It's my goal in precalc this year to have the kids apply knowledge whenever possible. We haven't been able to do that a whole lot (at all?!) so far in trig, so it was nice to get to graphing sine and cosine curves.
We've spent a couple of days graphing with periods, amplitudes, and horizontal and vertical shifts, so I'd assume that 90% of the kids could do that without a problem. Give 'em an equation and they can graph it.
Yesterday I gave each student a strip of paper with the name of a city and their average daily high temperatures for each month (found them here). I also gave them a full size piece of graph paper and a half-sheet of directions/questions for them to complete.
It was amazing the thinking they had to do. Everyone seemed comfortable with having 12 as the period of the graph, but how exactly do you fit that into the equation? (Some kids put 12... or 1/12...) What happens if you set January as month 1? Uh oh - a horizontal shift! And what trig function is it showing? Sine? Cosine? Has it been reflected?
Something that had become almost automatic for them turned back into a thinking game.
Today I told the kids (after they'd turned in their results) that I didn't want to grade them all. After being accused of being lazy, I passed the graphs and equations out (making sure that no one got their own) and told them to grade it for me. (Had to give them some directions, too, of course.) They plotted the data points on their calculator, typed in the equation, and had to figure out what was wrong.
More thinking?! Oh my goodness.
First were the problems actually graphing the functions on the calculator. Check your mode. Did you remember to type the variable? Adjust your window?
Then the identifying of the equation errors. Graph isn't wide enough? So what's the problem? Could you fix it?
Graph isn't tall enough? Is in the wrong place? How could you fix it?
I also had the kids rate the equations on a scale of 1 - 5. I was pleasantly surprised to see that they were very generous with their scores. Someone who hadn't even included a sin or cos in their equation (or a variable...) was given a 4 out of 5. Wha?! And then one girl was given a 4.5/5 because (although her equation was virtually perfect).
I'm happy to say that I think most kids have a better understanding of the different transformations of a sine or cosine equation and what exactly they do to a graph. Hopefully.