Surfing the Waves
Calculus
| Tammy Boehme Las Plumas High School Oroville, CA |
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Pre-Test |
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Introduction
Students will use the Internet to learn the definition of the derivative as the slope of the tangent line.
Calculus
Topic: Derivative
Grade Level:9-12
Link to Student Lesson
Standards
Addressed
Calculus
4. Students demonstrate understanding of the formal definition of the derivative of a function at a point, and the notion of differentiability.
4.1 Students demonstrate understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions
Instructional Objectives
After this lesson, students will be able to
- explain the meaning of the slope of the curve
- state the definition of the derivative of a function
- calculate the slope of the curve at a point using the definition of the derivative
- use the definition of the derivative to write a general equation for finding the slope of the curve
- write the equation of the tangent to the curve
Preview: During this lesson (Exploratory 1) students will preview the Power Rule for differentiation. Later they will derive the Power Rule by calculating the derivative of xn for several values of n. The exploratory may be omitted without effecting the instructional objectives.
Pre-Test
Prerequisites for this lesson are expressed in the Pre-Test.
Student
Activities
Day
1
Students use online resources, including
animations, to learn the definition of the slope of a curve and how to
calculate the slope.
Students do 8 problems online, using the definition of the derivative of a function at a point to calculate slope of the curve. Students get immediate online feedback regarding their solutions.
Homework is an (as needed) assignment from their textbook that reinforces the work from class.
Day 2Students are given the definition of the derivative of a function at a given point and introduced to derivative notation. Eight online problems requiring students to write the equation of the line tangent to a curve through a given point are assigned. An online example is available as well as animations that reinforce the notion of derivative. Students get immediate online feedback regarding their solutions.
Homework is an (as needed) assignment from their textbook that reinforces the work from class.
Day
3
Students are offered a chance to review
previous lessons before taking a quiz.
After the quiz, students are assigned a worksheet that requires them to calculate the slope of a single function for many different values of x. This is to prepare them to generalize their results in an exploration that follows.
In the exploration, students will derive
the general equation for the slope of the tangent of several functions.
An animation is available to students to reinforce the concept.
Homework is an (as needed) assignment from their textbook
that reinforces the work from class.
Day 4
Students create their own quadratic function, graph it, choose 3 x values and, for each one, find the equation of the line tangent to their curve at that point, then graph the lines on the same axis as their quadratic function. Below the graphs, students set up a second graph, aligned with the x-values of the top one, and graph the derivative of their function on this second axis. An animation is provided as an example.
Using a Word document, students will explain the concepts covered in these lessons. The explanation must include diagrams, formulas, definitions and examples. Students will include a self evaluation of their understanding of the material, achievement of the objectives and current confidence level. This document is turned in with their graphs.
Students will be asked to complete an evaluation
of this online lesson.
Assessment
Students self assess their work on the internet.
Students are assessed by the teacher on the quiz. The teacher may assign review of the material, additional problems or conduct a mini lesson, if student is not successful. Students are assessed by the teacher on the Exploratory worksheet. All answers must be correct before the student is allowed to move to the next activity. In small classes, students can explain the pattern they discovered and be asked to apply it to a new function. The teacher can provide intervention mini lessons at this time if needed.
The final project is examined by the teacher for accuracy only. Mini remediation can be provided if needed. Incorrect work must be revised.
The grade for this lesson is based completely on a mini-test that students may take twice.
Results
Ten students participated in this online
activity. The quiz was scored by the student and corrections were made before
proceeding. No assistance was given during the 4 day project. The lessons were
summarized by the teacher on the fifth day, followed by the short exam. No remediation
was required during the four days. The fifth day summary served more as a
validation for the students learning. Two students were required to repeat a
similar exam.
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Completed on-line activities |
Quiz Score 10 pts possible |
Test Score (1st attempt) 6 pts possible |
Test Score (final) |
| Yes | 7 | 4 | 5 |
| Yes | 8 | 5 | 5 |
| Yes | 8 | 4 | 6 |
| Yes | 9 | 6 | 6 |
| Yes | 10 | 6 | 6 |
| Yes | 8 | 5 | 5 |
| Yes | 8 | 5 | 5 |
| Yes | 8 | 4 | 5 |
| Yes | 10 | 6 | 6 |
| Yes | 10 | 6 | 6 |
| Yes | 10 | 6 | 6 |
Reflections and Adaptations
Web
Resources & Supplementary Materials
California Standards for Calculus
http://www.kings.k12.ca.us/math/standards/calculus.html
Applets animating Calculus concepts
http://www.ies.co.jp/math/java/calc
Online Calculus course
http://fourier.math.temple.edu/cgi-bin/manager
Worksheet
1
Exploratory
1
Quiz
Mini-Test
Example and Rubric
View the Lesson