**abstract** Intangible, not visible or touchable.

**acute angle** An angle that measures less than 90 degrees.

**acute triangle** A triangle in which all three angles are acute angles; in other words, each angle measures less than 90 degrees.

**addition keywords** Words that indicate addition.

**addition method** Sometimes called the elimination method, it is a method for solving a system of two equations. One or both of the equations needs to be modified so that when the two equations are added, one of the variables is eliminated.

**addition property of equations** An equation is still true if the same term is added to (or subtracted from) both sides of an equation.

**adjacent keywords** Two keywords that are next to each other (even if a word such as "the" appears between the keywords).

**algebraic equation** A statement that two algebraic expressions are equal.

**algebraic expression** A collection of numbers, variables, operations, and grouping symbols.

**amount** The number that is a percentage of the base. It is the number in the numerator in a percent ratio:

^{a}/_{b}= ^{p}/_{100} (see also definitions of base and percent)

**analogies** Similarities in relationships between two items.

**angle** Two rays sharing the same endpoint. (A ray is a part of a line that looks like an arrow. It begins at one point and goes on forever in the opposite direction.)

**area** The amount of surface enclosed by a closed figure.

**base** The number that is in the denominator in a percent ratio:

^{a}/_{b}= ^{p}/_{100} (see also definitions of amount and percent)

**base (of a triangle)** One of the three sides of a triangle that is perpendicular to the height.

**board method** This method can be used to help translate a word problem into an equation if there is a total given in the problem.

**clearing fractions** A method of simplifying an equation by multiplying both sides of the equation by the least common denominator before solving it. This method results in an equation with only integers and no fractions.

**coefficient** The number in front of a variable. For example, 4 is the coefficient in the term 4*x*.

**combining like terms** The process of adding or subtracting like terms.

**commutative property of addition** The order of the terms does not change the sum.

**complement** The percentage that would add to 100%. For example, 42% is the complement of 58%.

**concrete** Tangible, easily understood.

**consecutive even integers** Integers that are adjacent in an ordered list of even integers. For example, 4 and 6 are two consecutive even integers.

**consecutive integers** Integers that are adjacent on a number line. For example, 7, 8, and 9 are three consecutive integers.

**consecutive odd integers** Integers that are adjacent in an ordered list of odd integers. For example, 1 and 3 are two consecutive odd integers.

**direct translation strategy** This strategy is used when you can translate each word, one at a time in the same order as written, into its corresponding algebraic symbol.

**distance problems** Word problems that involve traveling a distance. The formula *d* = *rt* is used for these problems.

**distributive property of multiplication over addition** The property that allows a number or term to be distributed (by multiplication) to the sum of two terms in parentheses, for example, *a*(*b* + *c*) = *ab* + *ac*.

**division keywords** Words that indicate division.

**draw a picture** Create a drawing that helps visualize the word problem.

**elimination method** Sometimes called the addition method, it is a method for solving a system of two equations. One or both of the equations needs to be modified so that when the two equations are added, one of the variables is eliminated.

**equation** Two expressions set equal to each other. The easiest way to differentiate between an expression and an equation is that the equation has an equal sign.

**equilateral triangle** A triangle with all three sides of equal length.

**estimate** An educated guess of the solution. This guess is made before setting up an equation and solving the problem.

**evaluation** To follow the order of operations and determine the value.

**even integers** Integers that are even numbers.

**even number** A number divisible by two.

**expression** A collection of constants, variables, symbols of operations, and grouping symbols.

**grouping symbols** Usually parentheses, but grouping is also indicated by brackets and braces.

**harmless parentheses** Parentheses that are not necessary but that do not do any harm in the expression.

**height (of a triangle)** A line segment that is perpendicular to the base of a triangle, with one endpoint being the opposite vertex.

**homogeneous units** When both numbers are measured with the same units.

**identify the variable(s)** The process of choosing a letter to represent the unknown value and describing that variable.

**identity property of multiplication** Any number times one has the same value.

**implied multiplication** Multiplication is implied when a number is placed next to a variable, or when a number is placed next to an expression surrounded by parentheses.

**integer** A counting number or a negative whole number.

**investment problems** Word problems that involve investing money at a simple interest rate.

**isosceles triangle** A triangle with two sides of equal length.

**keywords** Words that indicate a mathematical operation.

**leading keywords** If the first word in an English phrase indicates an operation, it is a leading keyword; it "leads" the expression.

**least common denominator** The least common multiple of all the denominators in the problem.

**least common multiple** The smallest of the common multiples of more than one number. For example, 12 and 24 are both multiples of 3 and 4. 12 is the least common multiple of 3 and 4.

**like terms** Terms with the same variables raised to the same powers.

**linear equation** An equation that can be put in the form A*x* + B*y* + C = 0.

**mixture problems** Word problems that involve mixing two or more items to create a new mixture. Often, these are chemistry problems, but they can include problems in which two different types of nuts are mixed to create mixed nuts, and so on.

**money problems** Word problems that concern money.

**multiple equation word problems** Word problems that can be translated into more than one equation.

**multiple operations** An expression has multiple operations when you see more than one symbol for addition, subtraction, multiplication, and/or division.

**multiples of a number** An infinite list of the products of a number and each whole number. For example, the multiples of 3 are 3, 6, 9, 12, 15, 18 . . .

**multiplication keywords** Words that indicate multiplication.

**multiplication property of equations** An equation is still true if both sides of the equation are multiplied by (or divided by) the same term.

**obtuse angle** An angle that measures more than 90 degrees and less than 180 degrees.

**obtuse triangle** A triangle with one obtuse angle.

**odd** A number that is not divisible by two.

**odd integers** Integers that are odd numbers.

**order of operations** When an expression has multiple operations, they must be performed in the following order: 1) operations within parentheses; 2) exponents; 3) multiplication and division, from left to right; and 4) addition and subtraction, from left to right.

**percent(s)** The part per one hundred in the percent proportion:

^{a}/_{b}= ^{p}/_{100} (see also definitions of base and percent)

**percentages** A given part in every hundred.

**perimeter** The sum of the lengths of the sides of any closed figure.

**perpendicular** Two line segments are perpendicular if they meet at a right angle, 90 degrees.

**Polya's four-step process** A process of problem solving published in 1945 by George Polya. The steps are as follows: 1) Understand the problem; 2) Devise a plan; 3) Carry out the plan; 4) Look back over the results.

**proportion(s)** Two ratios set equal to each other.

**ratio** A comparison of two quantities by division.

**rectangle** A four-sided closed figure in which the opposite sides are parallel and of equal length. All pairs of adjacent sides of a rectangle meet at right angles.

**right angle** An angle that measures 90 degrees.

**right triangle** A triangle with one right angle.

**scalene triangle** A triangle with all three sides of different lengths.

**simplified** An algebraic expression is simplified by using the distributive property and combining like terms.

**square** A square is a special type of rectangle in which all sides are equal in length.

**substitution method** A method for solving a system of two equations. One of the equations needs to be solved for one of the variables. That expression is then substituted into the other equation for the variable. The resulting equation has only one unknown.

**summation problems** A word problem with a total.

**subtraction keywords** Words that indicate subtraction.

**systems of equations** More than one equation that are solved simultaneously.

**term** The addends of an expression.

**translate** Change a phrase written in English to an algebraic expression, using the correct symbols.

**triangle** A three-sided closed figure.

**tried-and-true method** A method for solving word problems that has been used for many years. Some examples are Polya's four-step process, identifying variables, and estimation.

**turnaround words** Words that indicate a change in order from left to right. The expressions are turned around, and the second English phrase becomes the first algebraic expression (and visa versa). The basic turnaround words are, TO (including INTO), FROM, and THAN.

**units** The method of measurement used for the numbers in a word problem. For example: feet, inches, dollars, degrees, and so on.

**variable** A symbol used to represent an unknown number, often *x* or *n*.

**variable omission** An error that occurs when, in the process of translating a sentence in the word problem into an equation, one of the variables is left out.

**variable reversal** This error is made if the variables are switched with each other.

**vertices** The three endpoints of the line segments that make up a triangle.

**work problems** Word problems that involve two people or machines working together at different rates to complete one whole job.