Photo courtesy of .raindrops.
Every so often, at around 7:00 p.m., I’ll get a call from someone I know. ”I don’t understand my kid’s math homework,” they’ll say.
These folks aren’t dumb or bad at math. But almost always, they’ve hit a concept that they used to know, but don’t remember any more. And those things can trip them up — big time. So, I thought it might be helpful to review 4 middle school math facts that may give parents trouble.
Every number has two square roots.
This is the question that prompted this blog post. I got a call from a friend who didn’t understand this question in her daughter’s math homework: “Find both square roots of 25.” Both?
Most adults have probably forgotten that each number has two square roots. That’s because we are typically only interested in only one of them — the positive one.
Yep, the square roots of 25 are 5 and -5. In other words:
[pmath]sqrt{25}[/pmath] = 5 and -5
It should be pretty easy to see why this is true. (You just have to remember that when you multiply two negative numbers, your answer is positive.)
5 · 5 = 25
-5 · -5 = 25
1 is not prime.
This question came up in my own daughter’s homework last week — a review of prime and composite numbers. Remember, prime numbers have only two factors, 1 and the number itself. So, 7 is prime. And so are 13, 19 and even 3. But what about 1?
Well, it turns out the definition of a prime number is a little more complicated than what we may assume. And I’m not even going to get into that here.
But there is a way for less-geeky folks to remember that 1 is not prime. Let’s look at the factors of each of the prime numbers we listed above:
7: 1, 7
13: 1, 13
19: 1, 19
3: 1, 3
Now, what about the factors of 1?
1: 1
Notice the difference? Prime numbers have two factors, 1 and the number itself. But 1 only has one factor.
0 is an even number.
This idea seems to trip up teachers, students and parents. That’s because we tend to depend on this definition of even: A number is even, if it is evenly divisible by 2. How can you divide 0 into two equal parts?
It might help to think of the multiplication facts for 2:
2 x 0 = 0
2 x 1 = 2
2 x 2 = 4
2 x 3 = 6 …
All of the multiples of 2 are even, and as you can see from this list, 0 is a multiple of 2.
Anything divided by 0 is undefined.
Okay, this gets a little complex, so bear with me. (Of course, if you want, you can just memorize this rule and be done with it.)
First, we can describe division like this:
[pmath]r={a/b}[/pmath]
Using a little bit of algebra you can get this:
r · b = a
So, what if b = 0?
r · 0 = a
That only works if a is also 0, and 0 ÷ 0 gives us all kinds of other problems. (Trust me on that. This is where things get pretty darned complicated!)
So how many of you have thought while reading this, “I will never use this stuff, so what’s the point?” You may be right. Knowing that 0 is an even number is probably not such a big deal. But at least your kid will think you’re extra smart, when you can help him with his math homework.
What are your math questions? Is there anything that’s been bugging you for ages that you still can’t figure out? Ask your questions in the comments section. I’ll answer some here and create entire posts out of others.
Admittedly my eyes were glazing over a bit reading this (not enough coffee yet?) but I appreciate the explanation of why 1 is not a prime number. This came up this week for my 5th grade daughter who said 1 was not prime, while my husband and I said it was. We agreed not to push her on it, since her teacher told her it wasn’t, and my geeky math husband had heard there was some controversy over whether it was prime. Thanks for the explanation!
Yeah, I was a little worried that I got too technical. Thanks for hanging in there anyway! (I do admit to a love of number theory, which is both the most useful — in that it defines our number systems — and least useful — in that we don’t really need to think about it too often — branch of mathematics out there. I’ll never forget my numbers theory professor in college, Dr. Gary Peterson.)
And actually there is little controversy now that 1 is a prime number. I haven’t been able to find out the exact dates, but according to some sources I read, it’s only been in modern mathematics that 1 is considered prime. And honestly, that’s because it’s more convenient to do so. This really starts to matter once we get into abstract algebra, which most people never, ever have to consider!
Smart kid you have there! (And smart parents not to argue with her on this point.)
Laura
If you disagree with something your daughter’s teacher teaches, I don’t think the right approach is to not argue about it. I think that it would be better to investigate the issue with your daughter. This way, you don’t teach your daughter to blindly accept what she is taught, and she gets to learn how to research. If there’s controversy about the issue, find out why there is controversy, and investigate the arguments for both sides.
By the way, I’m a math geek, and I still forget that every number has two square roots.
Good point, Hubert. At the same time, I often leave issues like these between my kid and her teacher. If she wasn’t paying close attention and suffers the consequences of that, she’s more apt to learn a lesson.
And while I am certainly very inclined to investigate an issue that I and/or my daughter doesn’t understand, sometimes there’s just not time! *smile* Plus, it may be easier for us math geeks to feel comfortable researching these things. When my daughter asked me if 1 is prime, I had to wade through a ton of math-speak to get to the answer.
Oh, and I’m so glad that you said you forget that every number has two square roots. I forgot when my friend asked me and had to think a moment before the answer came to me! Thanks for commenting.