Square Roots and Cube Roots

I’ve posted a lot about Calculus this year, and a bit about Multivariable Calculus too. But I’m not saying too much about Algebra II. Sorry. This year something is off, and the students aren’t as successful as in years past. I’m not exactly sure what to do. I’ve asked the student-led tutoring program to lead an Algebra II study group (we’ll see if anyone signs up). I also might change my teaching practice to allow more time for students to work problems in class — because I need to see more of them working and catch their errors in thinking earlier — before they go and practice the material wrongly at home.  We’ll get through less curriculum though if I do that, and that itself is a problem, since we’ve pared down the curriculum so much.

Anyway, that’s generally where I am with Algebra II.

Specifically, I just wanted advice on how you guys teach cube roots (and fourth roots and fifth roots, etc.).

My ordering usually goes:

1. Turn to your partner, and explain to them what \sqrt{5} means to someone who doesn’t know anything about square roots.

Students generally say that it’s the number that when multiplied by itself will give you 5. I then say “if the person doesn’t know anything about square roots, you might want to give them an easier example, like \sqrt{25}… and explain how that is 5. But that \sqrt{5} isn’t a perfect square so you’d get some number between 2 and 3. Yadda Yadda. I also then talk about the geometric interpretation (the side length of a square with area 5). Then I go back to the “it’s the number that when multiplied by itself will give you 5.”

I do not talk about there being two answers to “the number when multiplied by itself gives you 5” and the principal square root business. Because I want to use this to capitalize on their understanding of cube roots.

2. Then I put up \sqrt[3]{8} and say this is 2. And to think about what this funny \sqrt[3]{} symbol means. They get it. I put up a bunch more, and they usually can solve them. I put some negatives under the cube root symbol too.

3. I then ask them what \sqrt[3]{} means, and they say “the number that when multiplied by itself three times gives you the value under the cubed root sign.”

4. I then throw up a bunch of problems, and three of these include \sqrt{49} and \sqrt{-49} and -\sqrt{49}.

This is where the trouble comes in.

Some students now say \sqrt{49} is \pm 7. Because 7 and -7 are numbers when multiplied by itself which equals 49.

Here’s where I use the whole: “Don’t lose what you already know! Would you say \sqrt{49} is -7 ten minutes ago? No. You’re right, that there are two numbers which, when multiplied by themselves, give you 49. So we can tell them apart, we say \sqrt{49} is the positive one and -\sqrt{49} is the negative one. So don’t lose what you know. When you see a radical sign, it just represents a single number. If there’s a negative in front of it, it represents a negative number. If not, it’s a positive number. Just like what you’ve always known.”

Okay, now I know the idea of “principal square roots” and all that. And I honestly don’t want to have this whole discussion about principal square roots with them, because every time I do, they come out more confused.

So here’s my question.

How do you introduce cube (and higher) roots? How do you engage with this idea of principle square roots so that students don’t leave confused? I just can’t get it totally right.

And just so I am being clear, I know the properties of square roots and cube roots and all that. I’m not looking for someone to explain that to me. I want a way to teach my KIDS these without confusing them all up. And I bet crowdsourcing is a good way to get ideas for next year.

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18 comments

  1. I usually introduce fractional exponents first and then move to the radicand notation. I have them start with fractional exponents that they can compute in their heads without a calculator so that they can develop meaning for why it might be logical to say that 32^(3/5) = 8 and then usually a student that’s seen either square roots or cube roots before will connect the dots and we build off that idea.

  2. Same intuition as you, but start with the other side. Start with squares (of positives, negatives, fractions, etc.), then, “What do these things all have in common?” (try to get them to say they’re positive). Then look at cubes. “Notice these don’t have the same property. What about 4th powers? 5th? 6th? Any patterns?” Then show how it plays out with roots.

    Another option (and this may be backwards) would be to start with the property that sqrt(x)*sqrt(y) = sqrt(xy). And if x = y, then sqrt(x^2) = |x| (easier to see if you use actual numbers). Then show how you need 3 of these to do the same with cube roots and the negatives come from that?

    1. Also if/when you get to fractional exponents, I say “Like a tree, power comes from the top and roots at the bottom.” Many in my calculus classes don’t remember how to do them and that phrase helps them remember.

  3. Why not introduce them the same way many people do log/ln? In other words, X^2 = 25, so what is X? For cubed and higher roots, X^Y = Z. We need to find X. Now, when doing so, we will usually rewrite this as… Note that this also means that sqrt(Z) is a number itself that can be manipulated in interesting ways (helping lead into exponent rules and fractional exponents).

  4. I like the questions that you are asking, starting with what they know and extending it to something new. However, the piece of this I’d change is where you say, “don’t lose what you know.” I think the kids who are suddenly reporting radical(49) as either 7 OR -7 are extending what they know. They didn’t happen to think of the -7 earlier because 7 the answer most obvious. However, now that you’ve introduced the idea of negatives, they’re thinking deeper down into what radical(49) might mean.

    Have you talked about functions yet? I’m seeing now how having an idea of functions might help this situation a lot. Once you have radical(49)=7, making the radical into a function makes it a lot cleaner. We have one input; we sure would like there to be only one output.

    Even if you haven’t talked functions yet, maybe you could challenge them to use what you’ve decided as a class, radical(49)=7, to represent the number -7. I imagine they would come up with:
    a. radical(-49)
    b. radical(49)
    c. -radical(49)
    You could talk through each:
    a. does (-7)(-7) give -49? nope
    b. does (-7)(-7) give 49? Oh, it does. But, we already decided that radical(49) =7. Could it be two things? I think a teacher should concede that it could be, given what we’ve said so far. But, maybe argue that it will make life difficult if we have a symbol that can mean two different things (or if you’ve talked functions, make the function argument here).
    c. -radical(49) preserves what we already said, but now allows us to express negative numbers as well

    The question you ask is great. How do we get kids guessing around at what radical(49) is and then pull them back when they realize that (-7) feels like it could be an answer? In thinking this through, I feel like the function argument is the most appropriate for the Algebra 2 level, and that’s how I’m going to try it this year.

    1. Another way to attack the “function” argument is to talk about inverses. We know the graph of y = x^2, if we switch x and y so that x = y^2, what should that graph look like? Then go back to the vertical line test and whatnot to say we have to get rid of half of the graph and why it seems more natural to take the positive half.

  5. This is nice.

    I really like Clint’s way. Start with exponents they understand (positive integers), figure out the rules for how those behave, and figure out from the rules, what the other exponents (at least rational ones) have to mean.

    This sounds like it could be hard/sophisticated, though. I think it worked pretty well for me with a calculus class.

  6. I like Damon’s suggestion best so far. Thinking that square-root-of-49 is either +7 or -7 is a big step forward for most of them, and I wouldn’t want to squelch it just to get the \sqrt symbol across.

    If you are doing complex numbers, you can point out that “the” square root is not easily defined, but that there are conventions that work well for real numbers. The same goes for cube roots, fourth roots, and so on, where the n roots are evenly spaced around a circle, but only for reals is there an obvious choice for which root to pick as “the” root.

  7. Sam,
    First I want to say thank you. I just ran into your blog and I’m delighted to see updates in my inbox. Second, I would support what Damon says above. I think that approaching this from a functional standpoint is crucial. I would also lean on the fact that the students almost inherently trust their calculators. Have them use the sqrt key over and over and observe that they NEVER see two answers to one question. I also saw a colleague introduce this notation: Instead of moving from x^2 = 25 to x = +/- 5 he instead writes x^2 = 25 then +/- x = 5 since we have previous definitions saying that sqrt(x^5) = abs(x) and abs(x) yielding +/- x as solutions based on the piecewise definition for the absolute value function.

    Just my $0.02

    Jim D

  8. I (re)teach exponents in precalculus in a similar way, taking on the persona of a precocious elementary school kid and having my students come up and explain what exponents mean in general, then special cases like powers of zero, fractional powers, negative powers (each challenges them to expand their thinking/definition of what an exponent is; “the number of times you multiply something by itself” no longer suffices). I also encourage my students to change root problems into exponential problems, because of the variety of properties that make exponents easier to manipulate.

    I like the discussion above about tying roots to the function concept. Since functions are defined/created, I wouldn’t feel bad telling the students that the squareroot function (and notation) is DEFINED to output the positive number that when squared equals the input. A conversation about WHY it is so defined would touch on inverse functions and the horizontal line test, as has been mentioned by other posters.

    I like having students put together definitions that work for them. It seems a little strange, though, to allow them to create the definition “the number when multiplied by itself gives you 5″ (which is a bad definition), and then later to encourage them to “not forget what they know,” i.e. their personal, incomplete definition of roots (which they shouldn’t “know”). I would lean toward catching them in the moment of saying “the number when multiplied by itself gives you 5″ and pose to them the positive/negative question, see what they do with it, and then explain how/why the root function is generally understood (positive only). Though not a justification, I remind them this is “why” the quadratic formula has a plus or minus in front of the discriminant, because the squareroot only outputs a positive number.

  9. Can I push you a bit on whether they really know what a square root is in the first place? If we start by defining root(2) as the “number that, when multiplied by itself, yields two,” then students might interpret this as yet another arbitrary math rule akin to, “the knight moves two up and one over.”

    What if we took it back a few thousand years, though, and defined a square root as just that: the root of a square? “If we have a square whose area is 9, what is the side length?” Not only could this lead to a cool conversation about irrational numbers, Hipassus, the interplay between faith/philosophy/math, etc., but it would also reinforce the terrestrial/geometric origin of the notation. Which is to say, it would help students see the roots.

    At this point, you transition to cube roots, volume, the connection between the exponent and the dimension. Then “fourth roots? Fifth roots? Do you think the Greeks would have even asked that? Why or why not?” You can also bring up the issue of negatives–is -2 a solution to root(4), and what’s the square root of -4?–which dovetails nicely with the Observable vs. Pure Math conversation.

    Ultimately, this is one of those moments where it may be worth stepping back and asking where the concept came from, and using it as a lesson on the history of mathematics/human understanding. This in turn can help to remove any semblance of arbitrary definition, reiterate the man-made character of mathematics (and thus allow students to agree or disagree), and hopefully solidify their understanding and retention. The rest, then, might fall into place.

    Anyway, here’s my stab at it. It’s for middle schoolers, so it limits itself to square roots.
    http://www.mathalicious.com/?p=2371

    1. This is a terrific approach and an important reminder that these numbers actually MEAN something.
      Thanks, I shared this with my middle school colleagues

  10. For some reason, Bowen’s comment wasn’t going through… so here is his comment via me.

    ———–

    Here’s what we do in our Algebra 2 book. I really like it and it’s been effective:

    – The first task asks students to complete a table for f(x) = 2^x starting from x = 6 and counting DOWN to x = -3. It also asks students to complete some geometric sequences: 1, 3, 9, __, ___, ___ but also 1, __, __, 64, __, __.
    – Then a review (from Algebra 1) of the laws of exponents; the laws of exponents help you decide what 2^0 and 2^{-1} should mean, and the definitions come after the experience. One part of this argument talks about the consequences of letting 2^{-1} = -2 or some similar ridiculousness.
    – Now the cool part: a look at arithmetic and geometric sequences. For example, here’s an arithmetic sequence; what number is the question mark?

    0, __, ??, 27, __, __, 54, __, …

    If multiplication is repeated addition, you get a couple facts from the above: 27 x 2 = 54… but also 27 x 2/3 = 18. 18 is the number 2/3 of the way along from the identity (0) to 27 in an arithmetic sequence.

    Here’s a geometric sequence; what number is the question mark?

    1, __, ??, 27, __, __, 729, __, …

    If exponentiation is repeated multiplication, you get a couple facts from the above: 27^2 = 729… but also 27^{2/3} = 9. 9 is the number 2/3 of the way along from the identity (1) to 27 in a geometric sequence.

    This idea works wonderfully for the types of problems we want kids to do in their head with fractional exponents. If I want 16^{3/4}, I make a geometric sequence starting from 1, then three blanks, then 16, and 16^{3/4} is the value of the third blank. Guess what, you get 16^{1/4} and 16^{5/4} pretty much for free, and they actually have meaning in terms of the fractions you used.

    The bridge to the laws of exponents comes when you fill in the series blanks with “r”, for common ratio. For example, to compute 22^{1/3} you’d get something like

    1, r, __, 22, …

    and common ratio gives you the equation r^3 = 22. So, whatever 22^{1/3} is, it has to be a number that solves r^3 = 22. At this point, you can also bring up the fact that {22^{1/3}}^3 has to equal 22 by the laws of exponents, and you’ve got the same concept in two different ways.

    I agree that the issue around positive and negative answers can still exist, but one resolution is that a mathematical symbol is supposed to stand for one number. |-3| is one number; so is 2/7; 0/0 doesn’t make sense as a symbol since it doesn’t stand for one number. The same should be true for sqrt(25); you have to pick one of the two solutions to x^2 = 25, and conventionally we pick the positive one.

    Later, if students are ready for it, spring the same issue with the cube root of 8. There are THREE solutions to x^3 = 8! So, yeah, why don’t we just let the cube root of 8 be -1 + i sqrt(3)? Because we don’t, duh ;)

    Good luck and I hope this is informative. I learned about this series concept from a much more experienced teacher, and it really has proven effective with my own students as well as students using our curriculum.

    – Bowen

  11. Sam,

    I think your language is suited to something like “guided discovery” (my made-up term). You are pushing kids to figure out what comes next. (I like this. I think I do a lot of it.)

    But in this case, the idea of principle roots is somewhat arbitrary. The math does not lead us there. It is a convenience. A convention. And when they hit the “7 works, and so does -7, look at that!” I would explain the convention, directly, lecture style. There is non-artificial way most students can make good sense out of this one.

    I don’t lean on rules and formulae, but sometimes that’s the best we’ve got.

    I like the discussion of function. But I would save that for another day.

    Jonathan

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