While students were working on this antiderivative from a Desmos Activity called Sketchy Derivatives, I heard several students ask how they know where to place the y-intercept.
We didn’t have time to answer that question during the first lesson of the unit, but we started with it during the second lesson.
Should the cubic start decreasing or increasing? How do you know?
[Yes, I do know that I can anonymize the names. However, by the time I thought of that, these two responses were no longer adjacent. It would be nice to be able to drag the responses to different locations in case you want to compare/contrast several specific responses at the same time. ☺]
Then we looked at these.
They all have the same basic shape. Is one more right than the other?
Students began to think about all of the curves that have y’=2x, and so while they know something about the importance of the constant, one won’t be more right than the other until we learn about area under the curve.
I had the Second Derivative Grapher from Calculus Nspired queued during the previous class, but I decided it would be better for students to “do” instead of “discuss” for the last few minutes of class. We looked at it next to make the relationship between the second derivative and concavity of the original function more clear.
Students put all of this information together to begin to analyze a function given the graph of its derivative. The results of our formative assessments seem to indicate that students have a better understanding of the relationship between f-f’-f” than they have in the past. I’ll know for sure later today, though, after their summative assessment. And so the journey continues …
Easing the Hurry Syndrome 
howardat58
October 18, 2016 at 2:06 pm
“Students began to think about all of the curves that have y’=2x, and so while they know something about the importance of the constant, one won’t be more right than the other until we learn about area under the curve.”
It is simpler than that. The vertical position of the f curve depends on the constant of integration, so this position can be fixed by fixing one point on the f curve. In a practical way this can be the y coordinate for x=0, as an engineer might explain.
Dan Meyer
October 23, 2016 at 8:53 pm
> It would be nice to be able to drag the responses to different locations in case you want to compare/contrast several specific responses at the same time
Good call!
Michael Fenton
October 24, 2016 at 11:46 am
Hi Jennifer! Thanks for sharing a window into your classroom.
Here’s a one screen activity we cooked up that might pair nicely with Sketchy Derivatives while addressing the goals you describe in the second half of your post. It uses some of the Computation Layer technology that Dan has written about in recent blog posts.
https://teacher.desmos.com/activitybuilder/custom/580e3553724e274109c6509c
Let us know what you like, and/or what you’d like to change.
Cheers!
jwilson828
October 25, 2016 at 7:58 pm
Thank you, Michael!
I like that the values of f(x), f'(x), and f”(x) are displayed. I wonder, though, whether the tangent line is distracting, since we are trying to get at the relationship between the function and its second derivative. What if the second derivative were graphed instead so that students have a better chance of making a connection between the concavity at a point and sign of the second derivative? It might be too busy to have both the second derivative and the tangent line graphed.