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 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 … and explain how that is 5. But that 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 and say this is 2. And to think about what this funny 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 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 and and .
This is where the trouble comes in.
Some students now say is . 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 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 is the positive one and 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.