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Calculus Graph Sketching Activity for Connecting f, f’, and f”

This calculus graph sketching activity for connecting f, f’, and f” was inspired by a worksheet shared by Mike Koehler at the AP Calculus AB Summer Institute I attended last summer.

calculus graph sketching activity connecting f, f', and f"

The original worksheet shared at the APSI included eight small coordinate planes with eight points labeled on each coordinate plane. Students were instructed to sketch a graph of the function f between any two points to meet the stated conditions regarding f, f’, and f”.

I decided this would make an excellent dry erase activity and group review challenge activity. I wrote myself a note on a sticky note and stuck it to the page in my binder.

Now that we are in Unit 5 of AP Calculus AB, I decided it was time to take this idea and make it a reality. I didn’t want the activity to take super long because I am already waaaaaaaaaaaayyyyyyy behind where I should be in the curriculum due to being off all of November and December for maternity leave.

This activity ended up going SUPER well, and my AP Calculus students really enjoyed it. I hope some of you are able to use it with your calculus students as well.

A special thank you to Mike Koehler for sharing this problem structure and expert calculus teaching advice at the AP Calculus Summer Institute I attended last summer at Northeastern State University.


  • Dry Erase Markers
  • Class Set of Coordinate Plane Dry Erase Templates with 8 Points Listed [Download at the bottom of this post]
  • Laminated Challenge Cards – I printed off two or three sets of these. There weren’t enough for each group to start with the same challenge, but I actually prefer for different groups to be working on different problems. [Download at the bottom of this post]
  • 1 Graph Sketching Challenge Tracking Card for each Group [Download at the bottom of this post]
  • Stamp to Mark Each Group’s Challenge Card. You could also use stickers or simply initial the box on the challenge card after checking each challenge. I recommend this teacher stamp kit from Amazon.
calculus graph sketching activity connecting f, f', and f"

Activity Instructions

I asked my students to work in groups of 2 for this activity. Each group had to work through eight different challenge cards which asked them to sketch a graph that met various requirements about f, f’, and f”.

calculus graph sketching activity connecting f, f', and f"

Each group had a dry erase pocket with the coordinate plane template. I instructed each group to send a member up to my desk to grab their first challenge card.


dry erase pockets

I cannot imagine teaching math without my dry erase pockets! They instantly make any activity more engaging and save me countless hours at the copy machine since I can use the same class sets of copies year after year.

Here are my current go-to recommendations:

calculus graph sketching activity connecting f, f', and f"

Students took the challenge card back to their desk and used their dry erase marker to sketch a graph of f which met the stated conditions on their cards. Students had to determine if the graph of f was positive or negative, increasing or decreasing, and concave up or concave down based on the facts given about the original function, the first derivative, and the second derivative.

For example, if f is positive, then I know the graph has to be above the x-axis. If f’ is negative, then I know that f is decreasing. Thus, I know that my graph has to connect A and D.

Finally, if f” is negative, I know the graph of f must be concave down. I can now draw in the function connecting A and D so that the graph is concave down.

calculus graph sketching activity connecting f, f', and f"

When students finish sketching their function, one member of the group brings the dry erase template, the challenge card, and their challenge tracking card up to my desk. I check their work and give them a stamp in the appropriate box if the graph is correct.

If the graph is incorrect, I send students back to their desks to re-discuss and correct their graph.

If the graph is correct, students grab a different numbered challenge card before returning to their group.

calculus graph sketching activity connecting f, f', and f"

Because I was short on time, I let groups work on multiple challenges at a time. If I wasn’t behind, I would probably discourage this so that students were collaborating and communicating more in their groups.

I bought this teacher stamp kit from Amazon several years ago, and it has been a great purchase.

calculus graph sketching activity connecting f, f', and f"

The stamps are self-inking, and I can change out the designs whenever I get tired of the design. Though, I have to admit that I generally just use the exact same two stamp designs.

teacher stamps