I'm looking for a good way to explain how multiplication and division by fractions applies in the real-world the mechanics are receiving reasonably straight forward.
How can $2$ divided by $1/2$ being $4$ be explained in terms of a real-world example? And the same for $2$ multiplied by $1/2$ being $1$?
We have two cookies. We divide them into pieces of 1/2 cookie each and end up with four pieces. Thus 2 divided by 1/2 equals 4.
We have two cookies. We take 1/2 of the collection which is one cookie. Thus 2 multiplied by 1/2 equals 1.
Each of those examples can be criticized. In the first example, one could claim that it shows that 2 divided by 4 equals 1/2, not what I claimed. But a/b=c is equivalent to a/c=b for non-zero b and c. Likewise, the second could be claimed to show 1/2 multiplied by 2 is 1. In the second example, remember that the word "of" usually means multiplication. And a·b=c is equivalent to b·a=c. I can think of other criticisms also, but these examples do seem to work well.
If some object moves at the rate of 5 kilometers per minute, how long does it take to go 100 kilometers? Divide: 100 / 5 = 20 minutes
If some object moves at the rate of $\frac{1}{10}$ kilometer per minute, how long does it take to go $\frac{1}{2}$ kilometer? Again: divide $\frac{1}{2}$ / $\frac{1}{10}$ = 5 minutes
I think students can generally grasp multiplication by fractions, both in why the answers are reasonable and in applying real-world examples. Once students understand that multiplying by a number less than 1 gives a product that is smaller than the original number, remind students that division is the inverse of multiplication, and as such, dividing by a number less than 1 (but greater than zero) gives an answer that is larger than the dividend.
Real-world examples using length and time are easy to work with in a story problem. An example: I have 2 feet of ribbon and cut it into lengths of 1/2 foot each. How many pieces of ribbon do I have? Answer: 4.
Another example: Judy can make a friendship bracelet in 1/3 hour. She spends 3 hours making bracelets. How many bracelets does she make? Answer: 9.
This last example may have several ways to solve. Some would argue, therefore, it is not a good example. However, in the US, the CCSS allows for and encourages the development of multiple strategies in solving a problem. So, even though one student might think, "Judy makes three bracelets an hour" and multiply 3 times 3, this is mathematically equivalent to 3 divided by 1/3 -- we even teach kids to multiply by the reciprocal! I would suggest, then, that a problem like this be used during lecture time, allowing students to discuss the way they approached the problem.
Take anything that can easily be divided into fourths. For example a cracker that has four sections or index cards. Make sure the fourths are clearly marked. Take 6 of them for your example. Tell the students that someone uses 3/4 of this everyday and you want to know how many days it will last.
Then count or physically separate each group of 3/4. There should be 8 of them. This is a great real world example of 6 divided 3/4 is 8.
Multiplication and division are the inverse of each other.I try to give a simple example. When we multiply a number say 10 by 2, we get 10x2=20 and when we divide the same number 10 by 1/2, we again get 20. It shows that when 2 and 1/2 are the inverse of each other, then multiplication and division are also the inverse of each other.So whenever, fraction comes in multiplication or division, then firstly remove the fraction by reversing it and then simplify.
Every fraction has two parts, numerator and denomenator. The upper part is the numerator and the lower part is the denomenator where denomenator should never be equal to zero. Every decimal fraction should be converted into numerator and denomenator fraction before applying multiplication or division. The basic rule of multiplication is multiply numerator with numerator and denomenator with denomenator e.g.a/b multiplied with c/d will give you ac/bd. In division, the second number should be inversed. e.g. on dividing a\b by c\d, we will inverse the second number c\d to d\c and then multiply. So on dividing a\b by c\d, we will multiply a\b with d\c and the result will be ad\bc. Similarly, we can do multiplications and divisions for multiple fractions.
