(TI-85 BASED)

TO SUPPLEMENT THE INSTRUCTION OF CALCULUS I

AT TEXAS TECH UNIVERSITY

By Gary
A. Harris

Department of Mathematics

Texas Tech University

Lubbock, Texas 79409

g.harris@ttu.edu

**Contents.** The following contains a set of supplementary TI-85 exercises which are to
be incorporated into the syllabus for MATH 1351, Calculus I,
taken by all students at Texas Tech University who are seeking
certification to teach mathematics at the secondary level. The
exercises are designed for three primary objectives:

- to provide basic machine instruction so that the students can learn to use the TI-85 effectively, with little or no in-class instruction required.
- to illustrate the basic concepts of Calculus I that we want our students to understand.
- to make our students become interactive with the material in this course, and perhaps devote more time out-of-class to the study of Calculus I.

Also included with the supplements is a "Quick Reference Guide" to the TI-85 graphing package suitable for distribution to the students. In addition, for those who might want to experiment with formal cooperative groups there is questionnaire suitable for use in the formation of such groups.

**Recommendations.** The use of the TI-85 should begin
immediately and proceed throughout the course. The supplements
should be treated as assignments for which the students receive
some credit. They are written more or less to follow the order of
the content as it occurs in the course syllabus, and are often
intended to foreshadow future material. Some should be used as
in-class exercises with students working on them in groups and
others should be viewed as out-of-class homework assignments. The
emphasis throughout the use of these supplements should be on
student interaction, collaboration, and discovery!

**Justification.** In 1985 the Mathematical Association of
America, with support of the National Science Foundation,
sponsored a conference to determine the state of calculus
instruction in the United States. The conclusions were published
in an MAA report [1], and spawned what has now become known as
The Calculus Reform Movement. Begun as a way to improve student
success in Calculus [2], the movement has evolved into a
rethinking of the entire Calculus curriculum. [3]

During this same decade tremendous advances in the power and availability of Mathematics specific technology ("super graphing calculators" like the TI-85 and "computer algebra systems" like MAPLE) have dramatically impacted the debate about calculus instruction. This impact is succinctly summarized in the recent MAA guidelines [4], which recommend the following: "Departments should examine carefully new ways of presenting material, with particular attention given to the use of computers, graphing calculators, or other modern technology in teaching, learning, and applying mathematics."

The Department of Mathematics began in the Fall of 1993 to incorporate the use of graphing calculator technology into our curriculum, requiring the use of the TI-85 in various sections of selected courses, and we began requiring it in all sections of Calculus I, MATH 1351, in the Fall of 1996. We know that the graphing calculator can be an effective aide in Calculus instruction [5,6], but for this technology to have a positive impact on our student's learning it is clear we must provide our instructors with supplemental material specifically coordinated with our textbooks and course syllabi. Also we must offer guidance into the use of these materials; in particular we must emphasize our basic philosophy that the technology is a tool to help students better understand concepts, not simply a computing tool.

To this end we have created a series of laboratory-style technology modules to supplement the instruction in MATH 1351 and coordinated their use in all sections of MATH 1351. These modules use the TI-85 graphing calculator and will be provided to the students through class handouts; as well as eventually via the TTU Department of Mathematic's home page on the World Wide Web. [http://www.math.ttu.edu.] They are designed to acquaint the students with the particular technology while graphically illustrating the fundamental concepts of elementary calculus: limits, continuity, differentiation, and integration.

The ultimate objective of this project is to use the graphics capabilities of the technology to increase the student's understanding of the calculus concepts listed above; however, it is reasonable to expect to achieve some other important objectives as well. One reasonable objective is to increase the amount of time students spend studying Calculus out of class. Another is to impact positively both student's and instructor's attitudes about MATH 1351, while creating an atmosphere of interactive and cooperative learning.

A recent TTU student survey [7] found that only about 33% of the students in 0301, 0302, 1320, and 1330 reported studying 3 hours or more per week for math. This finding is comparable to a national survey finding that only about 25% of students taking calculus spend more than 4 hours per week studying for calculus. [8] We do not believe that the average TTU student can master the concepts of calculus without spending at least 6 hours per week studying calculus. It is established that using technology tends to cause students to spend more time out of class studying mathematics [9], and these supplementary modules should contribute to this tendency.

Also the use of these technologies can have a positive impact on the attitudes of both students and instructors and create an atmosphere of interactive and cooperative learning. [9,10] Technical empowerment is no small issue in this regard, and we use it to impact positively the attitudes of both the students and instructors of calculus at TTU. The modules are written from the point of view of minimum instruction and maximum student discovery, and students are encouraged to work together on them.

**References.**

*Toward and Lean and Lively Calculus*,**MAA Notes, No. 6**, 1986.- Goodman, "Toward a Pump, Not a Filter,"
*Mosaic, 22*(Summer 1991), 12-21. *Calculus: The Dynamics of Change*,**MAA Notes, No.39**, 1996.*Guidelines for Programs and Departments in Undergraduate Mathematical Sciences*, Mathematical Association of America, February 1993.- Dunham, P.H., "Does Using Calculators Work? The Jury
is Almost In,"
**UME TRENDS, 5(2)**, May 1993, 8-9. - Dunham,P.H. & Dick, T.P., "Research on Graphing
Calculators,"
**MATHEMATICS TEACHER, 87**, 1994, 440-445. - "A Profile of Students in Introductory Math Courses at Texas Tech University," Texas Tech Transition Advisement Center, 1995.
- Wang, Chamount, "The Use of Computers in Learning
Calculus: Issues of Student Commitment,"
**UME TRENDS**, May 1995, 12-13. - Dick, T. & Shaughnewwy, M., "The Influence of
Symbolic/Graphic Calculators on the Perceptions of
Students and Teachers Toward Mathematics,"
**PROCEEDINGS OF THE TENTH ANNUAL MEETING OF PME-NA**, (1988) 327-333. - Emese, G., "The Effects of Guided Discovery Style
Teaching and Graphing Calculator Use in Differential
Calculus," (Ohio State University, 1993),
**DISSERTATION ABSTRACTS INTERNATIONAL, 54, 450A**.

The following sections are available in Adobe PDF , PostScript (PS) and WordPerfect 6.0 format.

1. Introduction to
basic screen calculations and editing |
(PDF, PS, WPD) |

**Objective.** To familiarize the
students with the basic on screen calculation capabilities of the
TI-85. In particular they should learn how to enter an
expression, using standard algebraic notation, onto the screen
and edit when necessary. Of particular interest is the 2nd ENTRY
command which recalls the previous screen entry, and allows it to
be edited, thus allowing for various parameter changes and
computing a complicated algebraic expression without the need to
reenter the entire expression.

2. Introduction to
graphing: Part 1 and Part 2 |
(PDF, PS, WPD) |

**Objective (for part 1).** To
introduce the students to the graphing package on the TI-85. By
simple examples the students learn how to enter functions into
the graphing package to be graphed. They should observe that
functions can be entered in terms of the variable x, or in terms
of expressions involving previously entered functions. They also
learn how to "turn functions on or off" and begin to
see how to use the zoom features to find an appropriate viewing
window. Finally they should observe that the graphs of
polynomials eventually start to look like lines after zooming in
three or four times on a fixed point.

**Objective (for part 2).** To continue
the introduction to graphing of the TI-85 and to explore the
meaning of "steepness of a line." The students should
explore the effect of the change of the parameter m on the graph
of y = m x . They see how to use the TRACE feature of the TI-85
to determine the slope of a line. Finally they explore the effect
of the change of the parameter b on the graph of y = m x + b.

3. Building a
custom catalog |
(PDF, PS, WPD) |

**Objective.** To introduce the
students to the TI-85 catalog. The students will learn how to
build their own custom catalog from the TI-85 catalog. They will
investigate the greatest integer and absolute value functions,
and combinations and compositions of them with polynomial and
trig functions. Using the zoom features they will observe graphs
with corners and the two distinct types of jump discontinuities.

4. Graphical
investigations of limiting behavior |
(PDF, PS, WPD) |

**Objective.** To observe various
limiting behavior of graphs at finite values of x. The students
should observe the following types of limiting behavior: left and
right limits don't match, the limit exists but does not equal the
value of the function, the limit fails to exist by virtue of
oscillations.

5. The sandwich
theorem |
(PDF, PS, WPD) |

**Objective.** To illustrate the
sandwich theorem. The students will see the graphs of y = 1 and y
= cos x come together as x approaches 0 and squeeze the graph of
y = (sin x ) / x between them.

6. Chords and
tangent lines |
(PDF, PS, WPD) |

**Objective.** To illustrate the slopes
of cords approaching the derivative at a point. The students will
exploit the DIST command to observe the effect of cords drawn
from a point on the graph to another point on the graph
approaching the first point. The TRACE feature allows them to
compute the slopes of these cords, and compare these slopes to
the TI-85 calculations of dy/dx at the point in question. Finally
they will use the TANLN feature to add the tangent line to the
graph. They should also observe relationships between tangent
lines having slope 0 and behavior of graphs

7. Derivatives as
the limit of the difference quotient |
(PDF, PS, WPD) |

**Objective.** To enhance the concept
of the derivative as the limite of the difference quotient. The
students will investigate the slope of the graph of y = ln x for
various values of x. They will do this in several ways: computing
the difference quotient directly for small values of h, graphing
the difference quotient (as a function of thse step size x ) and
estimating the value as x approaches 0, and simplying using the
machine tool dy/dx. The students should arive at the conjecture
that d(ln x) / dx = 1 / x .

8. Graphing
derivatives |
(PDF, PS, WPD) |

**Objective.** To learn how to use the
TI-85 differentiating capabilities with the graph package. The
students will learn how to graph a function and its derivatives
on the same axis. This should enhance their understanding of the
relation between the graph of a function and its' first and
second derivatives.

9. Graphing
relations on the TI-85 |
(PDF, PS, WPD) |

**Objective.** To learn how to graph
quadratic relations on the TI-85 and view their tangent lines.
The students will compute slopes to quadratic graphs in two ways:
explicitly by solving via the quadratic formula and implicitly
via implicit differentiation. They will use the explicit
calculations to graph the quadratic graphs on the TI-85 and use
the TANLN package to draw the two tangent lines associated to the
given x value and compute the corresponding slopes, which they
will compare with the above calculations.

10. Newton and the
TI-85 |
(PDF, PS, WPD) |

**Objective.** To explore Newton's
method with the TI-85. The students should understand Newton's
method as an iterative process. They also learn how to write and
store a simple program in the TI-85.

11. Lab
exploration involving Newton's method |
(PDF, PS, WPD) |

**Objective.** To explore the effect of
the initial guess on convergence of Newton's method. The students
will see how sensitive Newton's method can be to variations in
the initial guess. They should develop a good understanding how
the slope of the graph affects the x-intercept of the tangent
line, and hence the values obtained from iterations of Newton's
method.

12. The TI-85 and
the area under a curve |
(PDF, PS, WPD) |

**Objective.** To use the TI-85 to
illustrate the area between two curves. The students should be
able to shade a region defined by to curves and limits on x. They
also see how to use the TI-85 numerical integrator with the graph
package.

13. The TI-85 and
Riemann Sums |
(PDF, PS, WPD) |

**Objective.** To realize area under a
curve as the limit of rectangular areas. The students should be
able to use the sum and seq commands to approximate the area
under the graph of a positive function by sums of areas of
rectangles. They should eventually understand the idea of using
the limit of Riemann Sums to define the definite integral.

14. Area
estimations with the TI-85 |
(PDF, PS, WPD) |

**Objective.** To use the TI-85 to
estimate definite integrals within specified error bounds.
Students should understand why the actual area under the graph of
a monotone function is between the left and right box
approximations. They should be able to use this information and
their TI-85's to approximate area to within specified error
bounds, and to check their answers with those given by the TI-85
numerical integrator.