How do I describe dividing fractions using pies

[mr magoo]

Here is what I have written down for describing math from multiplication up to dividing fractions, and as you see I can use pies to help describing the math but for dividing fractions I get stuck with how to use a pie analogy.

Here is what I have wriote to describe math using pie analogy below;

multiplication
____________________________
pieces per pie * total number of pies = total number of pieces
2 pies total * 4 pieces each pie = 8 total pieces of pie

division
____________________________
total number of pieces / pieces per pie = total number of pies
8 total pieces of pie / 4 pieces each pie = 2 pies total

adding and subtracting fractions
____________________________
create total pieces of pie in division, twice, so two divisions can be added together.

The total pieces of pie in both divisions must share the pieces per pie value.

Once the pieces per pie value is the same between the two divisions,
add or subtract the total pieces of pie from the two divisions.

2/1 + 2/2 = (2/1 * 2/2 = 4/2) + 2/2 = 6/2
5/3 - 4/5 = (5/3 * 5/5 = 25/15) - (4/5 * 3/3 = 12/15) = 13/15


multiplying fractions
____________________________
total number of pieces * total number of pieces / pieces per pie * pieces per pie =
total number of pieces / pieces per pie

1/1 * 3/4 = 3/4

3/3 * 1/3 = 3/9



dividing fractions
_____________________________
where is the pie analogy? I'm stuck. help me please.

2/7 / 1/2 = 2/7 * 2/1 = 4/7

4/7 * 1/2 = 4/14 = 2/7

______________________________

5/8 / 3/4 = 5/8 * 4/3 = 20/24 = 5/6

5/6 * 3/4 = 15/24 = 5/8
_______________________________

 

[mr magoo]

And I'm learning from online resources and am self teaching myself using the khanacadamy and mathtutordvd but they don't explain this well enough for me to use a pie analogy.

 

[mr magoo]

OK, I have watched more of the mathtutordvd and saw how the fraction makes the fraction smaller, then realized the multiplied fraction when dividing fractions is larger than 1/1, so the multiplication makes the product a larger fraction.

Here is the new text below, note I updated the dividing fractions part with this new knowledge;

multiplication
____________________________
pieces per pie * total number of pies = total number of pieces
2 pies total * 4 pieces each pie = 8 total pieces of pie

division
____________________________
total number of pieces / pieces per pie = total number of pies
8 total pieces of pie / 4 pieces each pie = 2 pies total

adding and subtracting fractions
____________________________
create total pieces of pie in division, twice, so two divisions can be added together.

The total pieces of pie in both divisions must share the pieces per pie value.

Once the pieces per pie value is the same between the two divisions,
add or subtract the total pieces of pie from the two divisions.

2/1 + 2/2 = (2/1 * 2/2 = 4/2) + 2/2 = 6/2
5/3 - 4/5 = (5/3 * 5/5 = 25/15) - (4/5 * 3/3 = 12/15) = 13/15 multiplying fractions
____________________________
total number of pieces * total number of pieces / pieces per pie * pieces per pie =
total number of pieces / pieces per pie

1/1 * 3/4 = 3/4

3/3 * 1/3 = 3/9
dividing fractions
_____________________________
from division;
total number of pieces / pieces per pie = total number of pies
2/7 = total number of pieces
1/2 = pieces per pie
4/7 = total number of pies

in 2/7 * 2/1, the 2/1 is representing 2/7.
and 2/1 is twice as big as 1/1.
so the answer of twice as big as 2/7 is 4/7.

How it works in pie description is using multiplication.
pieces per pie * total number of pies = total number of pieces
4/7 = total number of pies
1/2 = pieces per pie
2/7 = total number of pieces

this way division explains multiplication, then multiplication explains division.

2/7 / 1/2 = 2/7 * 2/1 = 4/7

4/7 * 1/2 = 4/14 = 2/7

______________________________
The below division works on the same principles as the above divided fraction.5/8 / 3/4 = 5/8 * 4/3 = 20/24 = 5/6

5/6 * 3/4 = 15/24 = 5/8
_______________________________

from division;
total number of pieces / pieces per pie = total number of pies
5/8 = total number of pieces
3/4 = pieces per pie
5/6 = total number of pies

from multiplication.
pieces per pie * total number of pies = total number of pieces
5/6 = total number of pies
3/4 = pieces per pie
5/8 = total number of pieces

 

[Mark44]

Trying to use an analogy of a pie for all arithmetic operations seems like a stretch to me, especially when you come to, divisions such as (2/7)/(1/2). What does "1/2 = pieces per pie" even mean?

An analogy is a device that is used to make some more abstract idea easier to understand, but if it's difficult to comprehend the analogy, then some other analogy should be found.

 

[mr magoo]

What you quoted "pieces per pie" was from earlier descriptions that I used there to show how dividing fractions fit into the earlier descriptions analogies.

I will work on the math some more and see if I can get a better analogy to describe what I tried to describe before.
_____________

 

[coolul007]

I don't know if this hinders or helps, I define division of fractions as a purely multiplication problem:
$\frac{2}{7}$ divided by $\frac{1}{2}$ is the fraction: $\frac{\frac{2}{7}}{\frac{1}{2}}$

keeping in mind $\frac{2}{2}$ is equal to 1.

I find a fraction that multiplied by the large fraction's denominator equals 1, in this case $\frac{2}{1}$

I now have the resultant multiplication operation of:

$\left(\frac{\frac{2}{1}}{\frac{2}{1}}\right) \left(\frac{\frac{2}{7}}{\frac{1}{2}}\right)$

Since the first fraction is equal to 1, the value of the result does not change:

$\frac{\left(\frac{2}{1}\right)\left(\frac{2}{7}\right)}{\left(\frac{2}{1}\right)\left(\frac{1}{2}\right)}$

performing the multiplication in the numerator and denominator the result is:

$\frac{\frac{4}{7}}{1}$ or $\frac{4}{7}$ and this is why invert and multiply works...

 

[mr magoo]

Thanks for that. I'm still learning the basic of math.

 

[Mark44]

For fraction arithmetic, you can focus only on addition and multiplication. Subtraction and division are defined in terms of addition of the additive inverse (the negative of a fraction) and multiplication by the reciprocal.

IOW, $a/b - c/d = a/b + (-c/d)$, and
$a/b \div c/d = a/b ~\cdot ~1/(c/d) = a/b ~\cdot ~d/c$

 

[Tobias Funke]

I don't think 2/7 divided by 1/2 is that different from something like 20/4. We need to know how many times 1/2 a pie goes into 2/7 of a pie. We've cut the pie into halves and sevenths, but it's not too easy to compare them just yet, so we can cut it into fourtheenths instead. Thus we have 2/7 represented by 4 slices of the pie that's cut into fourteenths, and the 1/2 is represented by 7 slices. 7 slices "go into" 4 slices 4/7 times.

Algebraically, we just did $$2/7\div 1/2=4/14\div 7/14=4/7.$$ I think coolul007's method should be used as well, but considering that maybe the majority of students (American, at least) never learn fraction arithmetic, there's nothing wrong with seeing it in as many ways as possible.

 

[Diax]

Maybe this will help someone some time.

2$\div$2=1

2$\div$2= 2 $\div$ $\frac{2}{1}$=1

2$\div$2 =2 $\bullet$$\frac{1}{2}$=1

A proof for why fractions are divided by multipling the reciporicals is floating on the outskirts of realization for me...