| Math Mojo - Making Math Meaningful |
This was the question:
Why do we carry or regroup in math?
Professor Homunculus' answer:
There are a lot of people out there who don't really understand why we regroup, or how to carry very well. It is a shame that all teachers don't make that very plain to understand. It would save a lot of people's frustration with math.
I will have to use an analogy to make it clear.
Imagine you
were a soda producer, and you had to send 4,000 bottles of "Croke"
Cola to a store in New York.
Would you just load the bottles in a truck?
If you did, you would have to count every single bottle, one by one, and when they arrived in the store, the clerk would have to count them all again, one by one. What a pain!
So they put them in packs. Let's say they put them in ten-packs. Now, you each only have to count 400 ten-packs instead of 4,000 single bottles. That makes it a lot easier. It's still a pain, though.
So they put ten-packs in cases. Let's say each case has 10 ten-packs. That makes a total of 100 bottles per case. So we can call them hundreds-cases. When you ship them, and when they arrive, only 40 hundred-cases have to be counted. Now we have it a lot easier to manage.
We use groups
as a matter of convenience. Grouping makes things easier to count.
That is the same reason we use groups in math. Look at the ten-packs as a tens
column.
Instead of
having to write 10 ones, you write 1 ten, and no ones. Same for hundreds-cases.
Instead of writing 100 ones, or 10 tens, you write only 1 in the hundreds-column,
and zeros (which stand for "nothing") in the tens and ones columns.
The whole point is to reduce the number of things you have to keep track of.
So far, I think you probably understand this.
Now, as far
as carrying, let's go back to the soda bottles.
Let's say that the store also sells "Peptic" Cola, "'Taterade,"
"Barf," and dozens of other sodas. They get deliveries from lots of
companies. Let's also imagine that all the companies use the same system of
ten-packs and hundreds cases.
Now comes
inventory day.
The store has:
1 cases, 9 packs and 3 loose bottles of Croke,
3 cases, 2 packs and 8 loose bottles of Peptic,
2 cases, 7 packs and 9 loose bottles of 'Taterade, and
1 case, 8 packs and 7 loose bottles of Barf.
After they add
them up, they could say that they have 7 cases, 26 packs, and 27 loose bottles
of soda altogether.
But it they used 20 of the 27 loose bottles to make two more packs, they would
have 7 cases, 28 packs, and 7 loose bottles. And if they now put 20 of those
28 packs into cases, they could say they had 9 cases, 8 packs and 7 bottles.
What do you think is easier to keep track of:
7 cases, 26 packs, and 27 bottles,or
9 cases, 8 packs and 7 bottles?
We carry the extra stuff an regroup it into bigger containers (or columns) when there become too many of them to keep track of in any one contianer (or column).
Convention has determined that any number more than 9 is too much to easily keep track of in any column.
What that means, is that we find it easiest to deal with things when we break them up in tens. Maybe it is because we have ten fingers. No one knows for sure. Other cultures have broken things up into other groups, and sometimes we do, too. When we use "dozens" and "grosses." There are twelve things in a dozen, and twelve dozen in a gross.
The amount of things yoiu break your groups into is called "the base." So our usual way is called "base 10." Dozens and grosses are "base 12".
Seconds in a minute and minutes in an hour are "base 10." They are actually a modified base 10, because after "hours" you get days, which are grouped in 24 hours. Now we are getting into something called "modulo," which is beyond the scope of this lesson.
If you want to learn about bases, there is a series that will take you from very simple beginning lessons to pretty complex stuff. You can view the first post at:
http://mathmojo.com/chronicles/2008/06/30/bases-what-are-they-1/
I hope this all makes some sense to you.
Hi-Ho!Professor Homunculus
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