Some days, when washing one's hair in the shower, you think about radicals.
No, not the kind of radicals that have counter culture beliefs (like carob is the same as chocolate), but the square root kind of radicals. The kind of radicals that make you ask the question what number times itself equals 9.
Well, that idea's a pretty straightforward one, 3 times 3 equals 9.. and well, -3 times -3 equals 9. But what about the square root of 12?
I work as a part time tutor at the local community college, and this week one of the sections of math has started work on radicals, so I've been fielding a lot of queries about how they work.
And, I found a couple of reoccurring questions, which is why I found myself wondering what was tripping people up as I shampooed my hair.
So I've been doing a lot of work like this:
We're thinking about the square root of 12, which we can think of the product of the square root of 4 and the square root of 3. This is handy, because we know the square root of 4 is 2. We (meaning math-y people) read the simplified form of the square root of 12 as 2 root 3.
But there's more. After we know how to simplify radicals, we start adding, subtracting, multiplying, and dividing them.
And this is where the students I was working with this week started to derail.
"What am I doing and why am I doing it?"
There was a collective feeling of being thrown in the deep end of the pool after being told that water is just hydrogen and oxygen.
I found myself throwing around words like "common terms" and stressing that relationship between the radicand (the number under the square root symbol) and the number in front of it is one of multiplication.
2 is being multiplied by the square root of 3. I can also think of this has having 2 root 3's.
Back to my shower, I'm thinking about all the furrowed brows I've seen this week, and it hits me. Mixed numbers. Whole numbers with fractions. Most of these students, when they've seen a number next to another "special" number, with no symbol between them, it's been a whole number next to a fraction. THAT relationship is one of addition.
This isn't really something I've ever thought about before. One of the great joys I experience with tutoring is learning from the students I work with. Working one on one with someone, or several someones, makes you examine your math knowledge with a fine tooth comb until you hit a snarl.
Consider this comparison:
When multiplying 2 root 3 by 4, we just multiply the coefficient (the number in front). I explain it to students that they can think of this as being similar to multiplying variables. 2X times 4 equals 8X.
But, when we think of mixed numbers and multiplication, we can't do this.
And I think this is one of the things that trips people up. We have taken a familiar form, created new rules, and not really acknowledged that we've changed the landscape.
What to do? Be frank. "Hi, we are throwing a bunch of new rules at you. This might feel overwhelming."
Also, use our language to our advantage. Say it out loud.
"2 root 3" sounds different than "1 and one third." Hear that 'and', that 'and' means addition. That is the difference here. It is all in the way we read it. "2 root 3" means "two times root three" the same way that "2X" means "2 times X". "1 and one third" means 1 plus one third. This is a great place to begin to untangle the snarl.