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Welcome to the first edition of Ask a Math Teacherwhich will feature real, live math questions from real, live people. How often will I do this? As often as I can. What kinds of questions can you expect? Whatever people ask. If you have a question, please post it to the Math for Grownups Facebookpage (after clicking “like” of course!) or email me at lelaing-at-gmail-com.

Today’s question comes from my friend and cookbook author, Debbie Koenig. You really should check out her blog and bookParents Need to Eat Too. Debbie posted this question on Facebook, which led to a two-day long post-a-thon. We finally got to the root of the question — and answer — and I thought you would like to hear about it.

My son’s math homework has me scratching my head–he’s supposed to “Draw a picture of 600 hundreds” in a space that’s maybe an inch and a half high. How does one draw 600 of anything in that small a space? And why is he drawing 600 of something? I have no idea how to help him. 

One thing that isn’t clear in this question is that her son is in second grade. This is a really important piece of information, because the answer is going to seem completely counter-intuitive to us grownups.

Some background: When children learn their numbers and then learn to count and then learn to write 3s, 7s and 4s (sometimes backwards), they are picking up teeny-tiny bits of number sense. When all of this information is put together, we call that numeracy. You can think of numeracy like literacy. It’s not just being able to count or add; it’s being able to understand how numbers work together in a much larger sense. As you can imagine, this is a big, hairy deal. It takes years and years to get to where we adults are. And most of us grownups take for granted the numeracy that we do have.

I say this because what this “hundreds” thing is getting at is place value, or the position of a digit in a number. Teachers can just tell students that the 4 in 9433 is in the tens place, or — and this is a muchbetter idea — students can learn a great deal more about numbers by really exploring this concept.

You see, place value is not some random construct. There are reasons that the first place to the right of a decimal is the ones place and the fourth place to the right of the decimal is the thousands place. Exploring this can help kids get better at multiplying or dividing and lay the foundation for decimals and even percentages.

So with that said, the first thing to do is ignore what you think hundreds means. Unless you’ve had some experience in math education, you’re probably not going to take the right guess. In second-grade math class, hundreds does not mean one hundred. It means the hundreds place.

The easiest way to get into this is by looking at a hundreds chart.

If you have one of these hundred charts, you have 100, right? How can you represent 600 then? I’ll give you a second to think about it…

Yep, with six of these buggers! Here’s a visual representation without the numbers:

So what Debbie’s son was being asked to do was draw something like the above. It’s important because it has to do with place value. Only most second graders don’t have a clue about that stuff yet. And what they have learned so far sounds like a big mistake to the rest of us — because the language being used is not what we expect. He’ll learn the word “place value” in due time and forget about these hundreds tiles and charts and suchlike.

Asking students to draw “600 hundreds” is helping students visualize place value and other important concepts. Teachers call these manipulatives, especially because they’re often real objects that students can pick up and move around. But on a homework worksheet, they’re a little harder to translate, especially for a parent who went to elementary school more than a few years ago.

So that’s the story of “hundreds,” at least as far as a second grader is concerned. I’d love to hear your thoughts! Do you know of other ways to get to the basics of place value? Do you, personally, think of place value differently? Share in the comments section.

P.S. I’m going to be speaking to parents of elementary-aged kids at my daughter’s school later this month. If you have questions that you think I should address, feel free to shoot me a quick note or post on the Math For Grownups Facebook Page. And if your school — in the D.C.-Baltimore area — would like to have me come down for a Math Chat, let me know. I’d love to meet you!

So the Sumerian system of numbers — as far as we know, the first in the world — came into being rather naturally and out of necessity. But the Egyptians took things one step further, and they did it very systematically. Priests and scribes invented a system of numbers that included tally marks and hieroglyphics. In doing so, they developed a base-ten system featuring different symbols for different numbers.

The Egyptian people were very fortunate. With few neighbors, they didn’t have spend time worrying about war or defending themselves from attack. They also lived in a very fertile area, making agriculture less troublesome than it might have been. All of this freed up their time to do things like develop a numerical system and make big advances in mathematics. (You know, the ordinary stuff we do when we live in peace and have lots of food and water.)

Hieroglyphics could be used to express a wide variety of numerical values — all the way to one million! The symbol for one was a tally mark, so four tally marks represented 4, and so on. But 10 was expressed as a horseshoe shape and 100 a coiled rope. A little tiny prisoner begging for forgiveness was the hieroglyphic for 1,000,000. (I’d love to know the story behind that one.)

Yes, I drew these myself. No, I am not an artist or an ancient Egyptian. But you probably knew that.

While these characters could be arranged to represent an almost endless set of whole numbers and even fractions, the Egyptians were missing a critical numeral: zero. This meant that with all of their advances, Egyptian numbers had no place value system.

All of this allowed the Egyptians to take huge steps in the development of arithmetic, including the four basic operations — addition, subtraction, multiplication and division — and using numbers for measurement. Without these advances, we would have no great pyramids.

As the ancient society moved to the much more portable and easier-to-use papyrus and ink to record words and numerals, hieroglyphs gave way to hieratic numerals. These are more akin to brush strokes, and allowed the Egyptians to write larger numbers with fewer symbols. It’s pretty easy to see that this sped things up quite a bit.

On Friday, we’ll visit ancient India, where the most amazing creation/discovery revolutionized the system of numbers. (Seriously, this was a big, big deal!)

Can you imagine having to use hieroglyphics to balance your checkbook? If you have questions about the Egyptian system of numbers, ask them in the comments section.

When the world began 4.54 billion years ago, it didn’t come with numbers. They didn’t appear with the dinosaurs or first mammals or even the first homo sapiens. That’s because numbers were createdas a way to describe the world. And that is a big-honkin’ deal.

Think about it: Numbers make our daily lives much, much easier — from knowing how much time you have before you must get out of bed to setting the table with the correct number of plates at dinner time. You simply cannot get through your day with encountering numbers — not just once, or twice or a dozen times, butthousands and thousands of times. (Do you see what I did there?)

So if numbers haven’t been with us since the beginning of time, where the heck did they come from? Well, that history is pretty challenging to tell, but this week I’ll give you a little overview, starting with the Sumerians.

Sumer was a region of Mesopotamia, roughly where Iraq is today. The Sumerians made so many discoveries and inventions that the region is often called the Cradle of Civilization. Before this time, people used tallies to count things and geometric figures showed up in art and decoration. But these representations were not really mathematical, and they weren’t used widely and systematically.

It was the rise of cities that really set things in motion. As Sumerians developed commerce, they developed one of the world’s first system of numbers. To keep things fair, people needed a way to keep track of sales and barters. First, they counted on tallies. But there were no numerals associated with the hatch-marks they were using to show the number of sheep in a herd or eggs in a basket.

(Here is a good time to underscore the difference between a number and a numeral. It’s a teeny-tiny distinction, but an important one. A numeral is a character or symbol that describes a number. A number is the actual value of the numeral. So 3 is a numeralBut if I say I have three kittens, well, I’m talking about the number of sweet, little, purring balls of fur curled up on my lap.)

The Sumerians took things a little further with their whole commerce thing — they started systematically subtracting. See, if I had five goats, I’d be given five special tokens. If I sold off one of them, I’d have to give back one of my tokens. To keep track of this natural back-and-forth of trading and selling, merchants began to keep clay tablets of tallies that showed not only the number of baskets or cows or whatever they had at any moment, but a sales history.

And so, arithmetic was born. Oh, and writing. Ta-da! (Those Sumerians were smart and resourceful.)

Now, as this process developed over time, the Sumerians settled on a base 60 system of numbers. We have a base ten system, which in very, very basic terms means two things: we have ten basic numerals that are used to write all other numbers (0-9) and our numbers are described in sets of 10 or multiples of ten.

But not the Sumerians. They liked 60, a number that should be very familiar to us, since it’s the basis of our system of time. That’s probably no accident, right?

Eventually, the Sumerians developed their own set of numerals, called cuneiform numbers. They looked like the inscription in the photo above.

So there you have it. The world’s first numerals — near as we can tell. Next up: The Egyptians.

(Disclaimer: I’ll be the first to admit that this history is a lot more complex than can be described here. And I’d bet my last dollar that there are a few historians out there who disagree with the generally accepted history of Sumerians and mathematics. There’s so much we don’t know about his ancient history.)

Got questions about the Sumerians or the development of numbers? Ask them below. Was anything in this story surprising or particularly interesting? I’d love to hear what you think.

There are numbers, and there are special numbers. Okay, so just like children, all numbers are special. But a few of these numbers have qualities that make them stand out from all of the rest. Some of them you’ll recognize right away, because they’re used in everyday math. Others may be completely new to you — or at least you haven’t thought about them for years!

Let’s take a look.

Zero
It may look pretty ordinary, but 0 is one of the most important numbers in the entire system. It’s called the additive identity, because when you add 0 to any number, you get that number back. As a digit, it is used as a placeholder in the decimal system. Without 0, 4.32 equals 4.032, which would really shakes things up!

It may seem strange, but zero is an even number. That’s because it is evenly divisible by 2 (0 ÷ 2 = 0). But dividing any number by 0 is undefined; you can’t do it! Zero is neither negative nor positive, and it’s neither prime nor composite. When you raise 0 to any number (square, cube, etc.), you get 0.

One
Another ordinary number, 1 is called the multiplicative identity. In other words, when you multiply any number by 1, you get that number. As a result, 1 is it’s own square, cube, etc. It’s often called the unity, and it’s the first odd number in the natural numbers. Like 0, it is neither prime nor composite.

i
Remember the rule that says you can’t take the square root of a negative number. Well, this is where i comes in. In fact, i is the square root of -1. It’s known as the imaginary number, but believe me, it’s very real. (Okay, it’s not real in the sense that it’s not part of the real number system.) That means that the square root of -25 is ±5i. The square of i is 1.

Imaginary numbers aren’t used in everyday math, but they’re a big deal in electromagnitism, fluid dynamics and quantum physics.

Φ 
Phi is another number that you might not be very familiar with, but many mathematicians would say that it’s the most beautiful of all numbers. That’s because it represents the Golden Ratio. Two numbers are in the golden ratio if the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller quantity. Whew! That complicated definition boils down to the irrational number 1.6180339…

The golden ratio is found in art, architecture, music and even finance. The proportions of the Parthenon are said to mirror the Golden Ratio or Φ, and Salvador Dali’s The Sacrament of the of the Last Supper employs Φ. Stradivari used the Golden Ratio to place the f-holes in his violins. And it seems that the financial markets mimic the Golden Ratio.

Nature abounds with the Golden Ratio. If you divide the number of male bees by the number of female bees in a hive, you’ll get 1.6180330… Measure of the distance from your shoulder to your finger tips and the distance from your elbow to your finger tips. Divide the longer measurement by the shorter, and — yep, you guessed it — you’ll get Φ.

e
Like i and Φ you may not be very familiar with the number e. Quite simply, e is the base of the natural logarithm. It is equal to the irrational number 2.71828…

Computer geeks love e. When Google went public, the company’s goal was to raise $2,718,281,828 or e billion dollars to the nearest dollar. In further homage to the special number, the company put up a mysterious billboard designed to attract potential employees, who were also enamored with e.

π
Of course no list of special numbers would be complete without π or pi, which is equivalent to 3.1415926… But do you know where π comes from and why it’s so important? The number is the ratio of the circumference of a circle to its diameter. In other words, if you divide the circumference of any circle by its diameter, you’ll get π. Cool huh? Pi helps us find the area and circumference of a circle. It’s also useful in trigonometry.

More importantly, π has it’s own day: March 14 (or 3/14), when eating pie is encouraged, as well as celebrating the most famous constant in all of mathematics.

Do you have any additions to this list? Share your ideas in the comments section.

So tomorrow the world is supposed to end. Okay, not quite. The rapture will begin.  Apparently Earth won’t be destroyed until October 21. So you have some time to get your $@*% together.

But this isn’t the only reason that the number 21 is significant. Here are some other interesting (in a pocket-protector kind of way) facts about this Very Important Number.

  1. 21 is the sum of the dots on a die: 1 + 2 + 3 + 4 + 5 + 6 = 21
  2. 21 is a Fibonacci number: 1, 1, 2, 3, 5, 8, 13, 21, …
  3. 21 is the third “star number.”
  4. 21 is smallest number of differently sized squares that are needed to tile a square.
  5. 21 is the legal drinking age in the U.S. (Not so math geeky, eh?)
  6. Blackjack baby!
  7. {221}-21 is a prime number.

Anything you want to add?  Or are you too busy packing your bags?