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Connecting f and f’ (Lesson 9.6 Day 2)

Unit 9 - Day 12

​Learning Objectives​
  • Connect graph features on the graphs of f and f’

Quick Lesson Plan
Activity: 3x4 Match-up



Lesson Handout


Experience First

Connecting the graphs of f and f’ is a cognitive leap for many students. The idea of the derivative being a slopes graph--where the outputs give the slope of the original function--is a challenging departure from anything students have done before. Luckily, there are many engaging and discourse-rich activities that will help students makes sense of this relationship. Today we will do two of them.


n 3x4 Match-up, you will need to print and cut the graph cards. The first page of cards should be on one color construction paper and the second page should be on a different color. The first page represents the original functions and the second page represents the derivative graphs. You will want each group of 3-4 students to have a set of cards.


This activity can be done in groups of 2, 3, or 4. Have one group member take the original graph cards and place them on their 3x4 grid in whatever configuration they wish so that the other team members don’t see it. We have students face each other and put up folders between their desks (think battleship). The rest of the team will get the derivative graph cards and the second 3x4 grid (you will need to print two of these per group). Their job is to match the configuration of their group member with the original function graphs by asking a series of questions that can be answered with a Yes, No, or a card location on the grid (ex: B3). Remind students that a match will not be identical graphs but graphs that represent f and f’. Questions could sound like:

  • Where do you have a graph that has a slope of -½ everywhere?

  • Do you have a graph that has a maximum at x=3?

When a group thinks they have matched their group member’s configuration, check their work. If some are in the wrong position, tell them to keep working on it. For students that get done early, have them switch roles or reverse the game, where the person who makes the configuration has the derivative graphs and the other group members must match the original functions to it.


A virtual version of this activity can be found here, where students simply match the f and f’ graphs in a Desmos card sort. Thanks Danny Whittaker, for making this activity remote-student friendly!


After 3x4 Match-up, we have students work in pairs on our modified version of the Desmos activity Sketchy Derivatives. Students will practice sketching derivative graphs and connecting features on the graph of f and f’. 

Formalize Later

Be prepared for students to struggle with this initially, especially as they navigate what kinds of questions to ask and learn to use precise language. Students will need to be very specific about which graph they are asking about so as not to confuse descriptions like “increasing” and “positive”.


We have not yet formally learned the derivatives of sine and cosine or secant nor do students have to know them to complete the activity. Encourage students to notice patterns between horizontal tangents and zeros as well as increasing and decreasing behavior. We will formally discuss derivatives of sine and cosine in lesson 9.7.

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