Practice Problems, Patterned Practice

A characteristic of problem collections in standard textbooks is that each question tends to stand alone, that attempts at varying the problems in a systematic way in order to elicit patterns are typically few and relegated to the introductory section in each chapter, to be dropped after the rules have been listed and boxed. The incentive to add my own problems is usually a wish for more practice of the kind where each item belongs in a set that has some internal logic, and in the section on evaluating powers with negative and non-integer exponents this is particularly useful, I think.

The very idea of exponents that are not counting numbers causes the students a fair amount of puzzlement and headache. The common misconceptions that negative exponents change the sign of the expression, and that a base raised to the power of zero must be zero, seem very resistant to instruction. So far, my main strategy is to repeatedly place such powers in a sequence and to make a big deal of how they MUST have the values they have in order to follow the pattern we see with positive integer exponents. We complete these tables again and again, graphing the results to see how the rules for negative and zero exponents are nicely continuous with the ones the students know intuitively.

This year I’ve spent more time on treating rational powers the same way. We have, for example, evaluated 27^x for sequences of x-values that are multiples of 1/3, and (81)^x for sequences of x-values that are multiples of 1/4. In completing these tables, the powers with whole number exponents become familiar reference points that students can check their rational powers against, and it becomes clearer that the rules for non-integer powers are not random or contrived, but that they are what they are because they have to be that way:



This also, incidentally, reinforces a basic lesson on fractions that continues to be necessary in Algebra 2: that fractions just are numbers, that they belong on particular places on the number line, and that they behave just as you would expect based on what integers they fall between.

My worksheet on graphing exponential functions for Algebra 2 is here. I’ve just started using Sketchpad for the grids, and am finding it tricky to get them to look neat. Oversized dots, tiny numerals and inconvenient scales continue to be issues, and kids complain a bit about these things. I’d appreciate technical advice! My introductory worksheet on exponential functions for Algebra 1 focuses more on the sheer prettiness of the graphs of 2^x and its reflections when they are drawn on the same grid in colored pencil (this particular sample from a freshman of last year who elected to do her Arts project on Escher's work):