I find it much better to give my students experience in construction and checking their own division problems than to buy a book of already made problems. This method, however, requires that you introduce the concept of remainders from the start.
We have already done some work with beans and cups and have learned that we could determine the number of beans altogether with a particular number of cups and the same number of beans in each cup. (multiplication) This time we start backwards, for division is just backwards multiplication.
We take a handful of beans or macaroni and we see how many beans or pieces of macaroni we can put in each cup, making sure we end up with the same number in each cup.
We started out with 27 and after playing around with different combinations, we ended up with 6 in four cups but we had three left.
So, we begin a new chart that has a heading for beans, cups, beans in each cup and remainder columns.
At this point, he can begin making up and recording his own problems, deciding the number of cups and beans to be used. At some point I encourage him to try to predict the answer to a problem before he works it. This encourages him to look for relationships between each problem and its answer in a search for patterns that will permit him to make more meaningful predictions.
To facilitate the search for patterns, I suggest that he keep one factor a constant, such as cups and change another factor, such as beans/macaroni in regular increments, such as 1 to start with.
When he has gotten all he needs out of this pattern search, we then look at a new way to write the remainders. The remainder beans are recorded as the top numeral of a fraction and the bottom numeral is the number of cups used. For example, if we have 5 beans/macaroni distributed equally in four cups with one bean as a remainder. The remainder can now be looked at as a fraction or 1/4 (1 bean left from the distributing in 4 cups.)
Now, just to make sure he hasn't just learned a skill with the material, and not able to extrapolate a broader application from what has been learned. The use of a third material to present the same series of experiences helps him to extend and review the knowledge they have already required.
|This matrix shows 7 x 3 equals 21 or 21 divided by 7 equals 3.|
Notice that the divisor is on the left, the dividend is on the right and the quotient is above, just like the typical way one writes a division problem.
We can use the crossed lines, just as we did for multiplication. This time we can record the lines across to the left of the crossed lines and the lines down at the top. The crossed line joins can be counted and replaced by the total number. This forms a division problem we are used to seeing.
We can play with this form just like we played with the previous form, writing any two parts of the problem and leaving the third to be discovered last.
He can now make up his own problems to solve any way he likes.