Algebraic Terms Sixth and Seventh-Graders Encounter
A dozen things for sixth and seventh grade teachers to keep in mind:
- Using Symbols: Symbols have previously been used in formulas provided for specific applications (e.g., A=s² for the area of a square) or as placeholders for unknown numbers in simple arithmetic equations (e.g., “9 = ? + 4”). Now we will extend these ideas to solving more complicated equations, to creating formulas for specific problems, and to modeling co-variation, where two variables change together in a way that keeps an equation true.
- Exponent: We introduce notation with integer exponents at this level both for its direct value in arithmetic (in writing factorizations, for example) and to prepare for its use in algebraic expressions. However, we do not address algebraic equations with exponents until later grades.
- Expression: An arithmetic expression consists of a set of numbers connected by operation symbols (e.g., +, −, ÷), perhaps with some parts arranged as fractions or as exponents. An algebraic expression is the same except that some numbers may be replaced with letters.
- Equation: An equation is a statement that the values of two expressions are equal.
- Inequality: An inequality is a statement that the value of one expression is larger or smaller than that of another.
- Solving: The solution of an equation or inequality is a description telling which values of the variable make it true (e.g., x = 3 is the solution of 5x + 3 = 18). Note that an expression (such as 3x + 2 by itself, which contains no equals or inequality sign) cannot have solutions because it is not a statement claiming that something is true.
- Variables: Arithmetic rules still apply in algebraic expressions, but we can use single-letter symbols wherever we can use numbers. We call these letters "variables" because different numbers may be substituted for them. A variable is called an unknown if we are trying to find a value that will make the statement containing the variable true.
- Constants and coefficients: The numbers in an expression are called constants, since their values will stay the same. Numbers that directly multiply variables are called coefficients. For example, the expression 7x + 9 includes the coefficient 7, the variable x, and the stand-alone constant 9.
- Implied multiplication: The most obvious change in notation in algebra is that the multiplication sign is almost always omitted in algebraic expressions. When numbers or symbols are not separated by some other operation symbol, they are multiplied together. So 235xy means the same thing as 235 × x × y, and 2(x + 3) means the same as 2×(x + 3).
- Factors: A factor is an item that is multiplied by another item. The 235xy expression above consists of three factors: 235, x, and y. The expression 2(x + 3) has two factors: the number 2 and the term (x + 3).
- Terms: The word “term” in algebra refers to items or groups of items separated by plus or minus signs. Each term may contain factors and/or items with exponents. Thus the expression 3x² − 2x + 5 has three terms: 3x², −2x, and 5. A term with no variables, like the 5 in this example, is often referred to as a “constant term," or sometimes as just “the constant."
- Parameters: When constants in an equation play roles (such as a price or starting point) that might have different values in similar situations, they are sometimes called parameters. For example, we can think of the formulas c = 1.85w and c = 2.45w, which compute cost from weight, as special cases of a general c = pw formula, where p is a price-per-pound parameter. In these cases, p = 1.85 and p = 2.45.