# 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.)

I’m a little bit nervous about hitting “publish” on this one. I feel like there are four likely responses to the post—for anyone persistent enough to ramble (as in walk) through to the end of these ramblings (as in babble):

1. Thanks for blathering, but I have no idea what you just said.
2. Thanks for nothing, everyone already knew everything you said.
3. Thanks for trying, but I think you need a mathematical intervention to work out some of your own misconceptions.
4. Thanks for sharing, that’s a nifty connection. I might use it one day with my own students.

If you made it this far, let me know which of those reactions best describes your own. Or go off script and drop a more thoughtful comment.

Either way, thanks for playing!

1. Leif says:

4, all the way.

Beautiful post. I was on the edge of my seat anticipating each next scroll. (Correction: I was sitting all the way in the chair, but I did tell my wife I was on a fun ride when she asked what I was doing while I was reading this.)

What you did with the non-“1”-in-the-numerator fractional exponents blew my mind!

2. Mary Bourassa says:

Definitely a 4. Very interesting. I love seeing math in ways I have never thought. Thank you.

3. Interesting connection between rational exponents and rational multiplication. I wonder if making that connection would help students when they get to logarithms, which turn multiplication into addition.

4. Debbie Boden says:

I’m giving you a 5 out of 4!! I love how visual this is, it’s SO much better than how I was shown to teach it (I don’t teach honors algebra that often though I LOVE it). I’m sending it to the honors alg teachers at our school and some alg 2 teachers at the high school. It’s AWESOME!!!

5. I think you should absolutely explore 1000^(2/3) with students to see how they extend the model that they’ve started building. And then, I’d love to hear what they come up with. I’d anticipate a mix of answers, including both 20 and 100…

For what it’s worth, I tend to break down rational exponents in my head just as your expect your students might. That is to say, I mentally rewrite 8^(2/3) as [8^(1/3)]^2.

6. Drew says:

They feel the same because they are the “same”; i.e. exponentiation is a group homomorphism from (R,+) to (R^+,*). Not sure of a good way to explain that concept to your algebra 1 students, but it may be a good way of reinforcing the notion of a group homomorphism to abstract algebra students. Probably especially helpful for math ed majors.

7. If I start with “1” as the base of everything, I can get to “1 multiplied by 2 three times” as 2^3, which makes it easier to get to 2^0.

8. This is so great! Thank you for writing it up.

9. Really like the use of the visual bar representation of fractions with other than 1 as the numerator, and the connection to the number of factors comprising the result

10. Patty says:

Love it! Thanks for sharing!

11. …In my experience, “notation” errors are **sometimes** a failure to connect a different way of writing or saying something with a concept they already understand. Many of my students, however, get the ones they “understand” correct simply because of some arbitrary mnemonic. It’s painfully common for them to switch between squares and square roots if they’re put together because they’ve almost always learned those magic spells separately.
I often go with the notion that square roots are division, esp. since they look like division… but I show squares (perfect and imperfect ones — the idea that making twentyfive tiles into 5 rows of 5 squares is not so bad but if I toss another one in the mix it had better be made of clay so I can mush it around and it isn’t going to be exact), and then that yes, you’re dividing, and in my exampes, shrinking things down to their roots, but it’s a special division where the answer has to be what you divided by.
I’m still processing the fact that Two students this year, confronted with the quadratic formula, wanted to know why if you a perfect square you multiplied the number but if it wasn’t, you just “put it next to the number” and put it in the calculator, which wasn’t the same thing.
And hey, it’s always *possible* that the reason my blog post about square roots really is because people are looking for that information, not “how to type in the square root sign” (except, oops, that’s usually what they were searching for ;))

12. I am totes late to the party, but this was riveting stuff. Many thanks for the post! I’ve never thought of this stuff in quite this way. The 10^(3/2) with the fraction bar above was a real kicker for me ❤
You specified at one point that you were using whole numbers and rationals. I'm wondering how this affects student understanding with irrationals and exponents or multiplication (e^(2*pi*i), anyone?). I'm also wondering how this pushes against the idea of multiplication as a scalar and not repeated addition. Would love to test this out with kiddos.
Again, thanks!