How to Teach Middle School Students to Read Square Roots?

Question

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?

Answer 1

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.

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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."

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

Answer 2

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".