## Saturday, August 17, 2019

### First Day, Once Again

I've seen some great advice for the first day of class.  (Here's the one I've read more than once. I've seen other great ideas, but I don't see them now.) I'd summarize my take on this article as:
• (Community) Start learning names, and get them learning each other's names.
• (Expectations) Don't spend much time on the syllabus; there are more important things to do. (Have them read it, and then you can quiz them on it the next day, or just ask for their questions.)
• (Learning, Expectations) If you use group activities (which are a very effective way to help groups of students learn), then you want to introduce students to this on day one.
• (Curiosity) An activity that helps them see what's coming in the course would be especially nice.

Our classes start in ten nine days. I am getting ready...

First day activity for Calculus I
I have them graph a parabola (y=x2), then draw a tangent to it at x=3. (Some don't know what that means, so I walk around checking.) And then estimate its slope. After they're done, I get to talk about what makes actually finding the slope hard - such a good intro to the course. And they've had time to review graphing a parabola.

First day activity for Linear Algebra
I have them solve a system of 3 equations in 3 variables. I  ask them to:
• Write down a description of the process,
• Solve the system,
• Now figure how to check whether your answer is correct. (Naysayers, has the group done enough to be sure that the answer is correct?)
• Extra: What does each equation represent geometrically? What does the solution represent geometrically?
Once again, a great introduction to the themes of the course.

First day activity for Geometry
I taught geometry earlier this summer for the first time. I had them draw a triangle (and make sure it was different than their neighbor's). Find the midpoints of each side (they could measure or fold). Connect each midpoint to the opposite vertex. I hoped most would do it well enough that the 3 connecting lines would intersect at one point. My goals were to highlight: vocabulary, shapes, construction (which we were not doing with straightedge and compass - yet), conjecture, and the possibility of proof.

First day activity for Precalculus
I've been thinking for the last few weeks about what I'd like to do for Precalculus. I have found exciting activities in the past that turned out to be way too hard, and intimidated the students. I have a lovely fractions activity, but that doesn't represent what we'll do going forward.

I am working hard to create an activity that looks at functions (and circles too) from 4 perspectives: equations, graphs, tables of values, and stories. I have 7 types of relationships (linear, quadratic, polynomial, rational, exponential, periodic, and circles). I don't have stories yet for the polynomial and rational. (My eternal gratitude to anyone who can give me a story I like for either of these.) And I won't show an equation for the periodic. (A trig function wouldn't make sense yet. But we'll get to discuss that.) So that makes 25 "clue sheets".

I have 40 students, who I'll put in 10 groups of 4. Each group will start with two clue sheets. [So 5 of the sheets will not be handed out at first. I can label those as graphs on the back, keep them at the front, and let student turn them over once they're pretty sure they didn't find a graph match to their set.] Each will describe a different type of function/relation from one perspective and ask them to do a few things. Then they pair up with a clue sheet for each pair, and go looking for the matching clue sheets (same function/relation, different perspective). They go back to their group and explain to each other what they found. (I'll have extras up front, so anyone done early can work on a 3rd function/relation.) When we're done, we'll have a summary of the function types we'll be studying all semester.

I dreamed some of this up late last night. When I started working today, I worried that it would be too hard. (I make up some crazy stuff sometimes when I'm falling asleep.) So my goal as I put this together has been to scaffold it enough. I am assuming some comfort with linear functions, and some familiarity (but not comfort) with quadratics and exponentials. They may not have encountered the others. (And most will not know any trig.)

I put my first second draft into a google doc here. Your suggestions may help me improve it. (I decided to leave out the rational function. 6 functions with 4 clues each would be 24. One story and one equation are left out. That's 22. The last two clues will sit up front.)

Edited (8/17): This is a great activity, but too complex for day one. I will do it on day two. On day one, we will review linear functions in a similar, but much simpler way. Here is my handout.

My Goals:
• Review plugging values in for x to find y. (Do they remember that b0=1? What else might trip them up?)
• Review graphing.
• See functions/relations in the context of modeling a situation.
• Identify functions/relations by type.
• See precalculus as a place to strengthen their understanding of all of this.

If you use this activity, please let me know what changes you decide to make and how it goes.

1. Maybe polynomial could be: a box is 2 inches taller than it is wide, and twice as long as it is tall. What's the volume? (Or the volume is 768 cc, or what is the volume for a box 10 in wide...)
My favorite rational functions are t=d/r . It usually takes you 1 hr if you drive 55 mph. How long does it take you if you drive some mph faster than that?

2. Thanks for coming by and offering some ideas.

The functions I chose, y=(x-1)(x+2)(x-3) and y=1/(x-2)+4, are the sorts of things we work on graphing. I'd want a story that would go with something similar. It's the graphs versus equations we focus on most.

3. Now I'm getting cold feet. Will this be too hard for the students?

4. Got it! I decided to do this complex activity on day two. I have now made a simple activity for day one.

I'll edit my post... 