This is a short post to archive some thinking I did on the subway home from work today. I had a Geometry class today and it was clear to me that their understanding of radicals was … not so good. And I don’t think it is their fault. I remember teaching Algebra II years ago and tried building up some conceptual understanding so puppies don’t have to die… and it was tough and I didn’t really succeed:

I also remember having this exact same conversation with my co-teacher last year. We considered the following “thought exercise.”

How would you explain to a student in Algebra I why $\sqrt{15}=\sqrt{5}\sqrt{3}$?

I would like to add the corollary “thought exercise”:

How would you explain to a student in Algebra I why $\sqrt{15}\neq\sqrt{10}+\sqrt{5}$?

We define (at least at the Algebra I level) $\sqrt{15}$ to mean “the number you multiply by itself that yields 15.”

I want to highlight the concept more than the notation, so let’s call that number $\square$.

So for us $\square$ is “the number you multiply by itself that yields 15.”
Now let’s similarly call $\heartsuit$ “the number you multiply by itself that yields 5.”
And let’s call $\triangle$ “the number you multiply by itself that yields 3.”

We know from this $\square \cdot \square=15$. Why? Because that’s the definition of “square” for us.

But we also know $\heartsuit \cdot \heartsuit=5$ and $\triangle \cdot \triangle=3$ for the same reason.

Thus we know $\heartsuit \cdot \heartsuit \cdot \triangle \cdot \triangle=\square \cdot \square$.

Here’s the magic.

Let’s rearrange:

$\heartsuit \cdot \triangle \cdot \heartsuit \cdot \triangle = \square \cdot \square$.

Study this a minute. It takes a second (or it might for students) to see that $\heartsuit \cdot \triangle = \square$.

Now remember I used symbols because I wanted to focus on the meaning of these objects, not the notation.Let’s convert this back to our “fancy math notation.”

$\sqrt{5} \sqrt{3}=\sqrt{15}$

So that gets at our first “thought exercise.”

I wonder if trying the same with the second thought exercise might work? The tricky part is that we’re trying to show a negative statement. I know… I know… most of you probably say “hey, just show the kids $\sqrt{1+4}\neq\sqrt{1}+\sqrt{4}$.” But that doesn’t stick for my kids!

So let’s try it: for us $\square$ is “the number you multiply by itself that yields 15.”
Now let’s similarly call $\clubsuit$ “the number you multiply by itself that yields 10.”
And let’s call $\spadesuit$ “the number you multiply by itself that yields 5.”

So:
$\square \cdot \square=15$.
$\clubsuit \cdot \clubsuit=10$
$\spadesuit \cdot \spadesuit=5$

Then challenge students do something similar to show that $\square = \clubsuit + \spadesuit$. They hopefully will start failing in their endeavor!

I predict they will start with: $\square \square = \clubsuit \clubsuit + \spadesuit \spadesuit$. Yay. That’s true… So from that true statement, they are going to try to show that $\square = \clubsuit + \spadesuit$.

But they can’t really go anywhere from here. They’re stuck. I still predict some weaker students may say: “But clearly we can just say $\square =\clubsuit + \spadesuit$. It’s like you have “half” of each side of the equation!” But it is at this point you can ask students to do two things:

1) Ask ’em to show the algebraic steps that allow them to make that statement. There won’t be valid steps. And in this process, you can see what other horrible algebraic misconceptions your students have (if any).

2) Or say: okay, let’s see if you’re right. If $\square =\clubsuit + \spadesuit$, then I know $\square \square=(\clubsuit+\spadesuit)(\clubsuit+\spadesuit)$. And as soon as you start distributing those binomials, they’ll see they don’t get $\square \square = \clubsuit \clubsuit + \spadesuit \spadesuit$ (our original statement).

Okay I just needed to get some of my initial thoughts out. Maybe more to come as I continue thinking about this…

1. Jerry says:

1. Thank you!!! I know I’ve been very unproductive with the MTBoS lately. :(

2. I loved this post! What a great way to help students understand surd properties and hopefully avoid making the same errors again in the future. Your students are very lucky to have you!

3. I see a lesson in here for Alg 1 (or pre-algebra) as we teach that you can multiply numbers under the radical. I am working with Alg Ii students who don’t know or think about the rules of radical use to solve radical problems. I still have to review simplifying radicals. I look forward to trying this idea as we review next week!

4. 1. Don’t they ever check their work with a calculator?
2. Offer them root(1) + root(1) = root(1+1) = root(2) and sit back.
3. Pythagoras: root(25) does not equal root(9) + root(16)

5. Not to rain on your parade but it seems like this line of reasoning would be interesting for kids who really enjoy math and how things work but would be equally confusing as any other method for students who didn’t want to put that much thought into it. Quite honestly I got lost in all the symbols and I know how radicals work. Maybe do the same thing but with a lot of numerical examples instead of symbols. That would make more sense to me personally.

1. mathnerdjet, I was thinking the exact same thing. Thinking in the abstract is complex (I think one reason why some students struggle in Algebra even if they are good at arithmetic). This is a great example for me to use and follow, but I think my kids would get more confused than they already were (and still not know how to correctly do this).

One idea I have had some success with is talking about metacognition and how the brain thinks and remembers. We see something and know how it works, and then apply it to other “somethings” that seem similar, and it works for us a lot of the time.

For example, at some point in our lives, we probably ate an orange for the first time. We could see the color and texture of the skin, and then observed the taste and texture of the fruit. We develop a schema for this type of fruit. Later, when we see a different kind of fruit, like a mandarin, or a tangerine, or a clementine, we can see that the skin has a similar color and texture, and the fruit has a similar taste and texture as an orange reinforcing our original beliefs. Eventually, we may come across a seville sour orange. The skin is near identical in texture and color. When we cut it open, the fruit looks to be the same texture as an orange (or mandarin, or tangerine, …), but the taste is super sour! These are not good for eating. Our schema has been broken, but it did serve us well for a long time.

Similarly in math, there are many examples where we have *something* on the outside of a group of objects, and that *something* applies itself separately to those different objects. For example:
2(x + 7) = 2(x) + 2(7)
(7x)^2 = 7^2 * x^2
sqrt(7x) = sqrt(7) * sqrt(x)

So we have developed a schema in our mind of how these things work. Unfortunately, then we find things that break our schema:
(x + 7)^2 ≠ x^2 + 7^2
sqrt(x + 7) ≠ sqrt(x) + sqrt(7)

By acknowledging the schema and talking about it, I think it helps the students to understand why they might think that way and be more conscious about applying that schema every time they see something similar.

As teachers, knowing about the schema can also help us have more patience with our students instead of wondering why they can’t “get it.” Our brain was wired for them to not get it, so we are fighting an uphill battle.

6. The way I summarize it, is that you can only “distribute” operations when they are adjacent to one another in order of operations. For example, it’s OK to distribute multiplication and division to addition and subtraction, so 2(x + 3) = 2x + 6. But, it’s not OK to distribute exponent to addition or subtraction because there are other operations wedged in between, and (x + 3)^2 is not x^2 + 9. (You are trying to skip too many steps, which causes incorrect result.) Similarly, once kids know that square root is an exponent, then you can apply the same logic to why (x^2 + 9)^(0.5) cannot be simplified to x + 3.

One thing I do to help kids who cannot keep all the rules straight is to teach them to use their graphing calculators to compare tables OR graphs of both formulas, in order to verify the equivalence. This is a powerful concrete way of seeing that it definitely isn’t true, and reinforces thinking about sqrt(x^3 + 9) and (x + 3) as completely different types of graphs. My international students also like to square both sqrt(x^2 + 9) and (x + 3) to show that their squares don’t have equivalent formulas, and therefore they cannot be equivalent formulas either.

7. Brilliant. A great way to help kids examine the underlying ideas. Thank you!

8. @Angela: Thanks!

@Clara: I look forward to hearing how it goes! I wonder if using these “stars” might be helpful: http://cheesemonkeysf.blogspot.com/2013/05/substitution-with-stars.html … I like the idea that on one side of the star, it says “sqrt(blah)” and on the other it says the meaning “the number that — when multiplied by itself — gives you blah.”

@howardat58: Yes, my kids do use their calculators, for sure. But this isn’t something that sticks with them. I have taught kids from 9th graders to 12th graders, and showing them “non-examples” never seems to work. I don’t especially think the ideas in my post will make this idea “sticky,” but I figure it’s a start.

@mathnerdjet: I agree with you to a degree. But I wonder — besides showing a bunch of examples to show “what works” and “what doesn’t work” — if there’s another way? Because what we’re doing doesn’t work or stay with them from year to year (at least in my school).

The whole thing that got me thinking about this is: when do kids learn the rules involving radicals? how do they get introduced to them?

I suspect that they are just “told” which rules work, and are shown a bunch of examples. I wanted to dig a bit further and see if we couldn’t justify — based on how we define radicals — *why* these rules work. I thought there was some good conceptual work that would be possible here.

@untilnextstop: I like the idea of having lots of explicit discussions of distribution and when it’s allowed (and why) and when it’s not. I’m about to have that discussion tomorrow, in fact, to dismiss the idea that cos(a+b)=cos(a)+cos(b)! I do think that graphing the two functions on the calculator is awesome. I still wonder if that makes it “sticky” or not.

Based on everyone’s comments, I wonder if it wouldn’t make sense to have ongoing “random” problems that ask students to prove that certain identities are false. Like sin(x+y)=sin(x)+sin(y). It is sometimes true (if x=y=0, for example), but one single counterexample destroys the identity. So having kids regularly try to “destroy” identities (or show certain identities are true) might be an ongoing practice to make things sticky.

Ex: Show (x-1)(x^4+x^3+x^2+x+1)=x^5-1 or prove that it is not true.
Ex: Show sin(x)=cos(x-pi/2) or prove that it is not true.
Ex: Show sqrt(x+y)=sqrt(x)+sqrt(y) or prove that it is not true.
Ex: Show sqrt(x^2+y^2)=x+y or prove that it is not true
Ex: Show abs(-x)=x or prove that it is not true

What’s interesting — something I’ve noticed from my kids this year — is that in the 3rd and 4th examples, more kids will do the 4th example of bad math than the third example of bad math — simply because the square and the square root are both there in the fourth example. (And obviously the square and square root undo each other!) When I realized that, I told my class my insight, I saw a few people nod in some semblance of understanding. How long that’ll last, I don’t know!

1. Interesting… I saw this post before… I have to think some more about this geometric approach!

9. Jennifer says:

I recently found your blog through Dan Meyer’s blog and love this concept of using symbols. Using hearts and squares are much less intimidating than radicals!

But I also wonder if this lesson might confuse or lose students or if they’d really be able to establish the connection.

1. I also wonder! This was a thought experiment another teacher and I had to find ways to avoid “bad math” without just saying “hey, that’s bad!” and show a counter-example or say “a puppy dies!” Something that would get at the *why*! But like you, I’m not *sure* it would work!