A toy man in a white shirt and tie stands next to a toy girl in a purple sweater. They are both looking at a popsicle stick which is sloping upward.

Introduction to the Derivatives (Calculus)

Activity aim:  Develop the applicability of the concept of slope to non-linear relations (curves). Students compute different rates of change: a velocity in one example and a rate of revenue increase in the other. They recognize the concept of slope that is the basis for computing rates of change in new contexts.

 Learning goals:

  • Making topic come “alive”…Keeping 26 students “interested,” and keeping up the level of energy.  Also, making affordances for individual needs.
  • Encouraging students to generate understanding for themselves through exploration of known quantity (slope) in known or familiar applications (height and revenue functions), but in new contexts (non-linear functions).

Supplies/materials:  Graph paper (optional).  Two problem applications on a  guided worksheet, written either on board or handed out ( See attached).

Activity Description:

  1. Warm-up: Review linear functions and the interpretation of the slope of a linear function in the context of basic applications (i.e. a canoe company charges a $5 non-refundable deposit and $3 per hour, etc./ Linear depreciation of new boiler, etc. etc.).
  2. Encourage students to check their answers and their interpretation of slope in each case OUT LOUD with a neighbor. Ensure each student has paired with at least one other student.
  3. Pose the key problem of focus for today’s lesson.  “I want you to similarly compute the rate of change of the following quantities at the indicated times or production levels.” Hand out guided practice worksheet (click here to view this worksheet in an MS Word doc).
  4. Introduce students to scaffolded exercise sequence on guided worksheet for the two problems posed. Students work at their own pace, and are encouraged to work together to answer questions on worksheet with same partner (max 3 people).
  5. Investigation, peer to peer discussion/learning, and work in groups.

 Teaching method(s) used:  inquiry-based exercise employing peer to peer instruction.


  • Inquiry-based learning that incorporates peer to peer instruction and possible peer reflection on problem analysis, time permitting, on applications involving rates of change: students are to compute rates of change as they see fit, following an application-based worksheet: They find the rates of change for these two different quantities for themselves, one an object moving in flight, and the other the rate of change of revenue gained by a production at different production values.
  • Peer Instruction (Test-teach-test or Mazur’s method)
  • Inquiry Based Learning

Lesson may also incorporate

  • Silent discussion
  • Writing to learn or reflective writing

NOTE:  The lesson is deliberately designed so that the objective function that results in the two distinct applications given on the guide worksheet are in fact precisely the same mathematical function. Thus, the analysis of slope required will be seen to be mathematically identical across the two different applications given.

 Justification for this particular teaching method:  Relation of class size (n=26) and the subject matter to the technique(s) you selected.

The concept of the derivative is central to any basic calculus course.  I chose to provide two problems that can potentially lead students individually in an inquiry-based approach to discover the underlying need for the concept of the derivative.

It is important to encourage as much independent reflection and examination as possible in order to develop an intuitive understanding of this concept. Thus, with a class of 26 students, I encouraged pairs, or possibly groups of three at most, in order to prevent any degeneration into passive learning that more likely occurs within larger group activities.  Each student must write out their reflection and solution justification, and be prepared to defend their position.

The peer to peer component offers the ability of students to think aloud with someone else, not necessarily to teach each other.  If they work on it together without my saying too much, I think they have best chance of getting deep understanding in the way that makes sense for them.

This activity was developed by Luis Zambrano as part of the Baruch College Center for Teaching and Learning Summer Seminar on Active Learning.

Image credit: Wired For Lego, Flickr Creative Commons

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