## EXPRESSIONS AND EQUATIONS

ENDURING UNDERSTANDINGS

- Expressions are powerful tools for exploring, reasoning about, and representing situations.
- Two or more expressions may be equivalent, even when their symbolic forms differ. A relatively small number of symbolic transformations can be applied to expressions to yield equivalent expressions.
- Variables have many different meanings, depending on context and purpose.
- Using variables permits writing expressions whose values are not known or vary under different circumstances.
- Using variables permits representing varying quantities. This use of variables is particularly important in studying relationships between varying quantities.
- The equals sign can indicate that two expressions are equivalent. It is often important to find the value(s) of a variable for which two expressions represent the same quantity. Finding the value(s) of a variable for which two expressions represent the same quantity is known as
*solving an equation*. - An inequality is another way to describe a relationship between expressions; instead of showing that the values of two expressions are equal, inequalities indicate that the value of one expression is greater than (or greater than or equal to) the value of the other expression.
- In solving an inequality, multiplying or dividing both expressions by a negative number reverses the sign that indicates the relationships between the two expressions.
- The equals sign can be used in defining or giving a name to an expression or function rule.
- Functions provide a tool for describing how variables change together. Using a function in this way is called
*modeling*, and the function is called a*model.* - Functions can be represented in multiple ways-in algebraic symbols, graphs, verbal descriptions, tables, and so on-and these representations, and the links among them, are useful in analyzing patterns of change.
- One important way of describing functions is by identifying the rate at which the variables change together. It is useful to group functions into
*families*with similar patterns of change because these functions, and the situations that they model, share certain general characteristics. - Some representations of a function may be more useful than others, depending on how they are used.

*Developing Essential Understanding of Expressions, Equations, and Functions for Teaching Mathematics in Grades 6 - 8.*Reston, VA; The National Council of Teachers of Mathematics, Inc.