In Algebra 2 . . . I used these cards:
Hi Students! Here's a deck of cards. Each one has a polynomial function and its graph.
The degree of each polynomial is determined by its highest exponent. Talk with your group and agree on the degree of each one.
The leading coefficient is the number in front of that term with the highest exponent. Talk with your group and agree on the leading coefficient for each card.
Now we are going to focus on the ends of the graphs. Look at the left side. Does it point up or down? What about the right side? What you are looking at is called the "end behavior". Group the cards according to end behavior. Show me when you're done.
Next I want you to look for similarities in each group. What do the degrees have in common? What about the leading coefficients?
In 5 minutes or so, all of the groups had identified what the groups had in common according to even/odd degree and positive/negative leading coefficient.
We did some practice once they had this figured out.
Oh, and then I played some music and showed them some polynomials one at a time and they showed end behavior with their arms. And there was dancing.
This used to take me 4 minutes. I would write four rules on the board and they would copy them down.
Today it took 24.
But it was fun. And later I saw somebody end-behavior dancing in the hallway.
This approach is better than just telling them, right?