Introduction
The base ten concept is necessary for knowing how to add, subtract, multiply, and divide large numbers. When I was younger I was only taught the steps necessary to do these operations without really appreciating the concepts behind them, and it's easy to take for granted the power of places to compress so much information into few numerals. It's rather simple, but the development of places and zero is the doorway to being able to access everything interesting about mathematics above the basic business-oriented functions it was used for anciently like measuring out land and pyramid dimensions (with a few, limited exceptions such as Pythagoras above). This lesson is part utilitarian--they do need to have this down solid to do more than basic arithmetic, but I also hope to convey the power that human constructs have in organizing and making sense of the universe. The concept of "carrying the one" may be a bit difficult to grasp for younger children. My 7-year old almost has it down. Mileage may vary.
Lesson Plan
One way of counting is to make one mark for each item you have. In this way, if we wanted to show one hundred, we would have one hundred tally marks (show them how this might look). As you can see, this would take a long time to both make and count every time you saw it. Some early cavemen did numbers this way, often by putting notches in bones or counting out pebbles, but they usually didn't have to count very high, so it was okay. Some ancient societies would only have numbers up to four, and then every number after that was " a lot." Ancient Romans would only name the first four children, then all of the ones after that became "the fifth," "the sixth," etc. Some people think that this group of four rule is because that's as many as you can count to before you get to all the fingers on the hand. V for five is almost as universal as I for one.
Another way might be to have a different symbol for every numbers. But here you would have to memorize every single symbol, so to memorize all the numbers up to 100 you would need 100 different symbols. Also, it would be very hard to add these symbols together. Some ancient languages would use letters as symbols for numbers, so in Hebrew the first letter equals 1, the second equals 2, and so on. People could then use groups of letters to show numbers. Some religions would believe in sacred numbers because they would add up to the name of God, or something like that. In the Bible it says that the mark of the devil is 666, so some people think that a bad person's name near the end of the world will have letters that add up to 666. This shows you how powerful numbers could be for religions.
The way most ancient societies counted things is groups of symbols. They would have a few symbols that would represent certain numbers. This way, they would only have to memorize a few symbols, it was easy to add, and it didn't take long to count up the number of symbols.
[Show examples from Roman numerals, Ancient Egyptian]
However, for really big numbers it was still hard to add a lot of numbers together, and still took a long time to write. The ancient Babylonians started to use places, so instead of having lots of symbols showing groups of different amounts, the Babylonians made each place equal sixty of the earlier place. They needed to do this because they were one of the first empires in the world, and had to figure out how much money they could get from the people they ruled, so they needed to use big numbers. Some Indian groups used places as well, but they would use it on a knot of cords, and wouldn't write it down, so people would have to bring the knot with them wherever they went to show people a certain number. Being able to just write down the number made things a lot easier.
Nowadays each place is ten of the place to the right of it, this is how most number systems today work. However, they had to somehow show if there was one place that was missing, so for example 406 is different from 46, so they came up with a symbol that meant that showed that four is four groups of one hundred, even though there are no groups of ten.
[Those color-coded blocks of 1, 10, 100, and 1000 are very helpful here to show them the idea of groups of ten]
The Babylonians came up with a placeholding symbol, but it wasn't the same as zero. Zero had not been invented yet. At first they used a space to show the gap, but it was sometimes confusing because the spaces often weren't big enough, so a sign that represented a space, or nothing became very useful.
[At this point an abacus is useful to show them the idea of groups represented by different symbols/colors, with a space equaling a space in the groupings].
The Greeks and Hebrews didn't like the idea of nothing, so they never used zero. Eventually the Indians used zero, and it caught on fast because with zero they were able to do very large calculations. The Mayans on the other side of the world invented zero independently.
With zero you have a lot of tricks for adding and subtracting large numbers [show them carrying the one, subtraction, etc.]. It would be very hard to do this without zero or places (try it with Roman numerals).
With the base ten system that we have today, you can do large calculations with very big numbers, like 10 billion, 543 million... + [another big number--for kids used to adding 3 +5 this blows them away]. Because it's 10 groups of a billion, 500 groups of a million, 40 groups of a million, 3 groups of a million.... So if you add the same groups together you get the two numbers added together.
We still do use a base one system though for computers, because one means a switch is on and zero means a switch is off. So for a computer 100 is four.
Future Lesson Development
The real significance of zero kicks in when we get to calculus. I'll use Xeno's paradox and the Greek's inability to solve it without zero as a launching point for the discussion of calculus, and how mathematics stalled for almost two thousand years because people didn't see the full implications of zero.
Reading for Parents
There are a lot of "history of numbers" books that are worthwhile reads. For zero, Zero: A Biography of a Dangerous Idea does an excellent job of describing the implications of place notation and zero in a non-mathematician, reader-friendly way.
Video Resources
The Story of One is a BBC documentary on the history of numbers that looks promising, but I haven't watched it yet so I don't know if it's age appropriate.
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