In Precalculus, it’s important to learn the shapes, equations, names, and domain & range of all those graphs that teachers spend weeks upon weeks on in Algebra I and II. See below for the specific names.
Here’s how the lesson used to go. I would hand them the worksheet and they I would write on the board the name and shape of the first one. As I started to get “creative”, I’d ask them if they remembered the shape.
So this year I had a spur-of-the-moment-night-before-idea to spice it up. The idea probably came from the 3 Acts lesson that just went so successfully not a few days before, but it’s equally likely that it came from being immersed in the MTBoS over the past year (well, “immersed” is a strong word for someone who mostly just reads others’ blogs and steals way more ideas than he gives).
Instead, I decided I would give the students a situation and have them come up with a graph of the situation. Working in groups of 3 on individual large (2′ x 3′) whiteboards, they would sketch a graph and decide on the domain and the range. I’d walk around and give suggestions or ask specific questions. Then after 5-10 min of this (depending on the difficulty of the graph), we’d come together as a class, briefly share the graphs, and I would sketch a class graph which they’d have time to copy down into their notes.
After sketching the graph specific to the situation, we’d then talk as a class and decide on the generic fuction and its respective equation, name, and domain & range (domain & range are almost always different for situations than they are for generic function).
Example of situations we’d graph and talk about are: Coffee temperature over time; Distance from the start points over time of x-country runner running at a constant speed; Runner runs out to a point and then turns around and runs back: distance from that point over time, again at a constant speed; Basketball shot: height over horizontal distance traveled. Yes, I do need more (& better) ideas for the rest of the equations, and I would welcome better ones for the equations I’ve already done so I can show the students other situations where these equations might arise.
We only got through 3 types of graphs in about 30-40 minutes, but I think it was much better spent time and the students have a better understanding of D & R, not to mention a specific situation where each graph applies. Now, when I ask them to create a function out of thin air, I think they’ll do better this year than the unfortunate students last year. Here’s a picture of our class’s work (my writing) on the board.