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Saint Francis DeSales
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Beginning Division

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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.
The answer, then, to this problem is six (beans) in four cups with three leftover.
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.)
At this point, we can begin recording our discoveries on a graph.
Using a 10x10 graph, we label the top cups and label each column with numbers 1-7, although I would have preferred it to go to 10. We label the side as total number of macaroni/beans and label each row 1-10.
He begins filling it out,  recording the remainders as fractions as needed.

Once this chart is completed, it is time to change materials.
We use tiles next. I take a random number of tiles and have him form them into a rectangle with an amount of rows of my choosing. In this case, we began with three rows. Sometimes it will come out even, but many other times it will not. The ones that do not fit in the rectangle are separated off and called the remainder.
Now we talk about how to record our problems. Using a piece of paper {or if you're lucky enough to have a table that you can erase easily, you can just write on the table : )}and tell him to record the number of rows we want to make on the left hand side. In this case 5 for the five rows. We then put the total of the number of tiles in the center. Now he records the number of columns in the rectangle and this is recorded at the top. Any remainders are written as fractions as before.
After doing problems in this way for some time, he can now write division problems on a piece of paper with a smaller typical division matrix.
We can now play games with problems making the unknown any part of the problem, expanding his awareness of the component parts of a division problem.
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.

2 comments:

  1. This Math Lesson Rocks!! I am going to put this work together immediately. Thank so much for sharing.
    Discovering Montessori
    www.godwhoasnohands.blogspot.com

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  2. Awesome. We are struggling with this concept right now. This will help especially with remainders.

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