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Teaching Math: A Video Library, 9-12

Conjectures Through Graphing

Content Standards
Conceptual Underpinnings of Calculus, Algebra, Functions, Mathematical Structure
Process Standards
Reasoning, Problem Solving, Communication, Connections
School: San Lorenzo High School; 1,500 students
Location: San Lorenzo, California
Teacher: Carlos Cabana
Years Teaching: 5
Students in Classroom: 17
Grade: 11 – 12
Conjectures through Graphing
Video Overview
Mr. Cabana reviews graphs with the class. Then students work in groups to solve four problems on derivatives of exponential functions. To solve the problems, they use graphing calculators, make graphs and tables, and discuss the slope of a tangent line. Mr. Cabana observes the group work and offers hints to guide student thinking. At the end of the class, Mr. Cabana reconvenes the class and one student presents his group's strategies and solutions.

The Class Challenge
Students focus on the following problems:

Find the derivative of any exponential function.
Look for a rule about derivatives and exponential functions.
Develop a generalization based on that rule.
Find an exponential function that is the derivative of itself.

Course and Curriculum
This elective calculus course is the fifth course in the school mathematics program. This was the first lesson on derivatives of exponential functions. Prior to this lesson, students studied natural logarithms and used graphing calculators to approximate derivatives. Three days before the lesson, they found antiderivatives for familiar functions, developed solutions by guessing and checking, and learned how to find derivatives for known classes of functions. Following the lesson, students began with the derivative, found the antiderivative, and learned applications of natural logarithms. They also investigated more complicated examples in real–life applications.
Students used graphing calculators in two previous courses. To ensure that students do not become too dependent on their calculators, Mr. Cabana asks questions that show students calculators can give wrong answers.
At the end of this unit, Mr. Cabana gave students an individual test in which they had to recall the process they used to find the rules for derivatives of exponential functions, apply the rules, and explain their reasoning. Throughout the year, Mr. Cabana has students work in groups, which encourages their enthusiasm by showing them they can solve problems without his help. He groups the students randomly and changes the groups after every unit. In this and most other San Lorenzo High School mathematics courses, students are taught to discuss questions in groups before asking their teacher.

A Pre–Viewing Exploration for Teacher Workshops
Investigating the Meaning of the Derivative
Objective: To understand the relationship between the derivative of a function and a tangent line to the graph of the function.
Materials: Graph paper or one graphing calculator for each teacher.
Note: The derivative of a function is computed by using the following rules:
If f(x) = cxⁿ, then f '(x) = cnx^n–1.
If f(x) = c, then f '(x) = 0.
Activity
Solve this problem individually.

Use your graphing calculator to graph f(x) = x² + 4x. Describe the graph of the function and locate the x intercepts.
Consider two points on the graph that are close to each other. Call the points A(x, f(x)) and B(x + h, f(x + h)), where h is the increase in value of x. For example, two points you might consider are A(0,0) and B(2,12). Draw the line that connects points A and B. Line AB is called a secant line because it intersects the graph in more than one place. Find the slope of this line.
Determine the equation that represents the secant line as it becomes the line tangent to the graph at (x, f(x)). Consider the slope in symbolic form. The slope of the line passing through the two points A(x, f(x)) and B (x + h, f(x + h)) can be written:
m=(f(x+h)–f(x)) / ((x+h)–x)
or
m=(f(x+h)–f(x)) / h
Using the symbolic form as a reference, what happens to the slope of the line if point A remains fixed and point B moves along the graph toward A? Remember that h represents the change in the value of x.
Use the slope equation in #3 to find the slope of the secant line to the graph f(x) = x² + 4x. Remember that if f(x) = x² + 4x,
then f(x + h) = (x + h)² + 4(x + h) = x² + 2hx + h²+ 4x + 4h.
Now you can determine the expression for the line tangent to the curve at any point. The secant line becomes a tangent point as B approaches point A and the value of h approaches zero. What is the expression that represents the line tangent to the curve?

Exploring Content

What is the relationship between the two functions: y = x² + 4x and y = 2x + 4?
How could you find the slope of the line tangent to the graph of y = x² + 4x at any point?
How would you describe the slope of the graph y = x² + 4x in the interval from x = –4 to x = 0?
How would you use the derivative of the function y = x² + 4x to find the slope of the tangent line at point (–4, 0)?
If you were to use a point to evaluate the derivative of a function and to compare that value with the slope of the tangent line at that point, what would you discover?

Exploration Answers
Activity

The graph is a parabola with minimum point at (–2, –4) and x intercepts (0,0) and (–4, 0).
m = (12 – 0) / (2 – 0) = 6
The value of h approaches zero and the slope is not defined.
m = (f(x + h) – f(x)) / h
= (x² + 2hx + h² + 4x + 4h – x² – 4x) / h
= (2hx + h² + 4h) / h
thus, m = 2x + h + 4
m = 2x + 4 because h becomes zero.

Exploring Content

The second function is the derivative of the first function. The first function is the equation of a parabola, and the second function is the equation of a line. The second equation can be used to find the slope of the parabola at a given point x.
Substitute the first coordinate of the point into the function y = 2x + 4.
The slope is negative from –4 to –2, 0 at –2, and positive from –2 to 0.
Substitute x = –4 into the function f(x) = 2x + 4.
The value of the derivative at any point of a curve is equal to the slope of the tangent line to the curve at that point.

Topics for Discussion
How did Mr. Cabana encourage students to make use of previous knowledge?

How did the students use techniques they already knew to solve the problems?
Discuss the students' realization of how they could apply their previous knowledge to find a function that is the derivative of itself.
How did students use their graphing calculators to investigate the derivative of exponential functions with a base other than e?
List the different types of feedback that Mr. Cabana gave the groups.
Describe how Mr. Cabana used hints to guide student thinking. What are the benefits and drawbacks of giving hints?
Consider the class environment and its effect on student learning.

What is the effect of the whole class brainstorming problem–solving strategies at the beginning of the lesson?
Discuss when and how much time Mr. Cabana gave to group work and to direct instruction. How would you structure these elements in your classroom?
Discuss when students chose to use technology and paper and pencil as they explored the problem of finding derivatives of exponential functions.
What do you think are appropriate and inappropriate uses of graphing calculators? Identify examples from the video and from your own lessons in which students used calculators appropriately.
How would you change this lesson if you did not have graphing calculators?
Discuss the relationship between process and content in this lesson.

How did the group interactions demonstrate students' intuition and attempts to validate their thoughts with mathematical evidence?
How did students use reasoning and their previous knowledge to develop generalizations regarding the derivative of y = e^x and y = a^x?
Group responses revealed that ideas came from more than one student and demonstrated students' attempts to understand the ideas. What can you do in your classroom to learn from similar responses and support such behavior?
Discuss the benefits and drawbacks of group investigations in mathematics classes.
In what ways did students' use of technology support reasoning, listening, and communication?
Discuss examples of how you have responded to problems with groups working together or staying on task.