TI 83 Workshop

Open the front of your TI-83. The cover fits onto the back in two ways: It serves as a stand or it covers the back. In your classroom, the cover should always be covering the back, where it provides extra cushioning and covers the battery panel.

Screen contrast is adjusted by pressing, and removing your finger from, 2^nd, then holding down either the up or the down arrow. (Your students will drive you up the wall for months as they, somehow, persist in getting this sequence confused.) The numbers in the upper right corner of the screen show the level of contrast. If you need the contrast at 8, 9, or 10, you probably need to charge or change your batteries.

Pressing 2^nd, ON invokes the OFF command and turns the TI-83 off. If you forget to turn the machine off it will stop itself after being ignored for approximately four minutes. No matter how the calculator gets turned off it will preserve its screen and all memory for later use.

Numbers are entered by pressing keys. Operations are keyed between numbers. ENTER instructs the calculator to simplify any expression.

Uses of function keys vary. The x^-1 and the x² keys are struck after entering expressions; note that exponents work on what they touch. The LOG, LN, square root, exponential, and trigonometry keys need to be pressed before the numbers on which they act; parentheses should be closed, but the TI-83 allows final right parentheses to be left off. The comma, parentheses, and exponent arrow keys ( , ( ) ^ ) are pressed, as needed, in sequence with numbers and operations. STO> puts the previously entered expression or result into the chosen memory; this requires use of either a list or a letter.

Normal Eng Sci. "Normal" numbers are presented with no exponents. "Sci" presents numbers in calculator scientific notation, showing the number with one digit to the left of the decimal point, all other digits to the right, and a two digit attachment that shows the power of ten by which the number is multiplied. "Eng" presents numbers in engineering notation, a form of scientific notation in which there can be from one to three digits to the left of the decimal point, and in which the powers of ten are expressed only in multiples of three.

Float 0123456789. Changing the default from "Float" to a chosen value instructs the calculator to present all calculation results with the chosen number of decimal places.

Radian Degree. This must be changed in order to make calculations in degrees. This is a great thing to check when students start getting bizarre answers to geometry and trigonometry questions.

Func Par Pol Seq. This row chooses functional, parametric, polar, or sequential graphing. It also controls the X, T,q, h keys. The independent variable in a regular coordinate function is X, T is independent in parametric graphing, q is the independent (radian) angle (arc length) measure in polar coordinates and functions, and h is the independent counter of recursions in sequential graphs.

Examples
Function: Set graphing mode to "Func." Key Y= and input 4x - 2 on the Y1 row. Key GRAPH. Try putting 2/3x - 2 on the Y1 row. GRAPH. Note that implied multiplication (3x) is done before the division of 2/3. Correct this by keying (2/3)x - 2 and graphing again. When your students make this error - and they most assuredly will - have them recognize that dividing by zero makes a mess of any line.

Other functions to explore include the polynomial, square root, trigonometric, logarithmic, and exponential functions.

Parametric: When an X variable and a Y variable depend not directly upon each other, but upon a third variable that influences them both, a graph is determined by this third parameter. In most cases, this parameter is some measure of time, hence the use of T as the independent variable; X and Y are dependent in parametric graphs.

As a standard example, consider the flight of a ball. The horizontal position of the ball is given by the product of its initial velocity, the cosine of its angle of inclination, and the variable T, a measure of time. The ball's vertical position is modeled using initial height, the sine of the ball's angle of inclination, and the constant force of gravity. These combine with T and some first semester Calculus to produce a function for altitude. To watch a ball fly, set MODE to "Par" and key Y=. Assume an angle of inclination of 30°. A ball thrown at 100 miles per hour travels 146 2/3 feet per second, so let X1 = (146 + 2/3)cos (30°)T. Say that the ball is released eight feet above the ground. Then, the vertical position of the ball can be modeled by Y1 = -16T² + (146 + 2/3)(sin 30°)T + 8. Key these in and graph the flight of the ball. Next, for X2 and Y2, model a ball at a different speed, released from a different height, with a different angle of elevation. Change the "Sequential Simul" MODE to "Simul" and watch both balls fly. To stop flight, key ON. To start over, GRAPH. To follow a path, key TRACE. How long does it take each ball to hit the ground? Note that, by changing any one coefficient (angle of elevation, or initial velocity, or initial altitude) students can explore, and even attempt to maximize, the lengths of flights of balls. They could even find ways to estimate their own throwing speeds. Note, also, that setting the angle of elevation at 45° can eliminate the need to use the sine and cosine functions, thereby simplifying the functions that students need use.

Polar: Polar coordinates are used to graph r, a distance, as a function of q, an angle measured in radians. Key Y= and let r1= sin(q). View the WINDOW, the GRAPH and the TABLE. Try graphing r1= sin(q)cos(q).

Sequential: Sequential graphing allows the user to TRACE through sets of sequential results. This allows use of recursive techniques such as Newton's Method and a powerful algorithm for calculating square roots.

To calculate a square root, set MODE at "Seq" and, one line down, "Dot". Keying 2^nd ZOOM accesses FORMAT. Here, choose "Web" as your graph type, then key to Y=. The recursive formula u_n+1 = .5(u_n + N/u_n) converges to the square root of N. To use this, choose N equal to some number, say 12, and make a guess, which does not have to be a good one. Once this is done, enter u(µ) = .5(u(µ - 1) + 12/ u(µ - 1)). For u(µMin) enter your guess. Key WINDOW. Key GRAPH. The x and y axes on this graph are referents for the y = x line, toward which the sequence converges; in other words, the recursion pushes two variables together, to where they both, in this case, reach the square root of 12. Key TRACE. Read that µ = 1, so this is the starting iteration; x = your guess; and y = 0, because no second estimate has been generated. Press the right arrow. Keep going. No matter what you initially guessed, you converge quickly to a six decimal place estimate of the square root of 12.

The Next Lines. The "Connected Dot" and "Sequential Simul" lines have already been explored. The "Real a+bi re^qi" line allows conversion between real number, complex coordinate vector, and complex polar vector operations.

Full Horiz G-T. The "Full" option leaves one screen on the calculator. "Horiz" puts the graph screen at the top and another screen (main screen, TABLE, Y=, for example) at the bottom. "G - T" places GRAPH on the left side of the screen and TABLE on the right. This is especially useful for presentations.

Exploring MATH
The MATH key opens the MATH menu, which contains four top-row headings, each of which allows access to a set of numbered options. There are ten of these under MATH MATH.

To use the first option, which allows work with fractions, go to the main screen, enter a decimal number, and key MATH ENTER. To undo the conversion, key MATH then arrow down to 2: >Dec. Options three, four, and five, give the cube, the cube root, and the xth root of a number entered on the main screen. To find the 6th root of 9, enter 6, access the xth root key, then enter 9. This can, of course, be done from the main screen by keying 9 ^(1/6).

Options seven through nine perform numerical estimates of the basic operations of Calculus. Use these keys by accessing the MATH screen, choosing the desired option, then inputting the functional expression, the variable with respect to which the operation should be conducted, and left and right independent variable boundaries; e.g., fMax(-X² + 5X - 14, X, -10, 10) estimates that the function maximizes when X is near 2.5.

The last MATH option is the "Solver". Choosing this opens a new screen that features the eqn:0= prompt. At this, type the expression to solve and ENTER. On the new screen, arrow to the row of the variable for which you are solving and key ALPHA ENTER to activate SOLVE. The TI-83 will use the estimate on the screen to produce its estimate for the nearest solution to the equation. Further solutions, if they exist, can be found by changing the displayed X; the calculator will continue to move along the horizontal axis until it finds another approximate zero. Note that this relatively unsophisticated solver can, in less than seconds, do more than most of us learned how to do in our first thirteen years of schooling.

The MATH NUM menu is almost self-explanatory. The first four options give the absolute value, nearest integer value, integer part, and fractional part of a number. To use, min, max, lcm, and gcd, choose the option and enter numbers, separated by commas, on which the TI-83 is to operate.

The MATH CPX menu allows work with complex numbers. Given a complex number, the first three options return its conjugate, real part, and imaginary part, respectively. "Angle(" gives the angle between the real axis and the complex vector; "abs( " returns the magnitude of the vector. The last two options convert between polar and rectangular vector expressions.

MATH PRB gives probability tools. Option "1: rand" generates a random number from a uniform distribution between 0 and 1. The permutation option, used by entering n, then "nPr", then r, gives the number of ways that n objects can be arranged into r spaces; order matters in permutations. The combination option, used in the same way, gives the number of r-element subsets that can be formed from an n-element set; combinations are the numbers in Pascal's Triangle, in the Binomial Theorem, and in most playing card problems.

The fourth PRB option is the factorial key; note that the TI-83 expands the mathematical world by solving for factorials of positive decimals and negative integers. The remaining three options generate random integers, random values drawn from the Standard Normal distribution, and random values drawn from a binomial distribution. Each of these can generate a list of values that can be saved for later work. For example, "randInt(0, 20, 15)" generates a list of 15 randomly selected integers between 10 and 20, inclusive. STO> the list in L1.

Using STAT
The STAT key opens the window to Standards-based math. It allows the analysis of data on which the learning of mathematical skills and thinking is based.

STAT EDIT begins with "1: Edit" which opens the spreadsheet-like list screen; it has at least twenty columns. To enter data, arrow to the top of L2, key in data, ENTER (or arrow down), and repeat until all data is in. Lists can also be filled, as on a spreadsheet, by formulas along the top row of the screen.

Once data is in it can be viewed in a scatter plot. Key 2^nd , Y= to activate STAT PLOT. ENTER on "1: Plot1". The first thing to do is to ENTER on "On"; this activates the plot. Arrowing to "Type" allows the choice of a scatter plot; an xy line, which is a scatter plot with points connected; a histogram, for use with one variable; a modified box plot, a one variable diagram that shows data outliers; a standard box plot, which shows the median, interquartile range, min, and max, of a data set; or a normal probability plot, a quick test that indicates the normality of data. Of these, the scatter plot and the standard box plot are the most useful. Note that plots can be toggled on and off on both the STAT PLOTS and the Y= screens.

The second and third STAT EDIT options sort lists in ascending and descending order, respectively. Choosing one of these options returns the user to the main screen, where the user uses the 2^nd option to choose a first list. This list will be sorted. No other list will change. If more than one list is chosen, the first list is sorted and the rows of the other selected lists are carried along in that sort. The "ClrList" option is self-explanatory and the "SetUpEditor" takes any existing lists and stores them.

STAT CALC might be the TI-83's most powerful menu. Option "1: 1-Var Stats" gives the mean, the sum, the sum of the squares, the sample standard deviation, the population variance, the number, the min, the first quartile point, the median, the third quartile point, and the max of the chosen list; it defaults to L1. Option "2: 2-Var Stats" gives the mean, sum, sum of squares, sample standard deviation, population variance, and number for an X-list and a Y-list, as chosen; the calculator defaults to L1 and L2 when no lists are specified. This option goes on to provide the sum of the XY products and mins and maxes for both variables.

Option "3: Med-Med" calculates the median-median line for an X and a Y variable; as usual, it defaults to L1 and L2. Option "4: LinReg( ax + b)" applies the least squares method and calculates a line of best fit for the data; a slightly different form of this is done in Option 8. Quadratic, cubic, quartic, natural logarithmic, exponential, power, logistic, and sinusoidal regressions are available in other options.

STAT TESTS allows hypothesis tests using Gaussian (Normal), Student's t, Chi-Square, and F distributions. Options with the "Int" suffix provide confidence intervals. To conduct a hypothesis test, input data to one or more lists. Next, key STAT TESTS and choose a procedure. At the Input line, choose Data to have the TI-83 calculate statistics using the contents of the list specified on the "List:" line; choose "Stats" if you already have the sample mean(s) and variance(s), or standard deviation(s). After inputting data at the remaining prompts, select a hypothesis at the "m1:" line. Finally, select "Calculate" or "Draw". The first of these returns numerical results, including a p-value. The Draw option shades the p area under the appropriate distribution and displays both the calculated statistic and the p-value.

The most important STAT TESTS option for school mathematics is probably "E:LinRegTTest", which derives a least-squares line, conducts a hypothesis test on the usefulness of the slope parameter (hence the usefulness of the line), and reports both r and r2 for the model. Further, the TI-83 stores the regression line under VARS, and the line can, like any other regression line, be called to the Y= screen for graphing.

As above, graph a function by entering an equation on the Y= screen, then keying GRAPH. More than one function can be graphed simply by entering them on different Y= lines. Try, for example, to graph Y1= 3X - 2 and Y2= -X^2 + 4. Key GRAPH and see both the line and the downward-opening parabola.

Seeing the two graphs on the screen raises a set of essential mathematical questions: Where are the graphs' zeroes? Where are the y-intercepts? Where do the graphs intersect? The TI-83 offers two ways to answer these questions. First, key 2^nd GRAPH to access TABLE. Finding y-intercepts is a trivial exercise for any student who knows what a y-intercept is; it's a discovery exercise for anyone who doesn't. Arrowing up and down the X column shows that the parabola zeroes at X = -2 and at X = 2. The line zeroes somewhere between X = 0 and X =1. If it's important to find the exact point, key 2^nd WINDOW to set the table. Since we want to check X's between 0 and 1, let the table start at 0, and then let "DTbl" be something small, like .01. Return to TABLE and find that the line crosses the X-axis somewhere between .66 and .67. Students should have the number sense necessary to draw a conclusion from this information. The graphs intersect somewhere between X = 1 and X = 2. Changing "TblStart" to 1, and reading, allow the student to decide that the graphs intersect between X = 1.37 and X = 1.38. Further adjustment of the table can either identify the exact point or, better, force the student to make a defensible decision about the level of answer precision to use in a given problem situation. The second method for finding intercepts, zeroes, and intersections involves examinations of the graphs themselves. On the GRAPH screen, key TRACE. At the top left of the screen the TI-83 shows the function on which it has set the cursor; to change from one function to the other press either the up or the down arrow. On the parabola, arrow left and watch as the cursor position and the X = and Y = values change. Note where the graph crosses the horizontal axis. Arrowing right helps identify the parabola's y-intercept, the point at which the two graphs intersect, and the right zero of the quadratic.

To find the best possible estimate of the point of intersection move the cursor, along either graph, to near the point of intersection. Key ZOOM. Option "1:Zbox" allows the user to control the zoom, so select it. To zoom, key ENTER; this anchors one corner of the box at the position of the cursor. Arrow horizontally and vertically to construct a box around the point of intersection, then ENTER. Repeat this process until you are satisfied with the estimated point of intersection.

The graphing capability of the TI-83 has a variety of other features. On the Y= screen, for example, go to the Y1= line and arrow to the far left. The default option here is a connected line. To change this press ENTER to activate a heavier line. ENTER again, and then some more, to see the shade above, shade below, path of a ball with line, path of a ball, and dotted line options. Arrowing to the "=" column allows use of ENTER to toggle the graph off and on. If, at this screen, you want to call a line from memory rather than typing in an equation try setting the cursor on an unused row, pressing VARS, keying down to and choosing "5: Statistics". At the next menu, arrow across to EQ and ENTER on "1:RegEQ". The last regression equation is now available for graphing. With this in place, clear (or turn off) all other Y= lines and press 2^nd Y=. Activate the first plot. Key GRAPH. ZOOM 9 activates "ZoomStat" and gives the best possible view of the scatter plot and its associated line of best fit. This is how student-collected data gets converted to the lines on which students learn about slopes and y-intercepts.

Next, reset the calculator and find where Y = -(5/3)X + 28 intersects Y = -X2 - X + 132. Note that the calculator does not automatically show these graphs. To find them, your students will have to know something about the functions with which they're working. They might begin to find them by keying WINDOW and changing "Ymax" to 30. Knowledge of graphs might suggest, next, that one point of intersection is up and to the left, and another down and to the right. Go to WINDOW and change "Xmax" to 15. Change "Ymax" to 50. In this way, the student applies an understanding of graphs with a variety of estimates to both find the points of intersection and strengthen the very estimation skills that calculator critics contend are undermined by technology. Used well, calculators employ number sense and estimation in ways that strengthen understandings of deeper mathematics and impress students with their own mathematical judgments.

Direct comments to: mibrown@nmsu.edu