Last week, I asked my Geometry students a very simple question. If any of them had any real memory for their Algebra 1 experience, it wouldn't have lasted 2 seconds. But they don't. They're a bunch of mediocre sophomores. Nice. Pleasant. I like them. But academically just not very impressive.
I guided them to draw a line segment with a slope of 2/5 on graph paper. We drew something like this.
Then I asked them to draw a line perpendicular to this line and figure out its slope. And because this class has memory skills that would make SpongeBob and Patrick look clever, I made sure we remembered what perpendicular meant. Then I set them to explore.
There is one simple key to this brief lesson that I like. I asked a question instead of providing a (magical) formula.
As I circulated the room, students asked me if their answers were correct. Most weren't. But they kept trying. And this was my moment of joy. A glimmer of perseverance in problem solving (CCSS mathematical practice #1). Just a bit, mind you. But it was there. They knew I wasn't going to bail them out immediately. Most knew they'd have to try again. And they did. And in the process of their trial and error, I think they absorbed the right answer more deeply than if I had given it to them.
Dan Meyer has recently been discussing real world, fake math and relevance. I have no doubt that you could teach perpendicular slopes with a better (any!) context. However, my students were engaged. I posed a question. I asked them to hunt for a solution. They were curious, almost to a man. They tried and experimented and guessed. My geometry lessons need LOTS of help. But this simple lesson worked for me; my students were engaged.
More on coordinate geometry soon.