Simplifying radicals

[bbaggett]

8.NS.2 asks students to use rational approximations of irrational numbers to compare the size of irrationsal numbers, locat them on a number line diagram and estimate the value of expressions. 8.EE.2 says to use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p and to evaluate tje square roots of small perfect squares and cube roots of small perfect cubes.

So….this does not look to me like 8th grade students should be simplifying radicals since 8th grade is the first time students have worked with square roots and cube roots (and the standards say specifically perfect squares and perfect cubes). But, I don’t see anything in the high school standards that says where simplifying radicals should be taught. My Algebra I teachers believe it should be in 8th grade. My 8th grade teachers believe it should be in Algebra I. What was the intention?

[Anonymous]

Expressions and Equations progression says: “Notice that students do not learn the properties of rational exponents until high school.” (a*b)^(1/2) = a^(1/2)*b^(1/2) in the form sqrt(a*b)=sqrt(a)*sqrt(b) appears to fit this description.

On the other hand N-RN.2 looks very appropriate. Part b in the task below uses the property:
https://www.illustrativemathematics.org/illustrations/608

Algebra I makes more sense.

[Bill McCallum]

Simplifying radicals is one of those high school topics that has evolved into a cancerous growth on the curriculum, starving other more important topics for resources. What is important is for students to understand and use the laws of exponents. When they get to algebra they should be able to see that $\sqrt{x^2y}$ is the same as $x\sqrt{y}$ (if $x$ and $y$ are positive). Seeing that $\sqrt{45} = 3 \sqrt{5}$ is a sort of rehearsal for this, and as Alexei points out comes quite appropriately N-RN.2. But treating such simplification as an end in itself, accompanied by long lists of problems, is an example of misplaced priorities. That’s why the standards don’t make explicit mention of it.

[John McGowan]

When one has a public blog, it must suck when some stranger from the internet asks you to explain a comment you made years ago.

This is a comment from a stranger from the internet asking you to explain a comment you made years ago!

I do not understand why you make the claim that an understanding of rational exponents are all the students need. I suppose one could do this algorithm:

root 45
= (45)^0.5
= (9 x 5)^0.5
= (9)^0.5 x (5)^0.5
= root(9) x root(5)
= 3 root 5

Am I reading this correct? Is this they type of algorithm you think is sufficient? If so, I can’t see why that is any easier or more intuitive than the standard simplifying roots algorithm. In fact, it seems like more work.

If I am misunderstanding you, can you explain how you think students should simply roots?

[Cathy Kessel]

It’s nice to see that people are still reading this blog.

I don’t know what is meant by “standard simplifying roots algorithm” but I suspect that it too relies on the laws of exponents (whether or not it uses exponential notation). Before students identify radicals as exponential expressions, they might use laws of exponents restricted to integers and the meaning of the radical sign in working with radicals, as discussed in the grade 8 section of the Expressions and Equations Progression, which is here: https://math.arizona.edu/~ime/progressions/.

You might be interested in the Number System and Number Progression (also here: https://math.arizona.edu/~ime/progressions/).

[John McGowan]

“I don’t know what is meant by “standard simplifying roots algorithm””

The standard method I am familiar with is this:

root(24)
= root(4 x 6)
= root (4) x root (6)
= 2 root (6)

I am unclear why this method is being devalued. Using rational exponents to simplify is no more intuitive.

[Bill McCallum]

If the goal is solely to get from root(24) to 2 root(6), then I don’t think the standards convey any preference whether it should be done using the square root symbol or using exponential notation. The main point I was making in 2014 is that this goal is not sufficiently important by itself to merit mention as a separate topic within the standards. Simplifying radicals, if it is to be included as an activity in a curriculum aligned to the standards, should have some higher purpose than merely simplifying radicals. One possible such purpose is to support understanding of the properties of exponents.

[John McGowan]

Thank you for replying. I expect it is strange to be asked to explain your thinking from years ago. This is an experience that the internet has given us, I guess.

I agree that algorithms should have some value other than learning them for learning sake. However, much of what we teach our math students is essentially vocabulary; they have to learn certain concepts and algorithms so that they can do the applied and complex problems later on. Memorizing times tables comes to mind. It’s a mindless task, but if students are not required to do it, then they do not develop numerical fluency, which prevents them from seeing patterns in higher math concepts.

Also, in this context, the laws of exponents could be considered the same type of unapplied algorithm. I don’t see much difference between that and simplifying square roots in terms of lack of application.

I am a classroom teacher, and thus my perspective is necessarily restricted to only the students I see. I assume that you have a wider perspective, since you have access to research that I do not. Based on the your initial response to this thread, back in 2014, I guess you saw a trend in “worksheet instruction”.

However, my problem, which has nothing to do with you, is that the textbook I am using, CPM, eliminated instruction in simplifying square roots based on the fact that it was no longer in the standards, but also didn’t replace it with any alternate method. When I asked them why, their head of curriculum pointed me at your blog. So here we are.

Thanks for taking the time to explain your thinking.

[Cathy Kessel]

Two comments:

In the Standards, the laws of exponents are not algorithms. From CCSS glossary:

Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. See also: computation strategy.

In the Standards, learning times tables is not a mindless task. See the Operations and Algebraic Thinking Progression at https://math.arizona.edu/~ime/progressions/.

[Bill McCallum]

I agree with Cathy that laws of exponents are not an algorithm, and maybe this is the source of some misunderstanding in this discussion. If, instead of having students do

root(24)
= root(4 x 6)
= root (4) x root (6)
= 2 root (6)

we insisted they do

24^(1/2)
=(4 x 6)^(1/2)
= 4^(1/2) x 6^(1/2)
= 2 x 6^(1/2)

then I agree that would be pointless. If they are to undertake this transformation at all, there is no reason to insist on one notation or the other. And, once students have learned that a^(1/2) = root(a), then it doesn’t matter whether you write root(ab) = root(a) root(b) or (ab)^(1/2) = a^(1/2) b^(1/2); either way you expressing a property that comes under the heading of “laws of exponents,” albeit in a disguised form in the first instance.

The question is, what is the purpose of this sort of problem in the first place? I can imagine it being put to service in a sequence of lessons whose ultimate goal is to get students to understand transformations like sqrt(a^2 – x^2) = a sqrt(1 – x^2/a^2), which you need to derive certain indefinite integrals in calculus. I can also imagine a curriculum writer deciding that calculus is too far off for that purpose to justify such problems in, say, an Algebra I course. Either way, this is a curricular decision, not a standards decision. The naked activity of simplifying numerical square root expressions, taken by itself, was not deemed sufficiently important to merit its own standard. If that activity becomes valuable in a curriculum designed to meet the standards, then so be it. That which is not mentioned is not thereby forbidden; it’s just that you have to have a standards-based argument for including it.

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