## Homework 12.1 Expressions

### Overview

The language of algebra is written in symbols, numerals, variables, constants, and expressions. In order to be successful in math, students should be able to translate its language into words and words into math language. Numerals and many symbols should already be familiar, as well as many operations with real numbers.

### Common Symbols and Numerals

Several mathematical symbols should already be familiar, such as + for addition, – for subtraction, x or ·for multiplication, and ÷ for division. Similarly, most students are familiar with = for equals, < for less than, and > for greater than. The integers 0, 1, 2, 3 … are common, as are fractions such as ½, 1/3, ¼, 2/3, and so on. By the time students begin studying algebra, they are familiar with working problems that involve basic operations, as well as some that aren’t so basic.

### Variables, Constants, and Expressions

Variables are letters or symbols used to represent any value that can change, depending on the needs in the expression. Constants do not change. They may be numerals or particular values. For example, the letter pi π is a constant that always means the same thing, 3.1416, give or take a few decimal places. Numerical expressions contain only constants and symbols for operations, such as 2 +2 = 4. Algebraic expressions are similar, but they also contain variables, constants and symbols for operations. While 2 + 2 =4 is a numerical expression, x +y = 4 is a simple algebraic expression. It contains two variables, x and y; one constant, 4; and two symbols for operations, + and =.

### Translating from Symbols to Words

In order to translate from symbols to words, understand what the symbols mean and analyze them. For example, the expression “x +2” can mean the sum of x +2 or x increased by 2. The expression “5 –a” can mean the difference of 5 and a, or a less than 5. The expression “8z” can mean 8 times z or the product of 8 and z. The expression “y/5” can mean y divided by 5 or the quotient of y and 5.

### Translating from Words to Symbols

In order to translate words into mathematical expressions, such as for a word problem or real-life application of math, analyze the statement in words and the relationships between them. Drawing a picture can be a useful strategy to translate a problem into an algebraic expression. Certain words and phrases mean particular operations. Are things to be put together? Addition might be the operation. Is one thing more or less than another? Subtraction might be needed. Are equal numbers of things to be grouped? Multiplication might be the key. If equal groups need to be separated, they are divided.

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