In my classroom, we break out the funny money and work some simple problems like this:
For the problem 41 divided by 3, we arrange four tens and one one on the desk. A real-world problem is provided: Three children set up a lemonade stand. At the end of the day, they have forty-one dollars. How many dollars will each child receive? How many dollars will be left over?
First, how many ten-dollar bills will each child receive? If we make a stack for each child, we can move one ten-dollar bill into each stack. In our algorithm, we show that each child gets one ten by writing a one above the tens place in the dividend. Then we multiply one ten by the number of students to see that we have used three tens. We place the three below the four in the tens place and subtract to see how many tens we have left (one).
Since we cannot tear up a ten-dollar bill and give one third to each child, we go to the "bank" and exchange our ten for ten ones. In our algorithm, we drop down the one in the ones place and look down at the eleven. This shows that we now have eleven ones.
Next, we divvy up the ones. You can see that each child gets three ones. In the algorithm, we divide eleven by three (divisor) and place the three (quotient) above the one in the ones place. Now we multiply three ones by three children and see that nine ones have been used. The nine is written below the eleven and subtracted, leaving two dollar bills.
When we're all done, we go through the entire problem again (much more quickly), pointing to each step and remembering what happened with the funny money. After working a few more problems like this, it's time to take a look at the algorithm without the funny money.
I model more problems, noting each step in the algorithm. As time goes on, my students are ready to use the algorithm with support from me. Just like teaching someone to ride a bicycle, I can feel when they're balanced and pulling away. Finally, they are sailing along, free as birds, confidently solving long division problems on their own!