Working with Fractions
A response to the question:
My students are have difficulty understanding regrouping when subtracting fractions, reducing fractions, simplifying fractions, multiplying fractions and whole number, multiplying fractions and mixed numbers, dividing fractions by a fraction, and dividing fractions by whole numbers and mixed numbers. Please send me some ideas, suggestions and strategies for teaching the concepts so that it is easier for students to understand.
It looks to me that these students do not really have a firm understanding of what fractions are. How much experience do they have with using models and manipulatives? Having trouble regrouping fractions when subtracting could be a result of not really understading equivalent fractions. It could also be that they are not clear on what is happening when one has to regroup during subtraction. (Are they able to apply this skill when subtracting wholes?) Using models (cuisenaire rods, pattern blocks, fraction squares or circles) could help them make the connections between regrouping of wholes and fractions.
How do you view reducing and simplifying fractions? I am thinking they are the same thing. If your students are having difficulty there, it might be that they are not comfortable with equivalent fractions. My students had to play around a lot with fraction models and fraciton charts before they were comfortable with the idea that the same amount can be represented so many different ways.
As for the operations on fractions (multiplying, for example), when I introduced my students (fifth grade) to this skill, I reminded them what we are doing when we multiply two whole numbers (eg. 5 X 3 is five groups of 3, 2 X 7 is two groups of 7). I tried to help them understand that multiplying is actually being able to see how the English sentence fits with the number sentence (eg. 2 X 1/3 is two groups of 1/3, 1/4 X 8 is one fourth of a group of 8, and 1/2 X 1/3 is half of a third).
Do your students have any trouble determining the equivalent fraction for a mixed number? If not, that is the key to multiplying mixed numbers and fractions. Once they can represent the amount 1 and 1/5 as 6/5, they are on their way to solving multiplication problems involving fractions.
When we divided fractions by fractions, we can still relate that to division with whole numbers. Think about what happens when we divide. We are looking for the number of groups of a certain size, or the size of a certain number of groups (eg. 28 divided by 4 means how many groups of four are in 28, or if you make four groups from 28, how many will be in each group.)
If you think about fraction division that way, 1/4 divided by 1/8 means "how many groups of 1/8 can you get from 1/4 (and the answer is two...), or you could think of it this way. 1/8 of a group is worth 1/4, so the group must be worth 8 of the fourths and that is 2. Also, 1/3 divided by 1/2 means how many halves can you get from a group of 1/3 (think of a model of a circle. If you have only a third of a circle, then you do not have enough for a half, do you?). You have 2/6, but you need 3/6 to make one half, so you have two of the three pieces you need to make the half, right. Therefore, you have 2/3 (of a half) or you could say 1/2 of a group is worth 1/3, so the entire group must be worth 2/3.
Dividing fractions by whole numbers and mixed numbers uses the same idea. 3/4 divided by 2 means, if you make two groups from the 3/4, how much would be in each group. 3/8 would be in each group, because making two equal groups means finding half, right? Or, you could think of it this way. 3/4 divided by 2 means how many groups of 2 can you get from a group of 3/4? There is not enough to make one full group, right? Not even enough to make half of a full group, so you know the quotient must be smaller that 1/2. In fact, to make a group of 2 you actually need 8 fourths, and you have only 3 fourths, so you have only 3 of the 8 you need, which is 3/8.
3/4 divided by 1+1/2 is like saying 3/4 divided by 3/2. You need to figure out how many groups of 3/2 you can find in a group of 3/4, and you know the answer will be less than one, right? Since 3/2 is the same as saying 6/4, you could use that fraction instead, and notice that while you have only 3 fourths, you actually need 6 fourths, so you have 3 of the 6, or 1/2 of what you need.
Or you could say, you have 1 and a half groups, and all together that is worth 3/4. So, 2/4 and 1/4 is 1 and a half groups worth. See where this is leading?
Whatever you decide to do, be sure to allow lots of time for working with manipulatives, models, and illustrations. Students who play with these sorts of things while working on number sense topics such as these seem to have an easier time making the generalizations they need to work with fractions and decimals. Hope this starts you off in a direction that will help you help your students.
-Gail, for the T2T service
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