# How to Teach Middle School Students to Read Square Roots?

This exact quote from my standard American Algebra 1 textbook states when first introducing rational square roots:

$\sqrt{49} = 7$ is read "The positive square root of $49$ equals $7$."

$-\sqrt{49} = -7$ is read "The negative square root of $49$ equals $-7$."

In the next lesson, the expression $5\sqrt{3}$ is used in an example of simplification, but no instructions are given on how to "read" it?

I think these are some possible ways:

1. $5\sqrt{3}$ is read "5 the positive square root of 3."
2. "5 square root 3."
3. "5 root 3."

Following the textbook's instructions from the previous lesson, the first way seems most appropriate; but the third way has the least words? Is there a "correct" way to read $5\sqrt3$ and are there any other possibilities?

• I tend to say "radical", e.g., "5 radical 3". – Daniel R. Collins Mar 28 '18 at 10:36
• No "5 times the positive square root of 3"? – Adam Mar 28 '18 at 11:40
• If I were being careful, I would read it as @Adam has indicated: "Five times the positive square root of three." More casually, if I am convinced that my students are keeping up and there were no possibility of ambiguity (because it isn't, say, a cube root that might be running around), I might shorten that to "five root three." – Xander Henderson Mar 28 '18 at 18:50
• @student I added some relevant tags. Change them if they are not appropriate. This is useful for non-American audience here; "algebra 1" tells nothing to me. – Tommi Brander Mar 29 '18 at 7:28
• Why not "minus radical 49". Do you say "arcsine x" or "the inverse function of the restriction of the function $\sin x$ over $[-\pi/2,\pi/2]$"? I even say "xn" for $x^n$ (way before introducing sequences). – Paracosmiste Mar 29 '18 at 15:46

I would say that the word of is pretty important here, and as long as you get that correct, the rest is less important.

Using the word of consistently when a function is being applied to an input, and talking about the fact that you are doing this, will help ward off a lot of the following kinds of mistakes that I see in my (much later) classes:

Question: Solve for $x$. $\sin x = z$.

Student solution: $x = \frac{1}{\sin}z = \csc z$

The mistake here is that it is unclear to the student that there are different kinds of mathematical objects: some are numbers, but some are functions/operations.

So my answer is that saying "3 root 5" for $3\sqrt5$ is a dangerous abbreviation, because it begins to confuse this issue earlier than it needs to be confusing (What kind of object is "root" anyway?).

As a result I would prefer at least "3 square root of 5."

Similarly "sine of x" for $\sin x$ and "ell en of x" for $\ln x$.

• I like this answer on the one hand, on the other hand, students and teachers are lazy and that's why "of" isn't used enough. – Amy B Mar 29 '18 at 15:50
• I tried to draw attention to that issue seven years ago. – skullpatrol Mar 31 '18 at 6:07
• Here is the most rigours treatment of the word that I've seen. While this shows how important it is. – skullpatrol Mar 31 '18 at 6:26
• @skullpatrol I don't understand what OED's definition of "of" has to do with this particular use of the word. – Chris Cunningham Apr 1 '18 at 4:24
• It was a general comment about the issue :-) – skullpatrol Apr 1 '18 at 6:36

I'm not aware of any "official" version, the text that you quoted not withstanding, so any response is going to be mostly just personal preference.

With that said, I wouldn't read either expression the way your textbook wrote them. "the positive square root" is redundant. The square root of a number, in this context, is always positive so I would just read the first version as "the square root of 49".

For your second expression, I wouldn't refer to "the negative square root" for the same reason, i.e. there's no such thing. This phrasing implies that there's a positive version of the square root and a negative version which isn't the case. I would read it as "minus the square root".

With respect to your last example, I agree that (1) seems to be the most consistent with what you posted from the textbook. However, it's also excessively wordy and the lack of a verb after the five makes it sound awkward. I would read it as, "three times the square root of five".

• I think "the negative square root of ..." is perfectly sensible. In the book I've used, we do talk about square roots of a positive number, differentiating between a square root of $4$, and the principle square root of $4$ (often, "principal" is dropped, but in its absence, the definite article "the" remains). Since there are two square roots of a given positive number, it seems fine to use the definite article "the" with both, provided our phrase uniquely determines the one we're referring to (the positive one, or the negative one). – pjs36 Mar 29 '18 at 17:56
• Saying "the positive square root" may be redundant, but that's not always a bad thing. It's very common for students to think that the square root symbol can mean either the positive or negative, and usages like "either square root" are pretty common. Emphasizing that the symbol always means the positive number seems well worth the extra word. – Henry Towsner Mar 29 '18 at 18:25
• ... well, out in the larger world, it is not at all reliably true that $\sqrt{5}$ means the positive square root of $5$... Personally, I think it is misguided to put much emphasis on situations where playing along with this convention without comment or context-checking is any sort of critical issue. – paul garrett Mar 30 '18 at 0:55