You know by now that Base Ten blocks and Cuisenaire Rods are my favorite math manipulatives. They're a hands-on way to model calculations in the Base Ten system, and you can combine them with centimeter-gridded graph paper or a centimeter ruler and some colored pencils even more hands-on enrichment.
Will actually did this particular activity back when she was beginning to learn long division, but I need to give her some review work in multi-digit multiplication next week, so I was looking over my past projects and resources and discovered that I haven't yet shared this with you--and it's really great, so I want to!
To start, review this lesson on teaching long division using Base Ten blocks and Cuisenaire Rods. I found the vocabulary suggestions especially helpful for connecting the modeling to the calculation.
The premise, however, with or without the vocabulary, is simple: 16 divided by 4 means "16, put into 4 rows." Using the Cuisenaire Rods and skip counting, or the Base Ten units and counting on, the kid adds an equal number of units to all four rows until she finds out how many units would have to be in each row for the four rows to total 16:
Then, ask the kid what other math facts are modeled using this same set-up. Give her plenty of time to come up with repeated addition, repeated subtraction, multiplication of both rows by columns and columns by rows (don't start off with a square number, as I have in my example--this wasn't the first problem that Will did), and division of both rows by columns and columns by rows. If you totally have to, you can ask leading questions after a while, but I don't like to unless the kid is clearly over it but still hasn't found everything there is to find.
Once the kid has the hang of this, you can give her several problems to model and write down math facts for:
If I'm not sitting next to her, I like to have her color in what rods she used. You can see that you can also use the units to help you count on, if you're having trouble keeping track:
You can also do quite big numbers, as long as they fit onto the graph paper, although then you'll run into the problem that big numbers can be tricky to count on by hand, and you'll start to see some errors that result from simple miscounting, like this one:
Thirteen times 19 equals 247, not 244. If I'd encouraged Will to use Cuisenaire Rods to complete each row, instead of the columns after the ten bar, then she possibly wouldn't have made that error, but I don't like to correct her when she's in the middle of working something out. We talked about it afterwards as 10x19+3x19, using two different graphs, and it was simpler for her to work out that way.
Will doesn't have a lot of patience for manipulatives, but she does love puzzles and problem-solving, so this was a great review activity for her. This may be the one that I ask her to repeat next week--she made some errors during her multiplying decimals unit this week that lead me to believe that she's forgotten how multi-digit multiplication physically works. So multiplying the units, then multiplying the tens, then adding together the two separate functions--doing all that physically with the manipulatives and seeing it actually happen is crucial for understanding what you're doing in the pencil and paper calculations. If you remember that, then there's no way that you'll forget to put the tens calculation a column to the left when you add.