# How can I explain dividing with fractions?

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We use Math-U-See are are totally stalled out on dividing fractions. I don't want to just say flip and multiply. The more I try to explain it, the more confused we both become. :blushing:

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We use Math-U-See are are totally stalled out on dividing fractions. I don't want to just say flip and multiply. The more I try to explain it, the more confused we both become. :blushing:

Why not say "flip and multiply"? How important--rilly,rilly important--is it to know *why*? Just do it.

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Are you on the Math U See loop? I asked about this recently, and got a great explanation from one of the authors. FINALLY, 30 years after being taught, I understand why multiplying by the reciprocal works! I appreciated learning the other way of dividing fractions too though--I never knew you could divide fractions without multiplying by the reciprocal!

Merry :-)

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Why not say "flip and multiply"? How important--rilly,rilly important--is it to know *why*? Just do it.

Oh no you didn't....:lol::lol::lol:

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How important--rilly,rilly important--is it to know *why*?

Pretty d*mn.

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We use Math-U-See are are totally stalled out on dividing fractions. I don't want to just say flip and multiply. The more I try to explain it, the more confused we both become. :blushing:

I followed the link and it explains mathematically why, but for younger kids that need a visual explanation I explain it sort of like this (hard to do w/o the examples in front of me.....but here goes)

Multiplying fractions explains what happens when you want to know what portion of a fractional part you have. I like using candy bars as examples b/c they are used to seeing KitKats or Hershey bars divided into pieces.

So....say I am told that I am given 1/3 of a candy bar. The 1/3 that I am given is notched into 1/4s. If I eat one of those 1/4s, how much of the entire candy bar did I eat? 1/3*1/4= 1/12 (I ate 1/12 of the original whole candy bar)

Dividing fractions tells how many fractional parts exist when I divide an object into pieces. So, say that I have 3 candy bars and I want to divide the candy bars into 1/8s. 3 divided by 1/8s means that 8 pieces are going to come from each candy bar giving me a total of 24 pieces which is why you say 3 (invert and muliply) 8.

Edited by 8FillTheHeart
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and we cut in in half

Then we each have 1/2.

I then say, "divide your 1/2 by 1/2 and have my student fold their piece to make crease. I have them cut along the crease. I do the same.

Then it's obvious that they have two 1/4 pieces.

We reesameble them to make a whole again.

Then I show them the invert and multiply algorithm and how doing it comes up with the same answer.

We do this with familiar fractions, dividing paper into thirds and fourths and dividing by halves, thirds and then doing the algorithm.

hth,

K

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:iagree: This is how I remember learning when I was a young girl. I think kids should become comfortable writing out the solutions to the "complex fraction" method before they're allowed to use the reciprocal rule. Thies helps them to develop an understanding of what happens when we divide fractions.

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I followed the link and it explains mathematically why, but for younger kids that need a visual explanation I explain it sort of like this (hard to do w/o the examples in front of me.....but here goes)

Multiplying fractions explains what happens when you want to know what portion of a fractional part you have. I like using candy bars as examples b/c they are used to seeing KitKats or Hershey bars divided into pieces.

So....say I am told that I am given 1/3 of a candy bar. The 1/3 that I am given is notched into 1/4s. If I eat one of those 1/4s, how much of the entire candy bar did I eat? 1/3*1/4= 1/12 (I ate 1/12 of the original whole candy bar)

Dividing fractions tells how many fractional parts exist when I divide an object into pieces. So, say that I have 3 candy bars and I want to divide the candy bars into 1/8s. 3 divided by 1/8s means that 8 pieces are going to come from each candy bar giving me a total of 24 pieces which is why you say 3 (invert and muliply) 8.

This is my plan...any excuse to eat chocolate...:001_smile:

Ellie - *I* had to see it before understanding so I know my dc will too. "Just invert and multiply" is the kind of teaching that led to the waste that was my high school math education...I earned good grades in Trig, Calc and Physics by memorizing stupid tricks like that without the understanding of what the heck I was doing. It's a bad habit to slide through without understanding the "why," and I don't want my dc to learn that from me.

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Liping Ma's book has a big fat chapter on this that made it clear for me for the first time (even though I did get the algorithm right).

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I LOVE this explanation! Thanks for posting.

Faithe

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Pretty d*mn.

But why???

I gotta say that as an adult I have rarely ever needed to divide fractions, but if I need to, I can. I truly do not see why it's important to *understand* in my heart of hearts why I have to *understand.*

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It matters because it is not hard to know. To fail to show why is to fail to teach it at all. To fail to show why is to deprive the child of something that educated people possess, that he could have, too, if you would just show him.

I have gotten so hot and bothered over the conceptual math people here on the boards, but the truth is that I did not understand that some people teach the algorithms only without ever showing why. I'd never heard of such nonsense, and it really was like falling down the rabbit hole to learn that some folks teach fractions without measuring cups (for example).

I am still gobsmacked by that, and half sorry that I ever learned it. Now that I'm looking for it, I do see the evidence and I am sorry for the kids.

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But why???

I gotta say that as an adult I have rarely ever needed to divide fractions, but if I need to, I can. I truly do not see why it's important to *understand* in my heart of hearts why I have to *understand.*

Many of the college students I see do not really understand why fractions work. Consequently, when they forget the algorithm that they are supposed to use, they remember *an* algorithm and simply apply it. For example, they invert and multiply when really they are supposed to be multiplying the fractions.

Purely in my opinion, memorizing algorithms fades more quickly than understanding what the algorithm means, why it works, AND how to use it. For the classes where I simply memorized algorithms without really understanding them (calc 3, organic chem spring to mind), I have no real recollection of how to solve those problems today. For the classes in which I understood what was going on and didn't need to memorize (calc 1-2), I could still solve the problems even after taking 7 years off from math.

In high school math, when the teacher taught the quadratic formula, I could do 100 math problems all of which involved the quadratic formula, and still get it wrong on the test. When the teacher taught completing the square, not only could I get all the test problems correct with that, I suddenly understood the quadratic formula.

YMMV. Maybe you're really good at memorizing. I'm not.

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Liping Ma's book has a big fat chapter on this that made it clear for me for the first time (even though I did get the algorithm right).

:iagree: I just finished this chapter, and loved it.

ETA: The way I made sense of this before reading Ma's book was like this:

10/5 can be thought of as "how many 5's are in 10?" The answer, of course, is 2 (10/5=2)

Similarly, if I had, say, 10/ 1/2 (10 divided by 1/2), I'd think "how many 1/2s are in 10?" There are 20 halves in 10. (10 divided by 1/2 = 20)

This does make sense, but Ma's book added many more wonderful examples to explain the concept of division by fractions; I highly recommend it. :)

Edited by yslek
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But why???

I gotta say that as an adult I have rarely ever needed to divide fractions, but if I need to, I can. I truly do not see why it's important to *understand* in my heart of hearts why I have to *understand.*

I am one of those people who absolutely MUST know WHY it works before I can implement it. It makes no sense to me to memorize something that has no basis. I HAVE to know why.

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But why???

I gotta say that as an adult I have rarely ever needed to divide fractions, but if I need to, I can. I truly do not see why it's important to *understand* in my heart of hearts why I have to *understand.*

I don't know what my dc are going to be when they grow up, but by not giving them a thorough understanding of basic math I can eliminate lots of possibilities. I'm fairly certain most careers in science/math will require an understanding of what it means to divide a fraction and how the algorithm works.

Besides, allowing a student to skim by without understanding just to complete the page, pass the test, and get the grade teaches them a very bad habit that will not serve them well...ever... (this touches on my reasons to HS in the first place btw;)) Allowing them to wrestle with the abtract concepts until they understand otoh, teaches dc to work and persevere until they own the concept. It's training the brain to think out of habit. It's not just the math, it's the principle.

I think it's insulting to ask a child to roll through math without understanding...like they aren't capable.:confused: It may be that you need to park and work sideways for a while before moving forward, but moving forward before the brain is ready is cheating the student.

I ask why not??? I can't think of a good reason...

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I think it's equally as bad to camp out on a concept a child isn't understanding and to stay there until he does.

Yes, it is a hard concept to understand. Otherwise we wouldn't be having this discussion in the first place. :-)

And learning why we flip and multiply is not the only place children can learn to persevere and all that.

Clearly I'm in the minority. It wouldn't be the first time, lol. Nevertheless, *I* would not beat myself to death trying yet another explanation. I'd teach dc to flip and multiply, give him lots of practice in doing it, and move on.

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I think it's equally as bad to camp out on a concept a child isn't understanding and to stay there until he does.

Yes, it is a hard concept to understand. Otherwise we wouldn't be having this discussion in the first place. :-)

And learning why we flip and multiply is not the only place children can learn to persevere and all that.

Clearly I'm in the minority. It wouldn't be the first time, lol. Nevertheless, *I* would not beat myself to death trying yet another explanation. I'd teach dc to flip and multiply, give him lots of practice in doing it, and move on.

Why not review math learned up to that point for a while, and come back to the tough concept in a month or two? Meanwhile, maybe I'll find some creative ways to demonstrate the concept from the back door when the math books and pressure are far from the dc's mind...I wouldn't want to beat myself to death or frustrate a child either. There is no rush to move on unless you are trying to stick to some external time frame (ps school S&S or preparing for a standardized test). I don't get how it's beneficial to move on when the dc doesn't understand? I guess I'll agree to disagree.

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Someone explained it this way to me and it clicked, finally.

When you have a division problem such as 10 divided by 2, you can think of it as 10 X 1/2.

This usually makes sense to everyone. Since 2 can be written (in the original expression) as 2/1, what you've essentially done is invert and multiply.

This works for any number (whole or fraction) as the divisor, expect of course, 0.

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I found this website: http://www.mathsisfun.com/fractions_division.html

I liked the way they explain dividing by fractions. Read everything, but the bottom part explains things nicely.

Just a thought when I do a multiplication problem isn't it a fraction? For example, 4 divided by 2 isn't it really 4 times by 1/2? If you do 4/2, that is 4 over 2 or 4x 1/2. 1/2 is the reverse of 2. Multiplying is the reverse of dividing and dividing is the reverse of multiplying. That is why if I divide 1/2 by 1/3, it is really 1/2 over 1/3 or 1/2 X3.

Sincerely,

Karen

http://www.homeschoolblogger.com/testimony

Edited by Testimony
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But why???

Because I'm a math Nazi who believes a child should know WHY before moving on to anything else. I was not taught why and I learned to hate and fear math. My son may think it is tough at times, but he is proud of what he is doing, and when that nickle drops, oh I love that smile.

I think of my ex, who was a great guy in many senses, but directions was not one of them. He grew up in Queens. To drive from point A to point B, he had to drive home first. Even with a map. Knowing WHY you do something in math means, to me, you don't have to drive home before heading to your goal. I can't expect him to ever stop running down the runway and actually lift off, mathematically, unless he knows why. IMO.

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I think it's equally as bad to camp out on a concept a child isn't understanding and to stay there until he does.

I find my son understands concepts long before he has perfectly memorized the algorithm or the math facts. When he forgets the algorithm, we discuss the "why", and he can usually re-invent the algorithm. This is, IMO, good experience for what being a grown-up doing anything complex and diverse.

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Thank you so much for these wonderful links and illustrations. For anyone who is interested, I've decided on a compromise.

Ellie, we are going to just flip and multiply and move forward. We have already spent three weeks stalled on this concept, crying and brain-blocked. Giving him permission to just do the process is liberating.

Meanwhile, I'll keep talking about and illustrating why. I think it is significant that different programs introduce dividing by fractions at various points. We are in lesson 10 of the Math-U-See program; while Life of Fred doesn't even discuss fraction division until lesson 30.

We won't leave fractions until I know he understands it, but I think working with fractions in other capacities may help this concept click. Again, most grateful for the weath of ideas and encouragment to pursue mastery.

Edited by bookfiend
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Liping Ma's book has a big fat chapter on this that made it clear for me for the first time (even though I did get the algorithm right).

I was able to read a large portion of this chapter on google books for anyone else interested.

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Oh no you didn't....:lol::lol::lol:

That's what I thought. :lol:

But I wanted to add that I think she has a point.

I have found that even when my children do understand a concept it still takes awhile to get all of the steps memorized. I think there could be a time that they could be memorizing the steps without the understanding, but we need to get back to them and reteach the concept to see if they are ready to "get" it yet.

Edited by Miss Sherry
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Thank you so much for these wonderful links and illustrations. For anyone who is interested, I've decided on a compromise.

Ellie, we are going to just flip and multiply and move forward. We have already spent three weeks stalled on this concept, crying and brain-blocked. Giving him permission to just do the process is liberating.

Meanwhile, I'll keep talking about and illustrating why. I think it is significant that different programs introduce dividing by fractions at various points. We are in lesson 10 of the Math-U-See program; while Life of Fred doesn't even discuss fraction division until lesson 30.

We won't leave fractions until I know he understands it, but I think working with fractions in other capacities may help this concept click. Again, most grateful for the weath of ideas and encouragment to pursue mastery.

To keep doing the process sounds like a good plan. It can take awhile to get all of the steps cemented in their memory even when they do understand the process. You can keep giving instruction to gain understanding as you go along and then check back again later to see if he understands. I suggest having him explain and demonstrate it to you at some point.

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YMMV. Maybe you're really good at memorizing. I'm not.

Plus, just memorizing the algorithm leaves a student inflexible. If they get the info in the right format (one that looks like the formulae they memorized), fine, but if it isn't, they are lost (this was what happened to me in math, and "being lost" in math is very alienating).

Here is a TED lecture on the subject:

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That's what I thought. :lol:

But I wanted to add that I think she has a point.

I have found that even when my children do understand a concept it still takes awhile to get all of the steps memorized. I think there could be a time that they could be memorizing the steps without the understanding, but we need to get back to them and reteach the concept to see if they are ready to "get" it yet.

Exactly. Just learning *why* won't automatically result in the ability to do it and to do it quickly.

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Pretty d*mn.

I don't like to post just to say this but:

:iagree:

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Exactly. Just learning *why* won't automatically result in the ability to do it and to do it quickly.

My children have had an easier time understanding concepts than memorizing the steps to do a math equation. Memorizing the steps is important. If you have to slow down to think about the "understanding" each time you will not be proficient.

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