I checked out my bookmarks last week, and found these ideas over at Dan Greene's blog (The Exponential Curve). I decided to start with the race between the Tortoise and the Hare (download the first .doc file for this and the following problem).
On Monday I wrote this on the board:
WarmUpSince he's so slow, the Tortoise gets a headstart of 16 feet; he "runs" at 1/2 foot per second. The Hare cheats and starts at the 2-foot line; he runs at 4 feet per second.
I started by asking if they knew the fable. Many of them told me 'slow and steady wins the race'. We joked a bit, and I said that sometimes the Hare did win, and that's why the Tortoise was getting a headstart. Then I suggested we find out when the Hare would catch up with the Tortoise.
I made a table of values for both of them at once, and many students found that confusing. So in the second class, I started by writing out full sentences. "After 1 second, the Tortoise is at ___ and the Hare is at ___." We figured that out, and did it for 2 seconds, and then put it all into a table. They found the animals' positions at 10 and 20 seconds in small groups, and saw that at 10 seconds the Hare had gotten ahead. So then they looked for the time at which the two were even.
I told them this method (Guess and Check with a Table of Values) was not in the book, and didn't really require algebra, but might get tedious with a real problem with uglier numbers. We kept exploring this problem by creating equations and graphing. The students had taken a test last week on graphing applications, but many are still struggling with the meaning of rate of change and interpreting the y-intercept. We used those ideas to go from the scenario to the equations, and I think some students started to see more connections between the pieces.
After we had the equations, we used the substitution method to solve (even though we had the answer already). I really liked how it went, and am so happy I'm not basing my lessons on the textbook. They're expected to do lots of homework to solidify the concepts, and I wish I would have given them more clarity about that at the beginning of the semester. I think I'll be able to do this much more smoothly next semester.
Today we did another problem of Dan's that felt similar to me, but probably not to them. I changed the names from Goofus and Gallant to Richie Rich and ... my first class suggested Tiny Tim, my second class went with Charlie Bucket.
WarmUpRichie Rich got $170 on his birthday, and then spent $15 a day on Starbucks and Hot Cheetos. Charlie Bucket only got $10 on his birthday, but he saved it and earned $5 a day with his newspaper route.
I asked what questions they had. The two morning classes came up with different questions, but both (eventually) asked when they'd have the same amount of money. (Thank you!) They still struggled, but I think more and more of them are getting it.
I'm mostly ignoring systems with no solution and dependent systems. I used to teach everything that was in the book, but these notions don't seem necessary on a first pass through systems of equations. I want them to start out grounded in real problems. (I'm hoping the cartoon characters keep it from feeling like pseudo-context...) As they get better at algebra, they can move toward considering the 'what-if' questions. (What if the equations represent parallel lines?)
Today was a good day.
[I still feel like I'm pulling them through the material, and they aren't getting much deep thinking. But I need to recognize that even that is a big accomplishment for many of them.]