I sometimes ask, “Who likes fractions?”
A few hands go up.
“Who prefers decimals?”
Many more go up.
When I ask why all these folks like decimals, they respond with, “They’re so easy.”
But the “easy” word runs off later in the semester when it’s time to multiply decimals, add them or (heaven forbid) multiply or divide them by 10.
I’m a fan of fractions. It took me a long time to figure out why.
The decimals and fractions comparison is for another post. But, it’s voting season and I’m 22/31 into this Math Learning Success series. So I’m making a case for fractions by mudslinging at decimals.
Addition vs. Multiplication
There are two different rules when addition and multiplying decimals.
When adding, line up the decimals (inserting zeros if you want). Then start adding like there weren’t any decimals. Put the answer’s decimal directly under the problem’s decimals, in the line.
When multiplying, there’s no need to line anything up. Ignore the decimals, multiply like they’re whole numbers and then determine the place to put the decimal using these steps:
- Count the number of digits to the right of the decimal in each of your starting numbers (include any zeros that you used when you ignored the decimals while multiplying).
- Add those two counts.
- Now, starting at the far right of your answer and moving left, count that many digits.
- Put the decimal to the left of the digit you stop on.
Addition and multiplication happen all the time, and often in the same problem. It’s easy to get these two mixed up.
Before you know it, you will be tempted to line up the decimals on your multiplication factors and then put the decimal under the other two.
Or you’ll ignore the decimal when you add and count the total digits to place the answer’s decimal point.
If you think “easy” describes these processes, then you may get over confident and sloppy. You could make the rule-scrambling errors above!
Multiplying by Powers of 10 that are Bigger than 1 (things like 10, 1000 and 10000000)
Here’s where the real trouble comes in. When you multiply by 1000, you “just move the decimal.” Sounds simple enough.
But which way? And how many times? And what if there’s not enough digits when you start to move over?
If you’re good at this, you may roll your eyes and answer quickly. But if you’re not (like me), here’s what happens in your brain when you multiply 6.28 by 1000:
Teacher voice: Just move the decimal in 6.28.
Me: But which way?
Teacher voice: “You memorized which way. I told you to.
Me: I can’t even remember my left from my right, much less remember which way to move a decimal when multiplying.
Teacher voice: Then figure out another way to remember it.
Me: thinking Multiplying anything by 1000 must make it bigger, so I need to go to the right.
Teacher voice: Great!
Me: But how many times do I go?
Teacher voice: You’re kidding right? Everyone knows this!
Me: Well, it could be 3 times because there are 3 zeros in 1000, but there are 4 digits in 1000, so maybe it’s 4…?
Teacher voice: Holy cow, you really have no clue.
Me: frustrated but still thinking If I multiply 6 x 10, I get 60. That’s one zero and there’s one zero in 10, so I’m going with ‘move the decimal 3 times because 1000 has 3 zeros’.
Teacher voice: Great, so do it. #sigh
For some reason, I am comfortable with what to do when there’s no more digits left. But many people aren’t.
Multiplying by Powers of 10 that are Less than 1 (things like 0.1, 0.0001 and 0.000000001)
Okay, let’s play a similar game. Multiply 6.28 by 0.0001.
Just another “easy” move the decimal thing, right?
Suppose we manage to remember (or work out) that we need to move the decimal to the left. Here are my questions:
- Do you move 4 places because there are a total of 4 zeros?
- Do you move 3 places because there are 3 zeros behind the decimal?
- Do you move 4 places because there are 4 places behind the decimal?
- Do you move 5 places because there are 5 total digits?
Dividing by Powers of 10
This gets worse when I’m supposed to divide by 1000 or 0.0001. Now which way?
At this point I stop saying THAT f-word and saying the other one: Fractions.
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