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:
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.
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.