The day that you first introduce derivatives is the most exciting day of Calculus!! (Well, at least WE feel this way as teachers!) Here's how to spread that joy to your students. Maybe they will discover why you love teaching this stuff so much!
To start out, take a day or two to just focus on the main idea. You do not have to jump right into chain rule right away. Give your class plenty of time to soak up the concepts. Focus on "the slope of a tangent line" and "change of rate" vocabulary and main ideas.
Let your Students be Surfers for the day.
When you are ready to jump into the lesson, start with a large graph. A quadratic function works well. Draw or find a graph of a parabola that opens downward, since that is the easiest to compare to a wave. Demonstrate how the surfboard can travel along the curve. Explain that its slope at any point is the derivative of the function at that point.
Resource #1: "Function Surfer" Applet for Derivatives
Resource #2: Printable Surfer Dude
Resource #3: Derivative Infographic
This Derivative Infographic is available in my store and displays the main ideas for understanding derivatives.
For one dollar, you can print it forever for all of your students every year until the end of time. Click the image to check it out. It prints two-per-page with an option of color or black and white.
It is great to hand out as a bookmark and can be laminated or three-hole punched for binders.
Connect With Physics
Show a position graph and explain what it represents. Write "position" on the board. Ask students to come up with a word for how quickly position changes. Write "velocity/speed" below the word "position." If they need help, act it out. Walk slowly, then fast across the room. Then ask "Ok, is there a word for how fast the velocity changes? Have them think about slowing down and speeding up when driving. Write "acceleration" below "velocity" on the board. Display or hand out graphs for position, velocity, and acceleration. You can even talk about "jerk" as an opportunity to go one more level to incorporate a third derivative.
The key here is getting students talking out loud or writing in complete sentences. Try to get a real discussion going. Encourage students to clearly verbalize the relationships. Have them look at the position graph and talk about it. You need them to speak out loud and hear classmates say plenty of statements like:
- "The car must be going faster here because the position is changing more quickly than it was."
- "When the velocity starts increasing more quickly, this means that the acceleration is increasing."
- "The car is driving at a steady rate (constant velocity) when the position graph is linear. The slope is constant, so the position is changing at a constant rate."
- "The car is stopped when the slope is zero. The position is constant."
Have students take time to really look in-depth at each graph and point to the different intervals and explain out loud. Make connections by labeling the position graph f(x), the velocity graph f'(x), and the acceleration graph f''(x). Start using the words "first derivative" and "second derivative" to introduce the new vocabulary.
Resource #4: Position, Velocity, and Acceleration Graphs
This PDF has graphs for distance, velocity, and acceleration for your students to analyze. Print one strip for each of them (two will print per page). Once they have looked and talked through the graphs, they can run their surfer dude along the curves to investigate the slopes.
When they are ready, you can try giving them the second page of the PDF, which has a distance graph to start with, but blank velocity and acceleration graphs. Challenge your students to complete the graphs.
Click the image to download.
Resource #5: Inquiry Activity: Graphing the Derivative of a Function
This three-page worksheet guides students to graph the derivative of a function. The activity is set up as a guided inquiry. Students do not need to know how to differentiate. This is a great introduction to the graphs of derivative functions.
When your students do the graphing in a hands-on way and make discoveries about the relationships, they will understand and remember the concepts.
In this activity, students answer critical thinking questions in complete sentences and make discoveries about the degree of f(x), f'(x), and f''(x). They use a straightedge and find slopes of tangents along the curve to graph the derivative function. The PDF for the "Graphing the Derivative of a Function" inquiry activity is available in my store. Click the images to check it out.