Exploring Math with the TI-89

Technology changes possibilities

The TI-89 is a graphing, programmable calculator with symbolic manipulation capabilities. I know that is a mouthful but we can break it down to three main features:

1) This device graphs not only functions of one variable; it also graphs functions of two variables, parametric curves, functions given in polar coordinates, and data from number tables.

2) Programmable means it has commands you can use for writing code that is directly executable on the device. You can save, edit, and execute programs in it. This gives you plenty of flexibility when it comes to calculations for complex problems, because you are not restricted to the built-in operations.

3) Symbolic manipulation means the device displays, and uses standard mathematical symbols like integral, derivative, roots, exponents, and fractions, in a way that is consistent with algebra rules. That means it is not only a number-crunching little machine, but it does many operations of algebra, and calculus, like factoring polynomials, solving equations, multiplying matrices, deriving functions, and finding definite integrals.

All these capabilities present opportunities for teaching math. For example, using this calculator it is very easy to show relationships between formulas of functions, and their graphs. Presenting these relationships in a book usually takes several pages. Students often have some trouble reading, believing, understanding, remembering, and applying the general rules for this level of algebra of functions. On the other hand, typing the examples from the book into the calculator, and seeing the graphs of the functions being displayed in real time in front of their eyes, gives them more confidence to “believe in the evidence.” Therefore they can be puzzled but they do not reject out of hand the result shown. On the contrary, given the easy-to-use editing capabilities of the function editor, they can experiment with the function formulas by changing their parameter values and seeing what happens with the graphic output. In my opinion this may make learning more interactive, fun, and effective.

A Number Divided By Another Number

One more instance of Math not being English

Lately I had a few students who had some trouble with fractions. Part of the problem was a very specific type of confusion at the time of setting up a division calculation. When asked “How much is 5 divided by 12?” for example, they would sometimes correctly calculate the result but most other times they would set up the division as 12 divided by 5. One of them asked me a few times if the result would be the same. When I tried to explain that division is not commutative, using some visual representations of fractions, he was not totally convinced. So I just pulled the calculator, I asked him to give me a couple numbers, and I did both divisions (let’s say, 17/4 and 4/17) with the calculator, showing him the results. We repeated the “experiment” with two other examples, and then he was convinced that division is not commutative. However, such discovery created some anxiety in him because he doubted he would choose the right order of calculation in any given problem. Actually, it took multiple repetitions on my part for him to finally learn how to set up the right calculation when a word problem involves the phrase “divided by.” At first he wanted to transliterate the written phrase (“105 divided by 15” for example) word by word, and number by number, in the exact same order into the division calculation by writing, from left to right, the 105 first, outside the division symbol, then the division symbol (the one that looks like a rotated “L”), and finally the 15, inside the division symbol. I would tell him that the order of the numbers in the phrase “105 divided by 15” is backwards to the order of the same numbers in the actual calculation but this only seemed to surprise him, and confuse him. Once more, I resorted to the calculator. I said: “O.K. just do the division.” When he asked: “In what order?” I said: “Do them both.” Once he had calculated both results by long-hand, I gave him the calculator, and I said, “Now do them both with the calculator.” As he punched the calculator keys, I directed his attention to what sequence in the calculator was giving him the same result as he had calculated long-hand. I said “Do you see when you enter into the calculator: ‘105, division symbol, 15, enter’ that gives you the same result as when you did ’15, division symbol, 105’ by long-hand?” When he saw this evidence he still said “It is confusing.” Then I said: “Yes, I know, but that is just the way it is, so you are going to have to remember it that way. In the calculator division the numbers go in the same order as they are in the phrase ‘105 divided by 15,’ whereas in the long-hand division, to get the same result, the numbers have to go backwards.” It took some repetition over a few tutoring sessions but he finally got it consistently right.