 + (subtract the y's)2. (To be followed next week by Standard Form of a Circle: Really Just the Pythagorean Theorem.) This is awesome because they heart them some pythagorean theorem.
First I asked how far away are these places as the crow flies. (We had to discuss what "as the crow flies" means.) (Thanks Visalia, CA, for laying out your streets on a square grid.)
Then we practiced a little treating the segment whose length we wanted as the hypotenuse of a right triangle, drawing the right triangle, using PT:
Then...dun dun duuuuuun:
We go back to the ones we can graph, and figure out how we can get the legs of the right triangle from the coordinates:
Then we are basically home free:
(average the x's, average the y's)
Same basic plan. Practice a few times, throw one in that doesn't fit on the graph, backtrack.
Subtract the y's, subtract the x's, divide. Yep we played the song. (We sang the song. We claimed we wanted to write our own song about the distance formula and film it and put it on youtube. We are probably not that ambitious.)
Which part of the roof would you rather stand on in a flood?
How do we compare their steepnesses numerically? They've seen slope before.
Do the y's or the x's go on top? Well...which of these slopes do you think should come out to be the bigger number?
Put it together:
I was a little uneasy about not making them copy down and memorize the formulas as they appear in books...am I setting low expectations? Will they get to college and fail because they are unfamiliar with subscripts? But then we were settling in for a quiz this afternoon, and someone said all panic-stricken "What's the midpoint formula?!" And someone else said, "Calm down, you just average the x's and average the y's." And I think that's all right.