My biggest goal for the start of school: I want to see students doing some mathematical thinking and communicating ASAP. And I want kids seeing how math involves digging into a context, making and sharing discoveries, and creating the next round of questions to explore. I intend to hammer (in a fun way) on the Common Core process standards in all classes this year, starting day one. Here are my current plans for first lessons:

**Algebra 1. **The What did you do this summer? matrix. Using a coordinate plane to graph activities on two axes: “miserable – fun” and “expensive – profitable.” Stolen from Kate, with the modification that activities can either cost or earn money (so we can talk about summer jobs and consider income and expense as opposite). We will start out intuitively, but at some point a scale will be required. Not a hard task, more an icebreaker to review graphing points and positive-negative numbers. I will have a large version up front, and have each kid make her own version. Discuss 2 or 3 examples, then ask kids to plot on their own paper the following: “swimming,” “going to a movie,” and “doing math.” Next think up and plot 4-5 activities of your own, and then we’ll share one each on the board. Some discussion questions: What activities or categories of activities might lie on each axis or at the origin? In what areas of the plane would you locate “a job,” “a good job,” “a hobby” (linear inequalities and half-plane solutions are coming up…)?

**Calculus**. Small groups receive windup cars (the kind you pull back then let ’em go). Task: calculate the top speed. Philosophically aligned with Shawn’s calculus starter: we are trying to get our heads around instantaneous velocity and how we measure it. They will need to come up with measuring tools and timers of some kind. (Subtext: in this class, you need to be active, resourceful participants.) We may, at some point, break out the motion sensors. Or a video-recorder. Somehow – hopefully by multiple paths – we will approach a position function. I will have some graphing stories on tap if they need a nudge. Then bonk! into the problem of trying to find the instantaneous velocity of an accelerating, then decelerating object (AKA slope of a curve). We are headed straight into position/velocity/acceleration in the first week, as I am trying hard for an introduction to calculus that lets students understand why it was invented in the first place. As we delve into motion and slope, we will refer back to the toy car, which starts out still, speeds up, slows down, and stops, as an example.

**Geometry**. Probably will start with the guided circle-folding activity per Mr.Meyer’s Geometry The Supplement opener. I have done this before and like the hands-on nature and the whole-class project at the end. I will follow up with some vocab/definition questions. I’d like to come up with a challenge problem involving origami for after, but I don’t have that yet. Another goal for the year: do more problem-solving in the beginning of Geometry, not just loads of vocabulary.

**Algebra 2**. In the first weeks we are going to head for arithmetic and geometric sequences, defined explicitly and recursively. But on day one I want to tackle counting diagonals. I like how this problem begins geometrically but is really about numbers, and how a complete solution can be elusive. I used it recently in a math circle-type setting with some precocious 5th-graders, who came up with a wonderful, detailed, student-language solution that involved “Cailin’s rule” (H + W – 1), the condition for the rule’s application, and a procedure for handling non-Cailin rectangles.

Here is a very simple Geogebra applet for introducing the problem (and checking student solutions.) In my classroom I project GGB and use a Mimio Teach device and stylus for an interactive display. I think that the juniors are going to struggle with this problem. At the high school level there is a lot of room to push for precise math language as we approach a solution. I don’t expect them to recall the concept or language of common factors too readily, but I am happy to review it with integers, since the algebraic equivalent is coming up.

I’m looking forward to hearing about how the wind-up cars lesson went.

Did you do it?

I did! It was neat to see different ways kids thought about the problem. An example: a group would start the car about 3 feet from a length of rope. They would start a stopwatch when the car got to the rope, and stop it when the car reached the other end. They intuited that the car reached maximum velocity somewhere within the rope-length. They understood that they were calculating an average speed, not the maximum. We could sketch a position function for the car and see that they were finding the slope of a secant (didn’t use that vocab yet). They knew that a shorter rope, properly placed, would provide a better estimate of the maximum speed.

It was a fun thing to do on the first day of calculus class. There was lots of action and sharing of ideas, and hopefully it helped prime them for the slope-of-tangent lessons coming this week.

We spent the rest of the week on polynomial functions and their graphs, to rebuild and refresh algebra skills. Tomorrow we return to the hunt for the derivative, using the slope tool in Geogebra. I’ll post about how it goes.

Thanks for checking back in, Sue!

What kind of wind-up cars did you use? Do you think I could do this in a (college) class with 45 students?

I borrowed a few from my son. They are the kind you pull-back, then let go. You could probably get some for cheap, just make sure they are not too cheap and crummy; you want them to go straight.

I had 5 groups of 4 kids doing it. I think you could do it with more groups, but it does take a bit of room. Larger groups might lead to students not contributing.

It would be cool to use a video camera with a timer, or a Vernier sensor, to generate a position function, but for the first day an intuitive sketched graph seemed OK.

Pull-back cars, huh? I

mightbe able to convince my son to lend me a bunch. And I might be able to get students to do this in groups outside of classtime. (I’m thinking of the students who need more thinking time on the meaning of the derivative. The ones who fail the test tomorrow.)