Hi! I have 2 questions:
1) Is 2/1 considered to be an improper fraction?
2) How do you explain to a kid why a fraction divide by another fraction gets a bigger fraction? My son cannot understand because a whole number divided by another whole number always get a smaller number.

Thanks.

For question 1.

An Improper fraction has a top number (numerator) larger than (or equal to) the bottom number (denominator). It is "top-heavy".

Examples include 3/2, 7/4, 21/15, etc. In your case 2/1 is an Improper Fraction. It is a whole number when you simplify it - 2/1 = 2.

A Proper Fraction is one where the numerator is less than the denominator such as 3/7 or 2/5.

A Mixed Fraction is the combination of a Whole Number and a Proper Fraction such as
5 2/3, 8 1/2, 4 3/5.


For your second question. Give me a short while to put together one or two visuals that might help.

Thanks for your prompt response. My son was given a question on fractions and the answer is 2 but it must be expressed as an improper fraction. He gave the answer as 2/1 but his teacher said it is wrong - must write as 4/2 or 6/3 etc... So is the teacher wrong or is there a mathematical rule stating that if the denominator is 1, then it is not considered to be a fraction to begin with?

Difficult question. Sometimes there is no wright or wrong. Mathematically speaking, a whole number can be written as a improper fraction with a denominator of 1. This is commonly referred to as an "Invisible Denominator" and is most frequently used when speaking about reciprocal fractions ie 2/3 is the reciprocal of 3/2 or 1/2 is the reciprocal of 2/1. You can search Google for Invisible Denominators and there will be numerous discussions and examples about it. That said, it really depends on the context of what the teacher was teaching the class. At times we limit the exposure of information and details to the child to make things simple. This is particularly common in science where there are deeper and "more correct" answers than what is provided because of the complexities involved.

Yeah, that's right. Sometimes, it is very difficult to convince my son and some teachers think that he is trying to make things difficult. But my son is not someone who is satified in just memorising formulae, he is a very visual and experimential person. If the teacher cannot explain a concept properly, he will lose sleep over it.

I have googled about this improper fraction topic and read briefly about invisible denominators but none of the articles explicitly answers his query. The question he was given is a very simple arithmetic one and the answer happens to be 2 but the instruction states that all answers must be expressed as an improper fraction.

Again it's context. Mathematically 2/1 is an improper fraction. This is similar to my daughters school. They would like her to use the word "units" instead of a 'u' and on each paper they mark it "wrong" but so far have not taken marks off. Mathematically there is no difference as they are both just a bunch of letters with implied meaning, since we do not teach the kids to explicitly define the meaning such as "Let units represent the number of bla bla bla" before we proceed. However, the teacher would like her to use more English such that it is easier to follow the questions and develops good habits for later. So the logic of why is understandable. However, mathematically there is no difference.

I have put together a few things that may help. This is a highly discussed topic, a simple search on Google will yield many discussions with none really coming up with a 'solution' that is immediately easily to follow for kids. IMO this is because we do not divide by fractions in our daily activities and thus we do not have many real life examples to use. Where examples exist we tend to create a new unit of measure to deal with it. Eg. We do not say how many tenths (1/10) are there in one meter. Instead we say how many centimeters are there in one meter. Similarly, we do not say how many fifths (1/5) are in a 2L container of milk. Instead we say, how many cups are there in the container of milk. And these two examples are easy where the numerator is one (1). Imagine if we were talking about dividing by 7/15....

Thank you very much. I shall try to explain the concepts using your example.