I’ve actually had some success this semester with something I didn’t think of ‘til it was almost over. Seems that ol’ radical ‘round the number is a tough chunk of symbolism to work with, and no wonder — if you think about it it’s presented differently than other operatoins in math.
One of my guys said, though, after my long visual/auditory/conceptual explanation of squares and the length of a side of a square if the whole thing had the area of 16… “So it’s a kind of division.” To which I replied, “yes, and you can remember it because the radical sign looks like divisoin. It means “divide this number by whatever will give you the same number as what you divided by.” I of course commended his brilliance as well, since I’d never looked at it that way.
And while perhaps all but the first word went on by, after that he (and the other folks there) knew you had to divide that number by *something* and no longer ignored the radical or left it on after they got the answer.
I always did like radicals…
Re: Square Roots!
The radical is the “square root” sign that looks like a division “gazinta” (if you were reading the division problem set up for long division, you could say 12 gazinta 4,314…) but with an extra little mark on the left side.
Re: Square Roots!
Actually the root or radical sign comes from an old-fashioned written lower-case r — the way I was taught to write my r looks like the sign.
It’s the one with a hook like a checkmark and then the bar over whatever it is that you’re rooting.
Yes, it bears strong relations to division. Squaring is just multiplying two same factors, and rooting is undoing squaring, splitting a number up into two *equal* factors.
This fact is very helpful in understanding the next phase, which is that multiplications and divisions can be simplified across the root sign, because it is a multiplication-division game and they cooperate well, but you CANNOT simplify addition or subtraction across it because they follow different rules. (You can play soccer and football on the same field, but if you want to play baseball it’s going to be strange)
Interesting little example and disproof:
Using r() to mean square root:
These ones work, as do all similar:
r(16) x r(9) = 4 x 3 = 12
r(16) x r(9) = r(16 x 9) = r(144) = 12, same, it works
r(4) / r(16) = 2/4 = 1/2
r(4) / r(16) = r(4/16) = r(1/4) = 1/2 (because 1/2 x 1/2 = 1/4 !! check it)
same answer, it works.
This is not a proof, but apparently suggests that multiplication and division are OK in or out of roots
***
Now, is r(16 + 9) =? r(16) + r(9)
Is r(25) =? 4 + 3
Is 5 =? 7
Definitely not! So you can NOT run addition through the root sign
Is r(16 - 9) =? r(16) - r(9)
Is r(25) =? 4 - 3
Is 5 =? 1
Definitely not! So you can NOT run subtraction through the root sign
This is a good example to keep on hand when explaining this topic.
Re: Square Roots!
Oh, yes, and the discovery that 1/2 to teh second power is 1/4… that’s a biggie, too. Usually showing 1/2 *of* a 1/2 helps but if only we had time to show it a few dozen times and really get the picture that multiplying by a fraction shrinks stuff for all those reasons…
You know me…I think in pictures….didn’t focus on math….I got lost in the verbiage… :oops: Is the radical the thing with that looks like an L on its side?