Help me with understanding division of fractions

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I'm reading Liping Ma's book and it's helpful. I am still stumped on the division of fractions. I know "how" to do it, as in flipping over the second and multiplying. What I don't get is "why." If I say I'm going to "divide" something, it gets smaller. It works that way with regular numbers. If you divide 4 by 2, you get 2. So why is it different with fractions? Why, when you divide by a fraction, does the number increase? If I was cooking, and wanted to split the recipe, I'd say I want to "divide the recipe in half." Which means my 1 cup of flour will become 1/2 cup of flour. But if I "divide by half" I'd end up with 2 cups of flour. Please help :)

Share on other sites Marilyn Burns's book Multiplying and Dividing Fractions is wonderful for that. She quotes a rhyme: "Yours is not to reason why; just invert and multiply," but then goes on to show exactly the why's, using folded paper and two or three short lessons. It demonstrates exactly what you are talking about, not only in words but also with a visual prop, which may be what you need for it all to click.

Another great verbal explanation is in the book Overcoming Math Anxiety by Sheila Tobias. Now, if my daughter or another child forgets why we invert the second fraction, I can work my way through why, on paper, easily. This is a terrific explanation of the formula or algorithm; but it doesn't explain the why's behind THAT like Marilyn Burns' demonstrations with paper do.

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If I was cooking, and wanted to split the recipe, I'd say I want to "divide the recipe in half." Which means my 1 cup of flour will become 1/2 cup of flour. But if I "divide by half" I'd end up with 2 cups of flour. Please help :)

You wouldn't end up with 2 cups of flour. There are 2 "1/2 cups" in 1 cup of flour. Make sense?

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Here's what comes to my mind...when you divide by a fraction - you are multiplying by a whole number. Essentially you are dividing that dividend into groups the size of the fraction. So for ex. if you have 8 candy bars and you are dividing by 1/2 then you are dividing 8 bars into groups containing 1/2 of each candy bar. So, 16 groups.

When you divide in half (as in the recipe) you are dividing by a whole number - multiplying by a fraction. You would think 8 candy bars divided by 2 - each group would have 2 candy bars. The answer would be four groups.

I think it helps to think of division as dividing something up into a number of groups of a certain size. If the divisor is a fraction - then each group will contain only a fraction of each number in the dividend. Therefore the number of groups (quotient) will be larger.

Does this help at all?

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I'm reading Liping Ma's book and it's helpful. I am still stumped on the division of fractions. I know "how" to do it, as in flipping over the second and multiplying. What I don't get is "why." If I say I'm going to "divide" something, it gets smaller. It works that way with regular numbers. If you divide 4 by 2, you get 2. So why is it different with fractions? Why, when you divide by a fraction, does the number increase? If I was cooking, and wanted to split the recipe, I'd say I want to "divide the recipe in half." Which means my 1 cup of flour will become 1/2 cup of flour. But if I "divide by half" I'd end up with 2 cups of flour. Please help :)

When you divide in half, you are dividing by 2, not by 1/2. You are multiplying by 1/2. So even with whole numbers, you invert and multiply, you just don't think of it that way.

In your example, to divide the 1 cup of flour by 1/2, think of dividing the flour into 1/2 cup piles. How many of those piles would you have? 2.

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When you divide x by y, you are asking, "How many y's are there in x?" So 1 div by 2 becomes, "How many times does 2 go into 1?" The answer is one-half.

1 div by 1/2 becomes, "How many times does 1/2 go into 1?" The answer is 2 because there are two halves in 1.

When the divisor (1/2 in the second problem) is less than 1, your answer will be larger than the dividend (hence there are 2 halves in 1; 4 halves in 2, etc.).

I am currently reading Ma's book as well and had to sort this out in my brain, too! Hope this helps!

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Thanks you all. I never liked math much in school, but because I was a good student I learned the answers but never developed an understanding of it or appreciation of it and want to do it differently with my children. Liping Ma's already given me a greater understanding of place value and multiplication, and I don't know where she found her US teachers but I am glad some of them aren't teaching my kids! :)

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Dividing is the opposite of multiplication. It undoes it, and so is its inverse function.

If you multiply something by 3, how do you get back to the original number? Divide by 3.
[We know that dividing by three is the same as multiplying by 1/3.]

If you multiply something by 1/3, how do you get back to the original number? You divide by 1/3.
[Multiplying by 1/3 is the same as dividing by 3. The inverse of dividing by 3 is multiplying by 3. So dividing by 1/3 must be the same as multiplying by 3.]

Likewise multiplication undoes division.

If you divide something by 3, how do you get back to the original number? You multiply by 3.
[We know that dividing by 3 is the same as multiplying by 1/3.]

If you divide something by 1/3, how do you get back to the original number? You multiply by 1/3.
[Multiplying by 1/3 is the same as dividing by 3. The inverse of dividing by 3 is multiplying by 3. So dividing by 1/3 must be the same as multiplying by 3.]

I'd say I want to "divide the recipe in half." Which means my 1 cup of flour will become 1/2 cup of flour. But if I "divide by half" I'd end up with 2 cups of flour. Please help
If you want to double a recipe, you multiply the ingredients by 2... triple, multiply by 3, etc. If you want to halve a recipe, you multiply the ingredient amounts by 1/2... which is the same as dividing by 2. When you're saying "divide by half" you really mean "divide by 2." Edited by nmoira
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Maria miller of math mammoth has an excellent video on YouTube explaining dividing by fractions.

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If you want a good explanation that's easy to understand, I would read Life of Fred, Fractions. He has a wonderful way of explaining things in a very easy to understand manner.

Karen

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