Help with multiplying vs. dividing fractions concept....

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Maybe someone who is more "mathy" than me can come up with a way to explain a particular concept visually.

I have no problem understanding the multiplication of fractions. Example: 1/6 x 2/3 = 2/18 = 1/9. If I were to draw this, I would make a pie with 6 segments, and then divide each of those segments (the sixths) into thirds (which would be a bunch of teen eighteenths), and take 2 of the 3 smaller segments from one of the (divided) sixths.

Now, how can you do this visually with a pie, using division? If I have the same example as above, only division [1/6 divided by 2/3], I will end up with 1/6 x 3/2 = 3/12 = 1/4.

How can I show a pie which has been divided into 6 sections, further divided by 2/3, and ending up with 1/4?

Thanks for any verbal help or links to websites showing this pictorially.

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1. If you need to visualize it I would begin by choosing a larger fraction divided by a smaller fraction so that you actually get a number of pieces.

1/2 divided by 1/4: you are trying to figure our how many 1/4 are in 1/2, that's what dividing is.

So you take 1/2 pie. You divide it into 1/4 pies and count: how many 1/4 pies are in the 1/2 pie: the answer is two.

Or take 2/3 of the pie and divide by 1/6 of a pie: you find that there are 4 1/6 pies in 2/3 of a pie.

2. Once that concept is understood, divide smaller fractions by bigger fractions:

1/4 divided by 1/2 is asking: how many half pies are in a quarter pie. the answer is of course : one half of half a pie. A better way of posing this question would be: how MUCH of half a pie is in a quarter pie.

1/6 divided by 2/3 is asking: how much of 2/3 of a pie is in 1/6 of a pie: There is half a third in one sixths, so there will be a quarter of two thirds in one sixth.

This will be very hard to illustrate- so I would turn this around and use multiplication:

1/6 * 4 = 2/3 so (1/6) / (2/3) =1/4

At this point, I would find the pie more confusing than helpful.

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regentrude gave a great explanation.

I wanted to add that I include a "story" that goes with it, usually having to do with a party and leftover pizza.

So if you have 2 1/2 pizzas left over from the party, and you want to give the child 1/8 of a pizza (one slice) in his lunchbox, how many servings will you have? (2 1/2 / 1/8), which = (5/2 / 1/8), which = 20. Hopefully you'll be putting those slices in the freezer!

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