The definition I use in my math class for the word expression is: "One or more numbers or variables joined with arithmetic operators and grouping symbols." When I present this definition, I need to identify the meanings of the words variable, operator, and grouping symbol.

Variables

My approach to teaching variables can be found here: Variables as Math Pronouns.

Operations

When I first started teaching, I didn't spend much time defining what an operation is, because...well...because kids have been doing operations since they were in elementary school, right? But I've learned that talking in precise terms about what operations are can be very beneficial to students.

Operation: An operation is like a math verb - it takes one or more numbers and performs an action on them - it combines or modifies the numbers to produce a new result.

If you have read my previous article about variables as pronouns, you will recognize that I'm extending the metaphor between language and math. 

I tell my students that most of the operations we'll talk about involve two numbers. The operation takes two numbers and swirls them around together in some way to produce one new result. Each of the numbers that you put into the operation vanishes and gets replaced by a single number. For example, plus is an operation, and if you have something like 5 + 7, the plus sign is the action that takes the 5 and the 7 and transforms them into a single number - twelve. When you've done the addition, neither the 5 nor the 7 are part of your answer; they've both disappeared and have been replaced by the twelve.

Aside: If this sounds like the description of a function to you, then that's good; our arithmetic operators are binary functions. For example: A(x, y) = x + y is the addition operator; you put two numbers into the addition function, and get a single result out. I don't mention this to the students (since they may not have ever heard of a function before), but looking at operations this way will help them be prepared for functions down the road.

Having talked through the idea of an operator, I ask students to list operations they're familiar with. They will come up with all the basic arithmetic operations: addition, subtraction, multiplication, and division. Someone will likely suggest "parentheses," and I'll tell them that parentheses are a grouping symbol rather than an operator, and we'll talk about those in a minute. Parentheses don't transform (change) the numbers inside, they just tell you to treat them like a group.

When we talk about division, I'll mention that there are multiple ways of writing division - either using the division symbol, or using a horizontal fraction bar - so it's okay for an operation to have more than one operation symbol.

Speaking of operation symbols, it's also likely that someone will say "exponents." This one is a funny operation, and it's worth spending a few minutes on it. What makes it funny is that unlike the operations students are most familiar with, it doesn't have a symbol. Students may be unsure about whether it's actually an operation, since there's no symbol for it. So I send them to their calculators, and point out that their calculator does have a symbol for it: the carat. So when they 5², if they want to think of it as having an operation symbol, they could think of it as 5 carat 2. It's an operation, because it takes the 5 and the 2 and combines them to create something new - the number 25.

Also, I encourage them to be very precise in their wording; an exponent is not an operation - it's a number. The verb is "raised to the power of", "raised to," or "to the power" or just "to the". If they want a single word to use to describe the operation, you can tell them that they can call it "exponentiation."

At this point they're probably done thinking of operations. I'll say to them, "remember I said that operations take one or more numbers? All the operations we've talked about take two numbers and combine them. Can you think of an operation that uses just one number?" If they can't think of one, ask them if "square root" is an operation. It takes one number instead of two (and that's okay, because our definition allows for that), and does something to it to produce a new result. = 6; = 9. You put one number into the square root operation, and you get a new number out. Again, this sounds a lot like how we think of functions.  

As an aside, it's what we call a unary operation; it only takes one number and transforms it. Also, in one sense, we could think of it as a binary operation; the word "square" indicates the type of root, and we could appropriately think of it as = 6. I don't bring this up with my students at this point, but it's a good distinction to be aware of; later in the algebra curriculum, we'll have to adjust our thinking on square roots.

Somewhere along the line, someone may suggest that "equals" is an operation, and I'll say to them, "That's not an operation, since it doesn't change any numbers, but we'll talk about what it is in a minute." Then, when we're done talking about operations, I'll briefly switch gears to talk about...

Relationships

Since my goal at this point is to get students ready to write expressions, and the equals sign is not part of an expression, I may or may not talk about relationships at this point; if they've brought it up I will; if they haven't, I might not. A relationship is a way of describing how two expressions are related to each other, but it doesn't change the value of the expressions. Relationships are things like "equal to", "not equal to", "less than", etc. I point out to students that the definition of expression does not include anything about relationships, so the moment they put in a relationship symbol, it's not an expression any more!

Grouping Symbols

Grouping symbols are used to indicate that certain numbers and operators should be treated as a single unit. Your students are likely familiar with the most common grouping symbol: parentheses. They may not realize that there are others. For example, there are square brackets and curly brackets, which are used to "nest" groupings as follows: 3 + 2{5 - 4[3 + 10(12 - 10)]}.

But there are other grouping symbols. I teach my students right off the bat that the fraction bar is a grouping symbol. The fraction bar groups everything in the numerator together, and everything in the denominator together. Writing is functionally equivalent to writing . Students must get this concept drilled into their heads before they start dealing with algebraic fractions! Interestingly, that means the fraction bar serves a dual purpose: it is both an operation and a grouping symbol.

I tell my students that there's another operation in their list that performs double-duty. Someone will usually spot it - it's the square root. Like the fraction bar, the square root symbol has a horizontal line. That horizontal line (called a vinculum, if anyone cares) is a grouping symbol. anything under that bar should be treated as a single group. Thus, is functionally equivalent to . I don't need to write the parentheses, because the grouping is inherent in the square root symbol.

Now, after spending quite a bit of time on these definitions and examples, students are ready to spend some time giving me examples of expressions. This is how I wrap up the lesson. Some of the students will play it safe and give me simple ones like 3 + 5. Others will want to get a little crazy and give me things like 

x +

3 - x

x +

5

- x^{3 + 4}. Let them have fun with it. Invariably, someone will include an equals and I'll immediately interrupt them and say, "Wait a minute! Is equals an operator?" To remind them that it's a relationship, and it has no place in expressions.

This whole process sets the stage for a very important ability students must develop in order to be good algebra students: the ability to convert English language phrases like "twice the sum of five and seven" into expressions.