Chapter 1 Introduction In this chapter we will: model relationships using variables, expressions, and equations. apply order of operations to simplify and evaluate expressions. learn algebraic properties then evaluate expressions by applying them graph points on a coordinate plane and make scatter plots. Using Variables (1.1) Algebraic expression: A mathematical phrase that include numbers, variables, and operation symbols. Variable: A symbol, usually a letter, that represents one or more numbers. Examples: 1) n +7 seven more than n 2)n 7 the difference of n and 7 Using Variables (1.1) Sample Problem Write an algebraic expression for each phrase. a)the product of seven and n b)the quotient of n and seven c)two times a number plus five d)seven less than three times a number Using Variables (1.1) Equation: A mathematical sentence that uses an equal sign. A true equation means the expression on either side of the equal sign must represent the same value. An open sentence is an equation that contains one or more variables. Examples: 1) c = 12n Exponents and Order of Operation (1.2) Simplifying a numerical expression involves replacing it with its simplest form. A numeric expression may include powers. Powers: A numeric expression that includes a base and a exponent. Base: a number Exponent: tells how many times the number is used as a factor. Examples: 1) (2)(8) + (2)(3) = 22 2)2 4 = (2)(2)(2)(2) = 16 Order of operation: 1.Parentheses (or any other grouping symbols i.e. brackets) When an expression has several grouping symbols, simplify the innermost expression first. A fraction bar is also a grouping symbol. When present do the calculations above and below the fraction bar before simplifying the fraction. Exponents and Order of Operation (1.2) Order of operation (cont.): 2. Exponents The base of an exponent is the number, variable, or expression directly to the left of the exponent. 3. Multiply/Divide 4. Add/Subtract Please excuse my dear Aunt Sally. Exponents and Order of Operation (1.2) Evaluating an algebraic expression involves substituting a given number for each variable and then simplifying he numeric expression using the order of operation. Exponents and Order of Operation (1.2) Exploring Real Numbers (1.3) Natural numbers: Any positive whole number. Whole numbers: Any positive whole number and zero. Exploring Real Numbers (1.3) Integers: Any positive whole number, zero, and any negative whole number. Two numbers that are the same distance from zero on a number line but lie in opposite directions are opposites. Absolute value: the distance a number is from zero on a number line. Exploring Real Numbers (1.3) Summarizing Real Numbers Exploring Real Numbers (1.3) Inequality: A mathematical sentence that compares the value of two expressions using an inequality symbol, such as. To compare fractions, it is better to convert the fractions to decimals first. Examples: 1)a < b a is less than b 2)a > b a is greater than b Exploring Real Numbers (1.3) Adding Real Numbers (1.4) These two properties are used in conjunction with two addition rules to carry out addition of real numbers. Identity Property of Addition n +0 = n; for every real number n. Inverse Property of Addition n +(-n) = 0; for every real number n and its additive inverse -n. Adding Real Numbers (1.4) The two addition rules are as follows: Adding Numbers with the Same Sign To add two numbers with the same sign, add their absolute values. The sum has the same sign as the addends. Adding Numbers with Different Signs To add two numbers with the different signs, find the difference of their absolute values. The sum has the same sign as the addend with the greater absolute value. Adding Real Numbers (1.4) Sample Problem Simplify each expression. a)-5 + (-6) b)13 + (-34) c) d) Adding Real Numbers (1.4) A matrix can be used to add real numbers. Matrix (plural: matrices): A rectangular arrangement of numbers in rows and columns. Element: Each item in a matrix A matrix is identified by the number of rows and the number of columns (rows x columns). Matrices are equal if the elements in corresponding positions are equal. Only matrices of the same sizes can be added. Add matrices by adding the corresponding elements. Adding Real Numbers (1.4) Adding these two 3 x 2 matrices involve adding the corresponding positions. Adding Real Numbers (1.4) Subtracting Real Numbers (1.5) To subtract real numbers we will use the concepts of real numbers and their opposites. There is a single rule for subtracting real numbers. Subtracting Numbers To subtract a number add its opposite. Subtracting Real Numbers (1.5) Multiply/Divide Real Numbers (1.6) These three properties are used in conjunction with two additional rules to carry out multiplication of real numbers. Identity Property of Multiplication 1 n = n; for every real number n. Multiplication Property of Zero n 0 = 0; for every real number n. Multiplication Property of n = -n; for every real number n. The two multiplication rules are as follows: When 3 or more negative numbers are involved, you must account for all the negative signs as you multiply. Multiplying Numbers with the Same Sign The product of two positive numbers or two negative numbers is positive. Multiplying Numbers with Different Signs The product of a positive number and a negative number is negative. Multiply/Divide Real Numbers (1.6) Sample Problem Simplify each expression. a)-3 4 b)(-3) 4 There are two rules for dividing real numbers: Dividing Numbers with the Same Sign The quotient of two positive numbers or two negative numbers is positive. Dividing Numbers with Different Signs The quoitient of a positive number and a negative number is negative. Multiply/Divide Real Numbers (1.6) The Distributive Property (1.7) The Distributive Property can be used to multiply a sum or a difference by a number. This property can also be used multiply some numbers using mental math. It can also be used to simplify an algebraic expression. Distributive Property For every real number a, b, and c, a(b+c) = ab + ac(b+c)a = ba + ca a(b c) = ab ac (b c)a = ba - ca The Distributive Property (1.7) Sample Problem Use the Distributive Property to simplify the following expression. a)34(102) The Distributive Property (1.7) Parts of an algebraic expression have different names. Terms: A number, a variable, or the product of a number and one or more variables. Coefficient: A numerical factor of a term. Constant: A term with no variable. Example: 6a 2 5ab + 3b 12 Coefficient Constant The Distributive Property (1.7) An algebraic expression in the simplest form will have no like terms. Like terms: Terms that have exactly the same variable factors. Use the Distributive Property to combine like terms when simplifying an expression. Examples: 1)3x and -2x 2)-5x 2 and 9x 2 3)xy and -xy The Distributive Property (1.7) Sample Problem Simplify the following expression. a)3x 2 + 5x 2 b)-5c + c The Distributive Property (1.7) Writing an algebraic expression from a verbal description requires an understanding of the mathematical terms, most of which were covered in section 1.1. Quantity: this term indicates that two or more terms are within parentheses. Example: 1)3 times the quantity x 5 3(x 5) 2) -2 times the quantity t plus 7 -2(t +7) Properties of Real Numbers (1.8) PropertyEquation Commutative Property of Additiona+b = b+a Commutative Property of Multiplication ab = ba Associative Property of Addition(a+b)+c = a+(b+c) Associative Property of Multiplication(ab)c = a(bc) Identity Property of Additiona + 0 = a Identity Property of Multiplicationa1 = a Inverse Property of Additiona + (-a) = 0 Inverse Property of Multiplication Distributive Propertya(b+c) = ab + ac(b+c)a = ba + ca a(b c) = ab ac (b c)a = ba - ca Multiplication Property of Zeron 0 = 0 Multiplication Property of -1-1 n = -n Graphing Data on the Coordinate Plane (1.9) Two number lines that intersect at right angles form a coordinate plane. Ordered pair identifies the location of a point on the graph. X-coordinate is also known as the abscissa. Y-coordinate is also known as the ordinate. 4 quadrants identified by the signs of the abscissa and ordinate. Graphing Data on the Coordinate Plane (1.9) Scatter plot: A graph that relates two groups of data. Can be used to look for trends in the data. Use of a trend line can help show the correlation in the data points more clearly. Trend line: A line running trough the data points and has as many points above as below the line. Positive correlation: This shows that both sets of data increase together. Represented by a positive slope. Negative correlation: This shows one set of data decreases as the other set increases. Represented by a negative slope. Graphing Data on the Coordinate Plane (1.9) Negative Correlation Positive Correlation Chapter 1 The End