Let’s take a look at three monomials (terms) that one might be expected to be able to simplify in a Pre-Algebra or Algebra 1 class:
If you were discussing traditional exponent rules as you might find in a math textbook, how would you approach these problems? I’ve put a hypothetical conversation below.
Teacher: “Multiply the coefficients and add the exponents of like variables”
Well-trained and obedient students: “OK”
Teacher: Take care of the exponent outside the parentheses first. Multiply those powers.
Well-meaning students: Wait, you just said to add them
Teacher: Not when it’s a power to a power.
Obedient but increasingly skeptical student: OK…
Teacher: Then subtract them with the ones in the denominators
Students, before teacher can even begin talking about this problem: this , now there are negative exponents?! I’m out.
Exponent rules stink. Except when they don’t. And they don’t stink when students discover them for themselves.
What if I told you that students can treat all three of those problems like they’re almost the same problem? No different rules to follow, no different steps, no wondering why sometimes you add, subtract, or multiply the exponents depending on their location?
The answer is found in Prime Factorization. The key is to laying the groundwork early; students need to understand how numbers are built according to the Fundamental Theorem of Arithmetic. They need to see how GCF and LCM can be found only using Prime Factorization. They need to see how fractions can be reduced only using Prime Factorization (no shortcuts). This can take a couple of weeks. But after that, exponent “rules” can be taught in a day, because you’re not teaching any exponent rules.
Check it out:
Because of allllll the work we did with prime factorization this section on monomials feels like review to my students, for the most part.
They’ll need a walkthrough of how negative exponents work, but they buy into it and understand it almost immediately — again, because of Prime Factorization:
Wait a minute, you say — to make the connection indicated in the above picture with negative exponents, won’t they have to know some exponent rules? And all of your examples have tiny exponents — what happens if they have to raise something to the 15th power? Or the 50th?! I bet they won’t want to do Prime Factorization then!
Here’s the thing — my students all know the exponent rules. Even though this whole blog post has been about not teaching the exponent, 90% of my Pre-Algebra kids understand the 4 or 5 basic exponent rules by the time they get completed this homework assignment. After a few examples, students are always asking me questions that start with “Can’t I just…” or “Why don’t we just…” or “Isn’t it easier to…” Because when true understanding is created (using Prime Factorization) the exponent rules are apparent and obvious to anyone who’s been paying attention.
Like this blog post? Check out the tags below (“For Teacher” “#MTBoS”) for more teaching tips.