March Mathness

You know, it would be pretty easy to give a lesson on logarithms using the NCAA bracket to show how they go from 64 down to the Final Four down to the championship game. But we're not there yet. I hope to be there by next year, because that would be pretty Sweet 16.

In the meantime, enjoy the exponential decay that is the NCAA tournament.

52 Pickup

Have you ever thought about a deck of cards? There are 52 cards in a deck. Four suits.

There are also 52 weeks in a year. Four seasons. Which really doesn't sound like very many as I get older, but it's not a coincidence.

On this first day of spring (well, at 5:58 this evening, actually), I find myself thinking about the deck of cards, but more importantly, it feels like a decent time to think about the 52 weeks of the year. Between now and this time next year, so much will have changed--we will have had another baseball season, the school year is about to come to an end and by this time, the next one will almost be over too. Football season, basketball season, the holidays, summer, burning leaves, and...it will all come all the way back around to where we are now. And we will wonder where it all went.

I watched a documentary about Ricky Jay earlier. He was a master of card tricks before his death last November. Here's a clip.



If you're looking for something based more on the history, the facts, and the probabilities behind a deck of cards, I'll post a TED talk tomorrow, on the first full day of spring.

The Alpha State

Dear Mr. C:

So, multiplying fractions seems to be pretty simple. You just multiply across the top and then multiply across the bottom. So I assume you're just going to do the same thing with dividing fractions, right? Just divide across the top and divide across the bottom? 

Thanks,
Joanie Mahoney

Joanie, whatever you do, stop before you do any more problems dividing fractions, otherwise you're going to get them wrong! Believe me, there is a twist in the problem-solving process that you're going to want to know.

The good news is that, as far as the simpler problems go, they are almost every bit as simple.

 

To divide fractions, you're going to need to remember one thing before you begin. You are going to need to take the reciprocal of the second fraction. We'll talk more about reciprocals later, but for now, just remember this means you will need to flip the second fraction over. 

Then, as strange as it may seem, you will multiply across the top, and multiply across the bottom!

 

The answer you get is two! 

Now, I know what you're thinking. How did I divide two fractions and then end up with a whole number? A whole number is bigger than both fractions, and I divided. It makes no sense! 

Or does it? 

I posed this very question way back when I was in graduate school to my math teacher. I told him that I actually thought this was not even based on any mathematical logic, and that it was just some kind of rule that some math people made up. 

His name was Dr. Goldberg. He asked me, "Jeff, what is 56 divided by 7?" 

"Eight," I said. Duh.

 

Then he asked, "How many times can you fit 7 into 56?"

"Eight," I replied.

"Now, how many times can you fit 1/4 into 1/2?"

"Oh. Two."

"Yes," he said, "now sit back down and take notes."

So yeah, I was humbled, but that may be also when I started loving math for the first time.  

Let's look at another one.

 

Don't forget, you have to take the reciprocal of the second number.

Then we're going to multiply the numerators and the denominators.

Looks like we're top-heavy again. Let's turn it into a division problem!

Of course we know that we can take that leftover fraction and simplify it a little bit by dividing the top and bottom both by 3.

And here's our answer. Of course, there are multiple ways to do this, but I just wanted to show you my favorite way. There are about six or seven side-lessons I could throw in here, but I've written them all down for future posts.

We Are Captain America

I think Chris Evans would agree.

Divide and Conquer

Mr. C:

You always say that "every fraction is a division problem." What do you mean by this? I always thought that every fraction was a fraction. I'm so baffled. 

Your biggest fan,

Cash Flagg

Cash:

Yes, one thing that I have always tried to drive home is the fact that every fraction is a division problem. What I mean by this is that it is true! 

Let's take a look at 13/4. First off, he's walking along, walking along, and uh-oh! He fell over! Now you have your division problem all set up.

 

Now, you just take 4 into 13. You get a remainder of 1, but don't treat it like a "leftover" remainder, turn it into a fraction...just a smaller one. put the 1 over the divisor, and now you have 1/4.

 

Let's take a look at one that isn't top-heavy this time. In fact, let's take that 1/4, and just turn that into a division problem. 

Just like the first fraction, this one is walking along and falls over...and it becomes an interesting-looking division problem. But you know what? It's still every bit as much a division problem. 

1 can't go into 4.

Or can it?

This is when we have to call in our friends, the Phantom Zeroes. Phantom Zeroes are always there, but you can't always see them. The only time they show up is when we need them.

This time, we're going to just show two of them, but the best thing is that you can always add more and more if you need to.

Don't forget to put the decimal there right after the 1. Make sure it rises straight up above the division line. 

You can see, when you work the problem out, you get .25, as in 25 cents! In other words, 1/4 of a dollar! You see? It's a quarter. 

Finally, let's look at this one. This is a very top-heavy fraction. But it's going to tip over just like any other fraction.

And it's going to work out just like any other division problem. 

I promise I did not mean for this one to come out evenly when I made it. It was a happy accident.

Any time you need to simplify a fraction, figure out what the mixed number version is, or figure out what the decimal form is, don't forget, every fraction is a division problem.

Always.

I'll Hold Your Breath

Dear Sensei,

The other day, I tried to just multiply my fractions by multiplying the big numbers, then multiplying the fractions the way I learned, but I got the answers wrong! Can you please explain what happened, because I have no idea. 

BNW

With this, I have some bad news, and some even worse news. The bad news is that yeah, that's not going to work. The worse news is that this is a very time-consuming thing to have to do.

Most math isn't hard, but it is time-consuming. In problems like this, you also get many times to make a simple mistake and throw the whole thing off. Let's look at an example.

To start, you have to make your mixed numbers into top-heavy fractions. We do this by multiplying the denominator by the whole number. Then we add that to the numerator.

So in this case, 3 * 3 + 2 = 11, and for the second one, 11 * 5 + 2 = 57. 

Now, we can multiply across the top and multiply across the bottom, but first, I see some cross-simplifying we can do. 11 and 11 can simplify down to 1 over 1, going diagonally. Also, 57 over 3 can simplify down to 19 over 3.

This one was really handy, because all you're left with is a 19 and some 1s.

 Now, let's look at another problem. You're going to follow the same steps...

Then when you get your answer this time, you have a 33 over 2. So turn it into a division problem...

And your answer is 16 1/2.

Don't worry, later on I'm going to go into more about fractions, simplifying, and some of those things that you could be excused for not knowing already when I went through this.

The bottom line, though, is that these problems will take a little while. They're not fun. They're not even all that rewarding, other than the fact that you're finished with them. I love Math, but man, these can be a slog; I get it. But hopefully even thought hey are time-consuming, you will find that they're not hard to do.

More later.

Lucky

Mr. C,

I'm sitting here thinking about stuff. I'm wondering if there is any way you can make multiplying fractions a little easier. It just takes forever!

Sincerely, 
Brandi Finegirl

Well, Brandi, the bad news is that for many problems involving fractions, there is no really good shortcut. You're going to find that most of them are pretty time-consuming. However, you can use a trick or two to make things a little easier on you.

Before we get to some really time-consuming fraction multiplication, I want to show you one way that can make the process a little bit easier.

At this point in doing math problems, we know that you can simplify a fraction like 3/9 by dividing the numerator and denominator both by 3, giving you 1/3.

Now let's see how this applies to multiplying fractions...

Here we have two fractions, neither of which can be simplified on its own. We have to multiply them. We could do this with no problem as it is. 

However, looking at the numbers, it looks like I could simplify 4 over 10. That could be simplified down to 2 over 5. Because we are multiplying, we can cross-simplify.  This wouldn't work if we were adding or subtracting, by the way.

And then, it looks like you could also cross-simplify 9 over 15. That can simplify down to 3 over 5. 

Now, you just multiply across the top, and multiply across the bottom just the same. 

Now, let's check if it would work the same way even if we didn't cross-simplify. 4*9 is 36. 15*10 is 150. 

Now we just simplify the end product...looks like we can divide the numerator and denominator both by 3, giving us 12/50. Then with that one, we can divide the numerator and denominator both by 2, giving us 6/25, just like the answer we got the first time.

Woohoo! It works!

The Payoff For All Their Hard Work

I'm not going to say much about this. She isn't the only one; she's far from being the only one. It's just so good to see the bad guys going down for once. It's good to see someone being held accountable for being greedy and sneaky. Right now, at least 50 people are being exposed.

Sad to say, but it's a great thing to see.

Watch Your Back

I hope everyone is remembering to Beware the Ides of March today! Yes, today, March 15, is the day Julius Caesar was assassinated just a few years ago, back in 44 (no apostrophe before that first four), for being a tyrannical leader by 60 conspirators in the Roman Senate.

Here's a fun fact that you may not have known about good ol' Julie, though. He is championed with introducing the idea of putting a Leap Day into the calendar every four years to keep things peachy with both the Julian and the Gregorian calendars.

Now, I have to say that I suspect that it was probably one of his advisors who came up with this idea, but I have no proof. But, if nothing else, he did at least promote and help put it into action. Who knows how off-kilter the seasons would be right now without that extra February day once every four years.

So thank you, Caesar! You not only brought some seasonal balance to the world, but you also taught us to watch our backs.

Irrational Day

I had huge plans to go into the origins of pi, the many uses of pi, and why irrational numbers are such a necessary part of mathematical application.

Then, the past two days have been really convoluted, exciting, and much more full than I had bargained for. So you're stuck with this lame image, a wish that you had a happy Pi Day, and that you will stay tuned next year. Next year...boy, I tell ya!

Musclebound

Dear Math Man:

I am a bodybuilder, and am always looking for ways to get more jacked. Recently, my wife and I have been looking for ways to become more mathematically pumped. Can you please give us some more multiplying fractions problems? 

Thanks in advance, bro.

K. Greene

Mr. Greene: Ask and ye shall receive! Here are some problems for you. Don't forget to simplify when you can! And the answers are below.

Don't forget, you're going to multiply across the top, then multiply across the bottom. If the top and bottom are both divisible by the same number, that means you can simplify.

By the way, I often hear students (and teachers) refer to this process as "reducing" the fractions, but this isn't really accurate. When you reduce something, you make it smaller. We are just making the numbers simpler. 

Okay, here are the answers. If you didn't get something correct, go back and try to see where you might have gone wrong.

Section Of a Section

Dear Math:

Can you help me learn how to multiply fractions? I know that when you multiply two numbers, you always end up with a bigger number. So I know I'm getting the wrong answer when I get a smaller result than either of the numbers I started with. Thinking about this makes me positively dizzy.

Sincerely, 

Lucille Austero

Ms. Austero, I have some good news. You should get a smaller number when you multiply two fractions.

Let's start off with this problem:

When you multiply 1/2 by 1, you know you are going to get 1/2, right? After all, anything times 1 equals itself.

But let's say you multiplied 1/2 times less than 1. Doesn't it make sense that your answer is going to end up being less than 1/2?

So let's apply our technique here for multiplying fractions. Just multiply across the top, then multiply across the bottom:

Then simplify (you don't want overly-noisy fractions, after all):

And your result comes out nice, simplified, and less than your original fraction.

It's the way it should be! We will practice this a few times this week, so you can get a feel for it. But just know, you haven't done anything wrong if you end up with something smaller than you had at the beginning.