An expression expresses something. Facial expressions express emo- tions. Mathematical expressions express calculations with numbers. Some of the numbers might be given explicitly, like2or 3 4 . Other numbers in the expression might be represented by letters, such as ,,P, or. The calculation an expression represents might use only a single operation, as in4 3or3, or it might use a series of nested or parallel operations, as in3 9 9. An expression can consist of just a single number, even0. Letters standing for numbers in an expression are calledvari- ables. In good practice, including in student writing, the meaning of a variable is specified by the surrounding text; an expression by itself gives no intrinsic meaning to the variables in it. Depending on the context, a variable might stand for a specific number, for exam- ple the solution to a word problem; it might be used in a universal statement true for all numbers, for example when we say that that for all numbersand; or it might stand for a range of numbers, for example when we say that 2 for 0. In choosing variables to represent quantities, students specify a unit; rather than saying "letGbe gasoline," they say "letGbe the number of gallons of gasoline". MP6. An expression is a phrase in a sentence about a mathematical or real-world situation. As with a facial expression, however, you can read a lot from an algebraic expression (an expression with variables in it) without knowing the story behind it, and it is a goal of this progression for students to see expressions as objects in their own right, and to read the general appearance and fine details of algebraic expressions. Anequationis a statement that two expressions are equal, such as10 002 20, or3 4 , or2 1 2 2.It is an important aspect of equations that the two expressions on either side of the equal sign might not actually always be equal; that is, the equation might be a true statement for some values of the variable(s) and a false statement for others.For example,10 002 20is true only if 500; and3 4 is not true for any number; and2 1 2 2is true for all numbers.Asolutionto an equation is a number that makes the equation true when substituted for the variable (or, if there is more than one variable, it is a number for each variable).An equation may have no solutions (e.g.,3 4 has no solutions because, no matter what numberis, it is not true that adding3toyields the same answer as adding4to).An equation may also have every number for a solution (e.g.,2 1 2 2). An equation that is true no matter what number the variable represents is called anidentity, and the expressions on each side of the equation are said to beequivalent expressions. For example2 1and2 2 are equivalent expressions.In Grades 6-8, students start to use properties of operations to manipulate algebraic expressions and produce different but equivalent expressions for different purposes. This work builds on their extensive experience in K-5 working with the properties of operations in the context of operations with whole numbers, decimals and fractions.