I can use the graph of the derivative to sketch a graph of the original function.
I can use the graph of the original function to deduce information about the first and second derivatives.
I can use the graph of the derivative to deduce information about the second derivative and the original function.
I can use the graph of the second derivative to deduce information about the first derivative and the original function.
I can determine when a function is concave up or concave down and where it has points of inflection.
I can determine when a function is increasing or decreasing and where it has maxima and minima.
We were on the first day of a new unit. I included two questions on the opener to ensure students know what we mean by increasing/decreasing and concave up/concave down intervals. As expected students were familiar with increasing/decreasing and not so familiar with concave up/concave down.
Based on the results, we discussed what it means to be concave up and concave down. Someone asked how we would be able to tell for sure where the graph changes concavity, which we get to learn during the unit.
We started the lesson with a few Quick Polls for students to determine which graph was the derivative, given the graphs of a function and its derivative. The polls were based on Graphical Derivatives from Calculus Nspired. I sent the poll, asked students to answer individually, stopped the poll, asked students to explain their thinking to a partner. If needed, I sent the poll again to see whether they wanted to change their response after talking with their partner. I had 6 polls prepared. I sent 3.
I listened while students shared their thinking. I selected three conversations for the whole class.
- A student who knew which was which based on the power rule, which she learned during the last unit.
- A student who knew that the slope of the tangent line at the minimum of the parabola should be zero, which is the value of the line at z=0.
- A student who noticed that the line (derivative) was negative (below the x-axis) when the parabola was decreasing and positive (above the x-axis) when the parabola was increasing.
Again, as I listened to the pairs talking, I selected a few students to share their thinking with the whole class.
- The first student who shared used the maximum and minima to determine which had to be the derivative, since the derivative is zero at those x-values.
- The second student thought about what the slope of the tangent line would be at certain x-values and whether the y-values of the other function complied.
- A third student volunteered a fourth student to discuss her thinking: she noted that the graph of the function (b) changed concavity at the max/min of the derivative (b).
After students talked, I sent the poll again to see if anyone was convinced otherwise.
Two students briefly discussed how they used increasing/decreasing and concavity to determine the derivative.
Next we began to solidify what increasing/decreasing and concave up/concave down intervals look like using Derivative Analysis from Calculus Nspired.
I asked students to notice and note.
Where is the function increasing? Where is it decreasing?
What is the relationship between the slope of the tangent line and where the function is increasing and decreasing?
Where is the function concave up? Where is it concave down?
What does the tangent line have to do with where the function is concave up and concave down?
Can you look at a graph and estimate intervals of concavity?
I was able to see what students were noting on paper and hear what they were noting in our conversation, but I didn’t send any polls during this part of the lesson.
Next we looked at Derivative Grapher from Calculus Nspired.
We changed the graph to f(x)=cos(x). We already know the derivative is f’(x)=sin(x). What if we were only given the graph of the derivative? How could we use that graph to determine information about the original function?
I had more for us to discuss as a whole class, but I wanted to know what they had learned before the class ended. I used a Desmos Activity called Sketchy Derivatives to see what students had learned – given a function, sketch its derivative; and given a derivative, sketch an antiderivative. The original activity was from Michael Fenton. I modified it to go back and forth between sketching the derivative and antiderivative instead of doing all derivatives first and all antiderivatives second, and I added a few questions so that students could begin to clarify their thinking using words.
We spent the last minutes of class looking at an overlay of some of their sketches.
Could you figure out exactly where to sketch the horizontal line?
Most students have the vertex of the parabola near the right x-coordinate. Should the antiderivative be concave up or concave down?
Most students have the derivative crossing the x-axis near the correct location.
The bell rang. Another #lessonclose failure. But thankfully, there are do-overs as the journey continues …