a. Interpret parts of an expression such as terms, factors, and coefficients.

Just like English has nouns, verbs, and adjectives, mathematics has terms, factors, and coefficients. Well, sort of.

Students should know that terms are the pieces of the expression that are separated by plus or minus signs, except when those signs are within grouping symbols like parentheses, brackets, curly braces, or absolute value bars. Every mathematical expression has at least one term. For instance, the expression 3x + 2 has two terms: 3x and 2. A term that has no variables is often called a constant because it never changes.

Within each term, there can be two or more factors, the numbers and/or variables multiplied together. The term 3x has two factors: 3 and x. There are always at least two factors, though one of them may be the number 1, which isn't usually written. But that 1 is always there...watching us.

Finally, a coefficient is a factor (usually numeric) that is multiplying a variable. Using the example, the 3 in the first term is the coefficient of the variable x.

The order or degree of a mathematical expression is the largest sum of the exponents of the variables when the expression is written as a sum of terms. For the example 3x + 2, the order is 1, since the variable x in the first term has an exponent of 1 and there are no other terms with variables.

The expression 5x² – 3x + 2 has order 2, whereas the expression 3xy + 5x²y³ – 7x + 32y⁴ has order 5, because the exponents of x and y in the second term are 2 and 3, respectively, and 2 + 3 = 5. No other term has a higher exponent sum.

Now that we have our words, we can start putting them together and make expressions. A good way to see if students really understand an expression like 3x + 2 is to have them translate mathematical expressions into English and vice versa. For instance, the expression 3x + 2 could also be written as, "the sum of 3 times a number and 2," or, "2 more than three times a number."

Clearly, it's much easier to write the mathematical expression than to write it in English (not to mention Pig Latin). The two are directly related to each other, however, and students should be able to translate back and forth. At first, students might want to make use of a "dictionary" like the table below to help them go from one language to the other.