Optimization - Graphically, Numerically and Analytically with the TI-92
by Josef Böhm
(bk teachware series "Support in Learning" - ISBN 3-901769-20-x)

This 48 page booklet is a recent addition to the bk teachware series which aims to provide practical guidance in the use of new technology in mathematics teaching. Other books in this series have been reviewed in previous issues of Maths & Stats (Vol 9, Nr 4, pp39-42 and Vol 10, Nr 2, pp36-38).

Josef Böhm, the author of this booklet, is well known as the founder of the DERIVE and TI-92 User Group, and as an enthusiastic and often inspirational conference speaker. The contents of this booklet originate from a TI-92 calculator workshop he has run for teachers, and this is reflected in its hands-on, directed style supported by a great many screendumps.

Böhm's premise - indeed the premise of most advocates of new technology in the teaching of mathematics - is that there should be a fluent and confident interplay between graphical, numerical and analytical approaches to problem solving, and that modern calculator technology allows the user to "shuttle" between these various approaches. The Texas Instruments TI-92 incorporates symbolic manipulation (also known as computer algebra) together with powerful graphing facilities, data editing and curve fitting, and is thus an ideal tool to meet these objectives.

However, the feature that makes this booklet really stand out is the use of the TI-92's interactive Geometry Tool (a variant of Cabri Geometre). Böhm expertly shows how this can be deployed with classical optimisation problems to provide a visual demonstration of how varying the physical properties of geometric constructions alters the calculated measurements. These dynamic geometry models can either be used by students on their own TI-92s, or projected onto a screen by the teacher for class demonstration and discussion.

The booklet starts with the by now almost hackneyed problem of cutting equal squares from the corners of a rectangular sheet of card, folding up the flaps to form an open box, and determining the size of square which would maximise the volume of that box. I must admit I inwardly groaned when I saw that example being used yet again, but it soon became clear how the familiarity of problem allows the reader to appreciate the novelty of the approach which the TI-92 allows. Böhm immediately introduces a dynamic geometry model of the situation, which is provided on the accompanying floppy disk (to be downloaded from PC to the TI-92 using the TI-GraphLink, which must be purchased separately). The user sees a net of the box (cut from a 45 by 25 rectangle), and can "grab" and move a point on screen which varies the size of the cut-out square. The length of the side of the square and the resulting volume are automatically displayed (Fig 1).

This is something which I wish I had had much earlier in my teaching career! For groups of students who consider themselves too sophisticated to make such boxes out of cardboard and measure them, and who would soon lose interest when sketching various nets and repeatedly calculating their dimensions and volumes, here we have an on-screen display which shows in "real time" how the volume increases and then decreases as the size of the cut-out square increases. . Immediate motivation to seek the optimum value!

Böhm proceeds to show how the various features of the TI-92 can be harnessed to progress the problem. A single keystroke (Ctrl+D) allows the current (x,V) data pair to be transferred to a Data Table. Thus by making slight changes to the geometry model and transferring the data, a spreadsheet-style table can be generated, and the result graphed (Fig 2).

The TI-92 offers a variety of regression models. Böhm demonstrates how a quadratic function, perhaps a first choice by some students who see the U-shaped data plot, seems at first to be a reasonable fit, but turns out to be inappropriate after careful examination of the values found by using the Trace facility, and by Zooming in to see how well the graph fits the data points. A cubic regression model appears more suitable (Fig 3).

The coefficients given by the TI-92 for the cubic regression model can be compared now with the algebraic formulation V = (45 - 2x)*(25 - 2x)*x. On establishing that these are identical, the computer algebra facilities of the TI-92 can be brought into play to help determine the analytical solution of the problem (Fig 4). This in turn can be compared numerically and graphically with the data table and regression function already found, thus completing the methodological circle.

In subsequent chapters, Böhm gives equally detailed treatment of three further optimisation problems: finding the least cost of a strut consisting of two variable lengths of different materials; finding the largest perimeter and area of a rectangle inscribed within a trapezium; finding the maximum profit given variable production cost and revenue functions. The first two of these are accessible to a similar student audience to the "maximum box" problem. The last problem was specifically chosen to be relevant to students of economics. Each of these problems also has a dynamic geometry model which can be loaded from the accompanying floppy disk.

These further worked problems allow additional features of the TI-92 to be introduced. Perhaps the most interesting of these is the Function Table allied to the graph editor. If a graph has been defined in the form y(x) = , then switching to the Function Table will show a spreadsheet-style array of x-values with their corresponding y-values. By editing the value of the table start and the x increment, the user can "zoom in" numerically to the solution. While this is a standard technique for spreadsheet users, it is nice to see the facility incorporated in the TI-92, whereby formulae obtained from manipulations in the algebra window can be copied and pasted directly into the table for numerical investigation. Again, this advocates the interplay between functions, data and graphs.

At the end of the booklet is a collection of fifteen additional optimisation problems, each of which has a corresponding dynamic model on the accompanying floppy disk. The problems are stated without solution, and cover fairly traditional scenarios such as maximising the area of a rectangle inscribed within a sector, as well as more practical problems such as minimising the cost of a route made up of two sections. They should all be accessible by students working at A-level or equivalent standard, and most could be approached numerically and graphically by students on "pre-calculus" courses. I found that all of the dynamic geometry files loaded without problem using the TI-GraphLink, and I enjoyed experimenting with them immensely. Böhm must be congratulated on the attention to detail he put into the provision of these.

One chapter in the booklet is devoted to a step-by-step guide on how to produce yourself the dynamic geometry model of the "maximum box" problem with which the booklet starts. For the reader who has previously dabbled with the TI-92 Geometry Tool, this will undoubtedly be a source of ideas and inspiration for further implementation. However, novice users would have a very steep learning curve to get to grips with this.

Overall, I can highly recommend this booklet. It represents a "masterclass", not only in the integrated use of the various technical features of the TI-92, but also in its underlying pedagogy of motivating an interplay between the numerical, graphical and analytical aspects of problem solving. By focusing on the topic of optimisation - one which is fundamental to "real" mathematics, but one which students often find difficult to fully appreciate - and by providing a wealth of ready-made dynamic geometry objects to encourage visualisation and investigation, Josef Böhm provides us with an eminently useable set of examples which reflect his infectious enthusiasm and acknowledged experience of teaching with technology. The richness of this slim booklet far eclipses the minor nature of its occasional typographical errors and quibbles with regard to layout.

If you are at all interested in how the latest calculator technologies can enhance mathematics teaching and learning (by reading this Newsletter, you are by definition!), and if you already have a passing familiarity with the TI-92, you really should buy and work through this booklet. It is one of the most rewarding publications of its kind that I have come across.