We began with using spaghetti and estimating the slope at each individual point. Among other things, I used this sheet from Math Teacher Mambo.
I told my students that the next step was to be able to sketch the basic shape of the derivative, sans spaghetti. No more estimating the slope at each individual point.
Students were having some trouble with this (they usually do, hmm). And then I thought about using color-coding, like so . . .
First we identified points where the slope was zero. We marked those in green on the original function and transferred the points to the x-axis of the derivative.
Next we identified regions where the function was increasing (positive slope). We shaded them in yellow and then shaded the corresponding region of the derivative ABOVE the x-axis only.
Then we identified regions where the function was decreasing (negative slope). We shaded those in blue and then shaded the corresponding region of the derivative BELOW the x-axis only.
Now, to sketch the basic shape of the derivative, you draw a graph that hits the green points and stays within the shaded regions.
It worked great! Students were all "Oh, now I get it!" Love those words!
We also needed a way to color-code places where the derivative was undefined, such as sharp points and discontinuities, so we added pink. A pink point translated to a vertical line on the graph of the derivative as a "don't touch this" signal.
Then we did a little more practice by matching some function/derivative cards. These are not my creation, but I am not sure where they came from to give credit.
There used to be lots of arguing and discussing during this activity, this time students breezed through it easy-peasy. I was just sitting there going ". . . but aren't you going to discuss . . . and argue and stuff?"
Next time I'll keep the spaghetti for sure. After that I'll have them try the card matching before the color-coding thing, and then back to the cards.