When I first learned that I would have to teach division with decimals, I panicked. Those rules confuse me, and I wasn't sure that I could teach it in a student-centered way so that I wasn't just telling the students what to do without rhyme or reason.
Then, I happened onto a terrific strategy for dividing decimals using reasoning.
When my students and I divide with decimals we focus on using reasoning to figure out where the decimal goes. We have three different strategies for reasoning about division problems:
1. Make an estimate using simple numbers
In the problem below, we estimated 97.5 / 6.5 as 100 / 5 or 100 / 10
2. Turn it into a multiplication problem
In the problem below we rewrote the problem as 6.5 x ___ = 97.5 or 10 x ___ = 100
3. Think about how many ____ are in ____ ?
In the problem below we thought of, "How many 6.5s are in 97.5?" or "How many 10s are in 100?"
Then, all we need to do is find the digits in the solution (975 divided by 65), and use our reasoning from above to place the decimal.
You can see all our steps in the image above. I like this way better than memorizing a rule (like moving the decimal), because the students can make sure that their answer always makes sense.