Anyone studying pre-algebra must know basic definitions of key terms. First and foremost, students must understand the idea of a variable: a variable is a letter, such as x or y, that serves as a placeholder for an unknown quantity. Some variables are directly preceded by numbers, as in 5b; these numbers are known as coefficients. Coefficients are multiplied by the variable -- that is, 5b is equivalent to 5 times b, with five being the coefficient. Numbers without attached variables are called constants; for instance, nine is a constant in 6h + 9. Constants, variables and their coefficients all fall under the same category -- terms. For example, x is a term, as are y, 5b, 6h and 9. A term may also include an exponent, as in k^3.
Pre-algebra students learn to add and subtract terms, but only certain types of terms can be added or subtracted, and many students struggle to recognize which ones. To help them with this, pre-algebra students should learn the definition of the phrase "like terms." Like terms are those that contain the same variables and exponents. For example, 3p and -5p are like terms, as are 6r^4 and 9r^4, because their variables and exponents match; in contrast, 3p and -5w aren’t like terms, nor are 6r^4 and 7r^5, because their variables and exponents differ. Pre-algebra students must be able to add and subtract like terms, for example, 3p–5p=-2p and 6r^4+7r^4=13r^4. They must also know that unlike terms cannot be added or subtracted. For instance, trying to subtract 3p–5w results in an answer of 3p–5w, and attempting to add 6r^4+7r^5 produces the same outcome, 6r^4+7r^5. Many students erroneously try to combine unlike terms, believing 3p–5w=-2pw or that 6r^4+7r^5=13r^9. Because the concept of combining terms is consistently revisited in pre-algebra, carrying these incorrect beliefs throughout the course greatly reduces a students' chances of passing.
Regardless of whether or not they’re like, terms or groups of terms separated by addition or subtraction symbols are known as expressions. It is extremely important for pre-algebra students to be able to distinguish between expressions and equations. Essentially, equations consist of expressions separated by an equals sign. For instance, 3z+1=7 is an equation, but 3z+1 is an expression. Equations are solved by manipulating the terms on both sides of the equals signs, resulting in an isolation of the variable, while expressions can only be simplified or evaluated. For example, the solution to an equation contains a variable and a number separated by an equals sign, such as z=2, but the answer to an expression problem contains only variables, numbers or a combination, as in 3z+1.
For many students, pre-algebra is one of the first math courses in which most problems can’t be solved mentally -- that is, writing down steps is necessary to obtain the correct answers. Even if an answer is wrong, showing work allows the student to learn from the mistake, and allows the teacher to award partial credit. It is important to write legibly; all too often errors are made because students can’t read their own writing, and in these cases, teachers can’t give credit, either. When solving equations, write steps vertically and keep equals signs in alignment. Put solutions on the bottom-most line. And don’t underestimate the importance of showing each and every step -- students who don’t show their work in this manner aren’t likely to succeed in pre-algebra or later math courses because the problems are too complex to solve otherwise.