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Dividing Basic Fractions
Spectral Swallow
Registered User regular
Registered User regular
So I'm trying to wrap my mind around dividing fractions and it is kicking my butt.
I mean I know how to (from back in school), but I'm trying to visualize it based on the rules and I'm getting two separate 'theories'.
So without further ado:
The 'rules'.
A.
B.
With A, that is literally what it is saying (based on the rules) and you end up with 6 full size blocks. But with B you end up with 6 partial blocks (and you're not following the 'rule').
Oy! Any help is greatly appreciated.
I mean I know how to (from back in school), but I'm trying to visualize it based on the rules and I'm getting two separate 'theories'.
So without further ado:
The 'rules'.
A.
B.
With A, that is literally what it is saying (based on the rules) and you end up with 6 full size blocks. But with B you end up with 6 partial blocks (and you're not following the 'rule').
Oy! Any help is greatly appreciated.
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Dividing by a fraction can also be written as a complex fraction like ( 3 )/ (1/2) where it becomes more clear that you are dividing the number your dividing by (1) first, which is identical to multiplying the top by the denominator.
I've always found that trying to treat fractions as special unique flowers leads to problems. The best thing I had ever had told to me about them was to always think of them as division problems you haven't solved yet. Do you have the same issue with dividing by decimals?
The expression A / B is asking you into how many groups of size B an item of size A may be divided. For example, if we have 3 / (1/2), we are asking into how many groups of size 1/2 an item of size 3 may be divided. The answer in this case is 6 (groups of size 1/2).
DevoutlyApathetic's method is a good sanity check on making sure that your answers are correct - it is always worth it to double check your work by reversing the process. EDIT: In addition, it is worth remembering, as he said, that dividing by fractions isn't any different from dividing by any other number.
Remembering that multiplication is inverse division is important too - dividing by a/b is the same as multiplying by b/a, you don't need separate rules for them.
Don't get hung up trying to figure out what division really means in this case; division is a very basic, broad, and abstract thing, and the rules are meant to help people conceptualize it. But the concept is just a useful fiction, it's not what division really is.
Never really had any problems with the stuff before, just when I started actually THINKING about what I was doing and trying to figure out which was actually correct.
With both of the versions shown though, throwing them into a multiplication problem yields the same result.
A.
B.
It's more just trying to figure out which version is 'more' correct.
I'm leaning toward A, myself, just based on the 'rule'.
The way you described is the way I remember learning it way back in the day.
But with the 'rule', we already know the number of 'groups' (1/2), and instead are trying to find the 'amount per group'. Which is where it's throwing me off.
It's almost like they're two separate problems. But not really.
Still I'd like to figure out why my mind is stumbling over something I learned 20 years ago...
"How the fuck can you have a group of 6 tennis balls if they're all cut in half! That's still 3 balls goddamnit!"
Try instead, "How many full size ice-cubes in this smaller tray can I make if I melt these bigger ice-cubes?"
Imagine it using volumes instead of objects. Dividing a single 10cc syringe of medicine into smaller doses in smaller 2cc syringes if you must.
Rather than dividing into groups, think of division as asking the question "how much does each one get?" So in the case of (2/3)/(3/4), you're saying "Ok, I have two thirds for every three fourths. How many would I need to keep that same ratio but have a denominator of one?" Obviously then, the way to determine that is to figure out how many for each fourth ((2/3)/3) then multiply by four. Which is precisely what you do when you multiply by the reciprocal.
Edit: To make it a little more concrete think of it this way: If three fourths of a person gets two thirds of a pie, how much pie should a whole person get?
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Edit: I think you're hung up on atomism, which (as said by others) is a philosophical concern and not one of computation.
My point is similar to tarnok's above me, in that, multiplication is easy and division is dumb. Invert and multiply is something I thought all math teachers taught, but I haven't seen anyone here mention it yet, so now I'm concerned that it isn't a common tactic?
Maybe it has to do with my complete lack of ability to do the visualization thing along with the difficulty I have with story problems? I always just have to do the numbers and not relate them to any real world objects, relabeling items in a problem with just letters and substituting back when I provide the answers after doing all the math.
Invert and multiply, anyone?
edit: rereading some of the above comments, I see some mention of the same technique, just not the keywords of invert and multiply, which for my brain makes fractions easy. I hope something here helps and gets that light bulb to go off!
Really, pizzas are a great unused resource here.
Imagine you have three pizzas.
Now, you can either divide by a whole number to find out the amount of pizza each person gets:
3 pizzas divided by six people equals one half pizza per person.
Or you can divide by a fraction to find out how many people you can feed with that much pizza each:
Three pizzas divided by halves feeds six people.
And that's the second example from above in a sensical way.
--LeVar Burton
If you're just doing a quick refresher on division, you'll be fine and dandy as long as you can actually perform the mathematical operation. If you are attempting to understand what division "is" (in relation to what numbers "are"), then you're probably better off taking a class on the philosophy of mathematics.
It is almost impossible to describe an abstract concept like mathematical division via the use of a single exemplar (as you are attempting to do with "groups" and/or "balls"). Why? Because you are actually working backwards - concepts are built via the use of multiple exemplars in synthesis, not the other way around. If you want to understand what the color "red" is, you can't simply point to a single red apple. You must point to a red apple, a red firetruck, a red LEGO brick, a red pencil, etc. etc. etc. and it is only through all of those things combined (including all "non-red" things) that your mind generates the distinct concept of "redness".
I get the concept behind the division and how to solve the problems, it's just the actual visualization behind following the rule that is throwing me.
To me putting any problem into the 'rule' yields weird results (as seen in A).
But I guess I'm not really going to worry about it too much more.
I understand the whole how many x's of y size does it take to make Z concept (from way back when), but it seems like the rule is just... stupid and unnecessarily complicated.
When I switch from basic math into philosophy I know it is time to call it quits.
So "never lose track of your units" might be the answer you're looking for.
Edit: PS, in this problem, each number has a unit: the units are "balls" (3 balls), "groups" (1/2 a group or 6 groups) and "balls per group (balls/group)" (1/2 a ball per group or 6 balls per group).
A: (balls) / (groups) = balls per group
B: (balls)/(balls/group) = groups.
Gosh the word "group" looks silly now.
Instead of thinking of the balls in B as 'balls', think of them as a pair of half-spheres, like below:
You understand that the number 1 can also be represented as 1/1 (one whole), or 2/2 (two halves).
The three whole balls you start with in B are groups of two halves (the 'sphere' in the above picture). When you divide by 1/2 (same as multiply by 2) you are counting how many 'halves' make up the 'whole'.
Take 3/2 divided by 3/4. That is equivalent to 3/2 multiplied by 4/3, then multiply straight across to get 12/6 and reduce to get 2. This works great for simple fractions, but when using larger and more complex fractions it can get rather cumbersome and tricky without a pen and paper
I'm not the OP and can't speak for him, but I cannot do a thing or remember a fact if I don't _understand_ it. I might be able to follow directions algorithmically but it's like trying to paint by numbers with a blindfold on. Once I _understand_ what's going on and why I hardly have to think about doing it anymore. Without an intuitive understanding of division of fractions algebra would have been a closed book to me.
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Though after reading all the posts here I'm 'better', but it still seems like the rule is stupid.
Lets work through a purely algebraic method. Hopefully this is will be a balance between rigorous, and exposition. I hope that this will be an interesting and different perspective.
We can define a simple set of rules of which all of the arithmetic we do. With basic arithmetic, we have natural numbers (1,2,3,...), integers (...,-1,0,1,...) and rational numbers of the form (a/b), where a and b can be any number, this includes the irrational numbers (Like prime roots, pi and very many others). We can give any numbers (x, y, z) several sets of rules:
Note that I'm skipping over basic additive rules, since we don't need to be concerned about them for this example (Actually, I'm skipping 12 other axioms of complete ordered fields, please don't hurt me!).
(1) 1*x=x
(2) (1/x)*x=(x^(-1))*x=1
It is ideal that we have multiplicative inverses for numbers, and it will be very helpful as such.
(3) x*(y+z)=x*y+x*z
For example: this is so we can say 6=3+3, and that 3*6=9+9=18, and so on.
(4) x*(y*z)=(x*y)*z
Here we have just established that we can commute.
I will use brackets to show what I am treating to be a number for the rules above (ie, z=(3/2)), in other cases it will be for clarity. For instance, much like the example in your first post, we have 12/3=(4*3)/3=4*(3/3)=4. Which is was a simple algebraic manipulation.
In the second example, which is a matter of tricky notation. I advise writing this out on paper: 3\div(1/2)=(2/2)*((3)/(1/2))=(2*3)*(2/2)=(3)*(2)=6
If we had resulted in number that was a fraction, we would leave it as is. It would be just a rational number, and it wouldn't need to exist as some other representation of itself, hence (a/b).
But wait! Aren’t these just rules? What have I really shown, if perhaps, anything at all? Just before I stated these 'rules' I used the word called axiom. Axioms are things that are assumed to be true. If we wanted to go more fundamental than this, we would be delving into abstract algebra.
Hopefully this doesn't seem to unsatisfactory. It is a very common thing to visualize these axiomatic operations using objects, or general concepts. But unfortunately visualizations can be completely wrong, or objects too for many other reasons, while seeming to be completely right. This is because we are not doing these operations with objects, but instead algebra.
My goal here was to generate some interest into using these things. Altweernative methods can become confusing over something that could be incorrect; that isn't to say that the algebraic way isn't confusing though. I understand that in basic maths people are often taught using objects and such, in which case I hope that the person who is teaching knows what they are talking about!