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.

Look for Solutions with Less Math and More Logic.

One instance where often “less is more.”

The following question is an excellent guideline for solving math word problems:
“How can I solve this problem by doing the least possible amount of math?”

Oftentimes there are several pathways from the setting of a problem to its final solution. Some routes are safer, while some are riskier, more error-prone. Some routes are faster, while some are time-consuming. Some routes are clearer, while some may be confusing.
Usually, the routes with more elementary operations (especially long division), and bigger numbers, tend to be lengthier, longer, and riskier, because adding, multiplying, and dividing big numbers or expressions requires a laser-focus attention. There are plenty of opportunities for doing silly mistakes during these calculations. Besides, it is easy to lose sight of the big picture when worrying about the accuracy of the calculations.
Factoring whole numbers and algebraic expressions is a good habit because it allows you to simplify some expressions before diving into the calculations, so you can operate with smaller numbers, gaining time, and accuracy.
Using logic is a very good habit, too. Many problems lend themselves to solutions that involve more reasoning, and less calculation. This is usually a good thing because these solutions tend to be clearer, and shorter.
Organizing all the information about a problem in a way that makes sense to you, is an excellent habit because this way you keep track of where you are and what you are doing all the time through the problem, and having all these references available makes it easy to retrace your steps, and identify any possible mistakes.
Go visual at any opportunity. Pictures, drawings, charts, graphs, and tables often are a huge help in writing down the right equations, or even in avoiding equations altogether sometimes.
There are many problems you can solve with a drawing and a little logic. Just because the problem is a math problem, that does not mean you need to write down an equation to solve it.
Focus on your possibilities, on what you can do. Organize the information in a logical way, using a drawing, or a table. Above all, try to spend the least possible amount of time and energy doing long, detailed, time-consuming calculations. Instead, simplify the expressions, and ask yourself logical questions about the problem.