Developmental Mathematics Revival!

Radicals, the Game

We have been working on simplifying radicals this week in my intermediate algebra classes, and I can’t help but think … why? Where does anybody need to (1) simplify square roots and cube roots that represent irrational numbers and (2) perform operations on radical expressions?

I am not opposed to including some radical concepts, especially connecting them to fractional exponents (so we can include exponential functions), and knowing that some radicals represent irrational or imaginary numbers is basic enough — and useful enough — to include. The issue is WHO CARES if we can write √(80) as 4√5 ?

Within our algebra courses, we use ‘simplifying’ radicals when we solve quadratic equations with irrational solutions (or complex). That is not what I am talking about; rather, in situations leading to a quadratic equation with either irrational or complex solutions, do we really need to express those solutions in ‘simplified radical form’?

Factoring is often listed as a, well, useless topic in algebra. Until we replace our algebra courses with something better, I actually do not mind covering some factoring — even trinomial factoring. The value with factoring is that it really deals with basic concepts of terms and equivalent expressions; much of our mathematical capacity is based on our being able to envision alternative but equivalent expressions.

I do not see the same payoff with radicals, either simplifying or operations (which also involves simplifying). When I explore with my students why they make mistakes with radicals, it’s not normally some basic issue that is a barrier; more often than not, it is the strange radical notation involved in the work. It’s not that they do not understand that an index of 3 means ‘cube root’, it’s the standard moves of the radical game that cause the problems.

Sure, a radical is equivalent to a product of radicals involving factors of the original. Students get that a number can be factored in different ways. It’s the particular partitioning of factors. Sure, we can teach prime factoring and circling groups according to the index, or we can teach a calculator trick to find the magic partitioning of factors. This partitioning is conceptually similar to partial fractions; with radicals, I think students feel like we are working backwards, as we are with partial fractions.

Seems to me that we’ve made ‘radicals, the game’. The focus ends up on the legal moves allowed, and the format the answers can be. The basics (meaning of radicals or fractional exponents, domain) tend to be de-emphasized. And, honestly, I could use these two weeks for other topics that would help my students more than this game.