How do you make sense of squaring a number less than zero When we square a number, we expect the result to be much larger than the original number.
But when we square a number between 0 and 1, we get a much smaller number.
Using money as an example, square 5 cents and we get 25 cents, BUT when we square 0.05 dollars we get 0.0025 dollars.
What is the best way to explain that?
 A: The idea here is that you need to square the units too. You see $\newcommand{\cents}{\text{ cents}} 5 \cents$ squared is not $25 \cents$, but is instead $25 \cents^2$ (you should read that as twenty-five square cents). Similarly, $\newcommand{\dollars}{\text{ dollars}} 0.05 \dollars$ squared is $0.0025 \dollars^2$. 
I don't know how to visualize or intuitively describe what a square cent or a square dollar are, but I don't have to be able to visualize them to work with them: that is one of the beautiful things about using mathematics to view problems abstractly. I do know that $100 \cents = 1 \text{ dollar}$, so by squaring both sides of this equation I do know that there are $10000$ square cents in $1$ square dollar, i.e $10000 \cents^2 = 1 \dollars^2$. This means that it is true to say $25 \cents^2 = 0.0025 \dollars^2$.
A: "When we square a number, we expect the result to be much larger than the original number": Only if our expectations are based on rather limited experience with squaring --- squaring numbers that were bigger than 1 to start with.  (It's sort of like someone who grows up in the Southern hemisphere and has never heard of the northern hemisphere, so he'd expect the weather to be much warmer in January than in July.) 
When you square $x$, the result is "bigger" than $x$ by a factor of exactly $x$.  That makes the square "much bigger" than $x$ if that factor $x$ is much bigger than 1.  If $x$ is only slightly bigger than 1, say 1.001, then $x^2$ is, correspondingly, only slightly bigger than $x$, like 1.002001.  If $x$ is exactly 1, then $x^2$ is "bigger" than $x$ by a factor 1, i.e., it's equal to $x$. And if $0<x<1$ then $x^2$ is smaller than $x$.  
A: If you want an intuitive way of understanding the squaring of positive numbers, then take some examples where they are just numbers, no units or conversions between them.
Examples: $(2)^2=2\times 2=2+2$, $3^2=3\times 3=3+3+3$
This is what the notation is short for so you would expect that squareing a number gives you a bigger number, at least when the number is bigger than one $1^2=1\times 1=1$.
$(\frac{1}{2})^2=\frac{1}{2}\times\frac{1}{2}$. From intuitive examples, like one half of $4$ is $2$ and, one half of $6$ is three, we expect the answer to get smaller. So one half of one half of something should be smaller.
It is  as it's a quarter of that something.
When the something you're interested in has units, you'll have to make a judgement as to whether it's going to be useful to include the units.
If your back garden is a square and you measure it to be $10$ metres ($10m$) in length and width, then it's are is $100m^2$, $100$ metres squared. This is a useful quantity to keep units in, because it gives us something useful, a way of measuring and comparing areas, and intuitively we expect an area to be different to a length.
In any case you would like to be multiply something with units by pure numbers without units. If I said the length of this table is three times the length of the table in my house, I don't want the three to have units, I just want to compare lengths of tables, and multiply numerical values and restore units afterward.
In your example of money, suppose person A says that if you gave them $x$ cents they would give you back $x^2$ cents, think of this as they would mulitiply your amount of money by a pure number, the same number as you gave them.
If person B say, if you give me $x$ dollars, and I will give you back $x^2$ dollars apply the same reasoning.
The result is this, person A is a good deal, person B is only a good deal if you give them more than a dollar and neither are a good idea if they aren't trustworthy people!
A: Let's expand this concept from squaring a number to multiplying two number $a$ and $b$. In general, you'd expect the product $ab$ to be larger than either $a$ or $b$ because children spend so much time multiplying integers. However, this is only true when both $a>0$ and $b>0$. There are two other cases to consider:

1. $a>0$ but $b<0$
Here the product $ab < a$ but $ab>b$. For example, $4$×$\frac{1}{2}=2$. You can see that the product, 2, is half of 4. On the other hand, the product is 4 times greater than $\frac{1}{2}$.
2. both $a<0$ and $b<0$
Here, the product will be smaller than both, as discussed in other answers, because the result here is best understood by setting $a=b$.
