Category: Basic Math Review

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The Metric System: What’s the big deal with bases?

I’m vacationing this week in sunny Radford, Virginia,and ike most parts of the United States, the metric system is not used here (to mark distances, anyway). But if you cross the border into another part of the world, there’s little doubt that you’ll be measuring kilometers rather than miles and grams instead of pounds. That’s because most of the world has embraced the metric system. (In fact, only two other countries — Burma and Liberia — have resisted the change along with the U.S.)

Ask any scientist or mathematician: the metric system is infinitely more intuitive and much, much easier to remember and understand. But why? The answer is simple: Base 10. What this means is even simpler: in base 10 the foundational number is 10. Take a look:

10 • 1 = 10

10 • 10 = 100

10 • 100 = 1,000

and so on…

Each time you add a digit in our number system, you are effectively multiplying by 10. That means that 99 is the last two-digit number in base ten, and 999 is the last three-digit number. In fact our entire decimal system is base ten. (But it wasn’t always like that.)

But here’s the thing — you don’t care (and you shouldn’t really care). We are so used to base 10 that we don’t even think about it any more. It’s like knowing how to ride a bicycle or drive a car; once you learn it, you don’t even give it a second thought, but if you’re asked about it, it’s hard (or impossible) to explain.

When you were in school, you probably were asked to convert numbers into different base systems — and this was probably pretty darned confusing. We’re not going to do that here for one simple reason: You don’t need to know how to do this. BUT it is important to know that different base systems are useful in a variety of situations and professions. For example, computers function in base 2 (or binary), which is simply a system of zeros and ones. Computer graphics depend on a hexadecimal system or base 16 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F). Crazy, huh? Even less technical situations depend on a variety of bases — dozens and grosses are base 12 (one dozen is 1 • 12 and one gross is 12 • 12).

Compared to these other bases, base 10 is pretty darned easy, right? And that’s why so many mathy folks don’t understand why our country hasn’t embraced the metric system. Yep, unlike traditional measure systems, the metric system is base ten. Let’s compare:

Traditional system: 12 inches = 1 foot

Metric system: 100 centimeters = 1 meter

Traditional system: 5,280 feet = 1 mile

Metric system: 1,000 meters = 1 kilometers

Just a glance at these conversions and even the most math-phobic person would probably agree: the metric system is much easier to maneuver.

But agreeing that the metric system is easier doesn’t help you with conversions when you’re traveling, does it? On Wednesday, we’ll take a look at those conversions. I’ll show you some really easy ways to estimate the conversions. Because who wants to do math on vacation?

What other bases can you think of? How do you use them in your everyday life? Share your ideas in the comments section.

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Basic Math Review Math for Grownups Math for Teachers

Formulas: Or is this going to be on the test?

Quick! What’s the formula for finding the circumference of a circle? Do you remember the Pythagorean Theorem? What about the distance formula?

If you’re around my age and not a math geek, chances are the answers are “I don’t know,” “No,” and “Are you kidding me?”

When you were in school, memorizing formulas was required. But as a grownup, that’s not necessary. In fact, you can find all sorts of shortcuts that make formulas unnecessary. Here are two examples:

1. Last week, during spring break, I offered to teach my daughter and four of her friends how to make circle skirts. We bought material, set up three sewing machines and two ironing boards and got to work. I found a really wonderful (and easy) tutorial at Made, which employs a great shortcut for cutting out a circle: fold the fabric into fourths and then trace one-fourth of a circle, which will be the waist. After that, measure the length of the skirt (plus hem allowances) and trace another one-fourth circle.

We needed the radius of the smaller circle, but really all we had was the circumference of that circle — the measure around the waist. Dana at Made has a quick and easy process for this: divide the waist measurement by 6.28. Ta-da! The radius!

But why does this work? Because the circumference of a circle is C = 2πr. 2πr is approximately 6.28r. That means that you can divide the circumference by 6.28 to get the radius. Neat, huh?

2. Yesterday, I was the guest on the 1:00 hour of Midday with Dan Rodricks, Baltimore’s public radio station’s noon call-in program. Dan asked listeners to find the surface area of a cylinder with a radius of 6 and height of 8. A caller reminded me that there is a formula for this: SA = 2 π r2 + 2 π r h. But lordy, I didn’t remember that!  Instead, I found the area of each base — both circles — and the area of the rest of the cylinder (using the circumference of the base times the height of the cylinder). I added these and got the same answer.

So what’s the point? You don’t need to remember a formula. If you can break the problem down into smaller parts, do that. If it’s easier to remember to just divide or multiply by something, go for it. Unless you’re taking middle school math or have to teach a math course, the ins and outs of the formulas are not critical. What you need to be able to do is use the concepts you understand to solve the problem. Sometimes that means remember the formula, sometimes that means finding a sneaky way around your bad memory.

Don’t forget to enter the Math for Grownups facebook contest! Just visit the page to find out today’s clue (and Monday’s and Tuesday’s). Then post where you’ve noticed this math concept in your everyday life. Good luck!

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Homework Help: 4 middle school math facts you probably forgot

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:

sqrt{25} = 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:

r={a/b}

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.