Expressions

Expressions use numbers, variables and operations to describe computations. The rules of arithmetic, the use of parentheses and the conventions about order of operations assure that the computation has a well-determined value. Reading an expression with comprehension involves analysis of its underlying structure, which may suggest a different but equivalent way of writing it that exhibits some different aspect of its meaning. For example, //p// +0.05//p// can be interpreted as the addition of a 5% tax to a price //p//. But rewriting //p// +0.05//p// as 1.05//p// shows that adding a tax is the same as multiplying by a constant factor. Algebraic manipulations are based on the conventions of algebraic notation and the rules of arithmetic. Heuristic mnemonic devices are not a substitute for procedural fluency. For example, factoring, expanding, collecting like terms, the rules for interpreting minus signs next to parenthetical sums, and adding fractions with a common denominator are all instances of the distributive law; the definitions for negative and rational exponents are based on the extension of the exponent laws for positive integers. The laws of exponents connect multiplication of numbers to addition of exponents and thus express the deep relationship between addition and multiplication captured by the parallel nature of the rules of arithmetic for these operations. Complex expressions are made up of simpler expressions using arithmetic operations and substitution. When simple expressions within more complex expressions are treated as single quantities, or chunks, the underlying structure of the larger expression may be more evident. //Connections to Equations and Functions.// Setting expressions equal to each other leads to equations. Expressions can define functions of the variables that appear in them, with equivalent expressions defining the same function. Students understand that: > > > > Students can and do: > > > >
 * Core Concepts**
 * 1) Expressions are constructions built up from numbers, variables, and operations, which have a numerical value when each variable is replaced with a number.
 * 1) Complex expressions are made up of simpler expressions.
 * 1) The rules of arithmetic can be applied to transform an expression without changing its value.
 * 1) Rewriting expressions in equivalent forms serves a purpose in solving problems.
 * Core Skills**
 * 1) See structure in expressions.
 * 1) Manipulate simple expressions.
 * 1) Define variables and write an expression to represent a quantity in a problem.
 * 1) Interpret an expression that represents a quantity in terms of the context.