# Mistakes, Radicals, Rational Exponents, and Partitioning?

A strange thing happened a few days ago. One of my Algebra 1 students stopped by Thursday afternoon to receive extra help on a topic. (That in itself is not the strange part.) He pulled up a chair, we discussed what he had been struggling with, and from that I typed out a review handout for him to work on while I helped another student from another class.

I asked him to complete as many of the problems as he could during the next 10 minutes or so. Then we’d chat about what he got right, what he got wrong, and what he skipped. A little while later he had this:

## Trouble with Radicals, But Not with Rational Exponents?

When we first sat down, I figured he would have trouble with rational exponents. Nearly all of my students (this year and in the past) who struggle with this topic have almost no trouble with square roots in radical form, some trouble with cube roots in radical form, and (if they have any issues at all) massive problems with rational exponents.

This student’s struggle was more or less the exact opposite of what I typically see. He had trouble with radicals (square roots, cube roots, anything in radical form), and almost no trouble with rational exponents. (My conjecture on #21 is that the times table in his brain has a blank spot at 8 times 8.)

## Concept vs Notation

The evidence clearly shows that this student doesn’t have a conceptual deficiency. Instead, his struggle is with…

Remediation for notation is usually fairly simple. We talked for a minute or two about how radicals (unknown and unfamiliar to him) relate back to rational exponents (known and familiar). Several minutes later, he came back with this:

Things are looking up, even if they’re still not perfect.

## Shifting My Approach

This exchange has me rethinking the direction of the conceptual/notational connection I’ve been trying to draw out for years. In working with an expression raised to the 1/2, I’ve always angled our conversations toward (and silently rejoiced inside when a student shouts):

I treated square roots like the native language, the most helpful representation, and rational exponents as this foreign thing that needs to be converted back to familiar territory.

It’s true that students are more familiar with radicals (at least in my experience with middle schoolers), but I’m quickly starting to believe that rational exponents are dramatically more informative when it comes to thinking conceptually (and when it comes to working procedurally).

## August, Every Year

When students enter my classroom, our first discussion about exponents (which invariable happens within the first couple weeks of school) goes more or less like this:

Me: What does $2^{3}$ mean?

Students: Eight!

Me: No, not “What is its value?” What does it mean, what does it represent?

Students: Oh (why didn’t you say so). 2 times itself three times.

Me: What?! You mean…

$2$
(two)

$2\times 2$
(times itself once)

$2\times 2\times 2$
(times itself a second time)

$2\times 2\times 2\times 2$
(times itself a third time)

Me: There. 2 times itself three times. Wait… That’s…

Students: No, you got it wrong. That’s 2 to the fourth!

Me: But you said…

Students: Yeah, but we didn’t mean…

Me: Grrr…

### TWO MINUTE TIME WARP…

Me: And that’s why it’s more useful to say it like that. So, how would you state the meaning of this: $10^{4}$

Students (in unison, with a three-part harmony): Four factors of 10!

Me: Perfect!

I want them to express powers this way for a number of reasons. At the very least, saying it the other way is flat out wrong. But describing $a^{b}$ as “b factors of a” has proven immensely useful in developing properties of exponents (which, for what it’s worth, I don’t hate as much as many in the MTBoS, probably because I’m easily entertained, and maybe also because my simple brain enjoys finding and justifying simple patterns).

## Okay, So… What Exactly Are We Talking About?

By now, of the 12 people who started reading this post, and the three who are still reading, at least two of you are wondering: What does this have to do with rational exponents and your struggling student?

Well, several weeks ago in Algebra 1 (the above student’s class) we had our first discussion of rational exponents. As usual, I was trying to elicit from them the idea that “to the 1/2” can be thought of as “square root,” and so on.

But a few students—bless their little hearts—wondered: Why?

And another student—bless his heart—applied our beloved “b factors of a” phrasing to come up with this:

What would that even mean? I knew, and you know, too, because we’ve seen this movie before (or at least accidentally read a spoiler in some blog comment or Facebook news feed overpopulated by comments you were never interested in reading in the first place; I digress).

But my students didn’t have half a clue what “half a factor of…” would mean, and I was on the edge of my seat to see where they would take this. (Correction: I was standing. But I fully expect I was standing on the edge of wherever it was that I was standing.)

## What They Saw

After a few more minutes of discussion, here’s what they saw and (more or less) how they described it.

• If you need to find “some number” to the 1/2, write the number as two identical factors. Then “take” one of them. That’s your answer.
• If you need to find the value of “some number” to the 1/3, write the number as three factors of the same number. Then “take” one of them. That’s your answer.

## What I Wondered

I’d never have a conversation on rational exponents take that turn, so now I was curious… What would my students do with other rational exponents? The next day, on their Topic 2 assessment, I invited students to attempt two challenge problems on the back:

1. Find the value of $16^{1/4}$
2. Find the value of $100000^{1/5}$

The results were mixed, but a majority of those who attempted the problems were spot on. Here’s a sample:

I guess in some ways this doesn’t differ much from the classic treatment:

$16^{1/4}=2$ because $2^{4}=16$

But it somehow strikes me as different, as offering more potential for extension, at least in the form my student wrote it on the review sheet that inspired this post. And now I’m wondering another thing. Would these same students—without any additional instruction from me—be able to evaluate $8^{2/3}$?

My guess is some of them could, and I expect they’d treat it like this:

Write 8 as three identical factors, take 2

In fact, I’d wager that with a brief class discussion, most of them would be equipped to handle any of these:

Write 16 as four identical factors, take 3

Write 100 as two identical factors, take 3

Write 4 as two identical factors, take 5

Write 100,000 as five identical factors, take 3

## For My Next Trick, I’ll Be Misusing the Word Partition

This idea of partitioning a number into identical factors and selecting a portion of those factors feels an awful lot like multiplying whole numbers by rational numbers:

Write 20 as four identical terms, take one

Write 21 as three identical terms, take two

Write 8 as two identical terms, take five

I don’t know if you can use partition in that sense (the factors sense), but I couldn’t shake the notion that these two problem types have a lot more in common than I ever thought before. (Maybe now that I’ve rambled all over the page I’ll be able to get some sleep at night. Or was it the kids waking up in the middle of the night that was disturbing my sleep… Too tired to remember.)