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Basic Math Fractions
by Gail Englert

A response to the question:

How do you teach fractions and division using manipulations?

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I would like suggestions about teaching fractions and division using manipulations to an academically challenging 7th grade self-contained class.



Dear Patricia,

As an introduction to the instruction, you may want to remind your students about what we are trying to find out when we divide. For example, if we were dividing 138 by four, we would be trying to find out how many groups of 4 there are in 138, or how many are in each group, for 4 equal groups, depending on the context of the problem we were doing the arithmetic for. (You have 138 pairs of sock to put in boxes, and four pairs will fit into each box. How many boxes will you need, vs. You have 138 pairs of socks to store in 4 boxes. How many should you put into each box, to evenly arrange them.)

When we divide with fractions we are looking for the same sort of information. 1/2 divided by 1/4 is really asking, "if you have half a pie, how many fourths of a pie do you have." Since 2 fourths is equivalent to one half, the answer is two. It forces us to think about the fractions in relation to the whole.

How about one-fourths divided by one half. That is asking how many halves there are in one fourth. If you draw a whole, and shade a fourth of it, you can see that half of a half is in a fourth. So the answer is one half.

Here is where a pitfall enters. Some teachers have taught their students that when you divide you get a smaller answer. That is true for whole numbers, but not for fractions and decimals necessarily. When students divide one half by one fourth, and get the answer "two" it does not seem reasonable to them. Having them actually cut a half into fourths (fourths of the original whole, that is, not cutting a half into four equal pieces) will help them see what is happening when fractions are divided.


Thank you for your fraction tips.

Now, I also am working with a 7th grade boy who is just this year memorizing most of his multiplication problems. He is working on simple one digit division problems and has alot of difficulty remembering the correct number for the quotient. He gets very frustrated. What do you suggest as the best way to proceed with him. Would it be better to have him work with a calculator or a multiplication chart? Any other suggestions?


Perhaps he really doesn't "see" what he is doing. Have you tried letting him use counters to work out these problems? For example, if the problem is 29 divided by 3, have him take 29 beans, or other materials, and divide them into three groups. Ask him how many are in each group. Then have him divide them into groups of three, and ask him how many groups he made. If you ask him to solve very simple word problems (that could be solved with division) while he uses these manipulations, you will also be helping him figure out a context for this "skill" to be used in. Once he makes the connection that dividing is making groups from a whole set, he may have less trouble figuring out a reasonable estimate, and remembering the correct answer.

On the other hand, if he is having trouble remembering the quotient because he still hasn't mastered the multiplication facts, your strategy of using the chart is a good one. You might want to have him look for patterns in the chart, and then relate what he sees to models using beans or some other manipulations. He might begin with 24, and look for all the ways he can make equal groups. Then he could find these groups on the multiplication chart. Hope this gives you some ideas to start with.

-Gail, for the T2T service

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