Chapter 7 Objectives Define basic terms in algebra: integer, number statement, expression, and coefficient Learn the relationships between positive and negative numbers Solve problems that include absolute value Complete the integer operations of addition, subtraction, multiplication, and division Calculate the square root Use the order of operations to solve problems Write expressions for word problems Solve equations

Pre-Algebra Basics Pages 162 - 184

Pre-Algebra Basics Page 162 Algebra = the study, understanding, and use of symbolic reasoning, mathematical properties, processes, and calculations to problem solve for a variety of unknown situations. This unit lays the foundation for algebraic concepts. In health care, the use of algebra is seen in the use of formulas to solve problems. One such formula is the conversion between temperature systems. If you do not follow the order of operations, for example, when you are calculating with the temperature formula, you will not reach the correct solution. In addition, dimensional analysis is an algebraic formula used to convert among measurement systems.

Page 163 Integers We use integers to solve many everyday math problems. The integers consist of the positive whole numbers (1, 2, 3, 4, 5, …), their negatives (-1, -2, -3, - 4, -5, …), and the number zero. Zero is neither positive or negative; it is neutral. Integers form a countable infinite set. The number line is a line labeled with the integers in increasing order from left to right. The number line extends in both directions: Examples – page 136 -1 1 2 3 4 -7 -6 -5 -4 -3 -2 5 6 7

Integers (Cont’d) Page 164 The (-) sign is used to show a negative number and the (+) sign is used to show a positive number. Remember that any integer on the right is always greater than the integer on the left. We can use >, <, & = to represent the number relationships. Examples – page 164-165 – Practices 1-3: evens Group Work – page 164-165 – Practices 1-3: odds

Pages 165 – 166 Absolute Value The absolute value of an integer indicates its distance from zero on the number line. The absolute value of a number such as 3, is shown as 3 . The straight bars indicate the distance of a number from zero. This is why absolute value is never negative; absolute value only asks “How far?”, whether the number is in a positive or a negative direction on the number line. This means that both -3 and 3 = 3 because both are three units away from zero. Examples – page 136 – practice 4: evens Group Work – page 136 – practice 4: odds

Integer Operations Pages 166 – 168 Adding Integers with the Same Sign To add integers having the same sign, add their absolute vales and then use the same sign as the numbers you are adding. Adding Integers with Different Signs To add integers having the different signs, subtract their absolute values and then use the sign of the larger number. Examples – page 167 – practice 5: even Group Work – page 167 – practice 5: odd *** Examples – page 168 – practice 6: even Group Work – page 168 – practice 6: odd

Integer Operations (Cont’d) Pages 168 – 169 Integer Operations (Cont’d) Subtracting Integers Negative – negative = negative + positive Subtract absolute value and use the sign of the larger number Positive – negative = Positive + positive Just add Examples – page 169 – practice 7: evens Group Work – page 169 – practice 7: odds

Integer Operations (Cont’d) Pages 169 – 170 Integer Operations (Cont’d) Multiplication of Integers Positive x Positive = Positive Negative x Negative = Positive Negative x Positive = Negative Positive x Negative = Negative Examples – page 170– practice 8: evens Group Work – page 170 – practice 8: odds

Integer Operations (Cont’d) Page 171 Integer Operations (Cont’d) Division of Integers The division symbol is usually not used in algebra. A fraction bar is used to show division Positive ÷ Positive = Positive Negative ÷ Negative = Positive Negative ÷ Positive = Negative Positive ÷ Negative = Negative Examples – page 171– practice 9: evens Group Work – page 171 – practice 9: odds

Exponential Notation Page 172 Exponential notation = a useful means for writing a product of many factors. Base = the number being multiplied Exponent = the number of times that the base is multiplied Simple rules: Any number raised to the first power is always equal to itself If a number is raised to the second power, we say it is squared. If a number is raised to the third power, we say it is cubed. Any number raised to the zero power is equal to 1 1. Numbers without a power are said to have an invisible power of 1 4. Except 0 *** Examples – page 172– practice 10: evens Group Work – page 172 – practice 10: odds

Square Roots Pages 172 – 173 The square root sign is √ To find the square root of a number x, one finds the factor that when multiplied by itself one time, equal the number x inside the square root sign. Examples – page 173 – practice 11: Even Group Work – page 173 – practice 11: odd

Order of Operations Page 173 Mathematics have developed a standard order of operations for calculations that have one arithmetic operation. Following the order of operation allows for only one correct answer for each problem. Perform any calculations inside parentheses fraction bars, exponents & square roots Perform all multiplications & divisions, working from left to right Perform all addition & subtractions, working from left to right

Order of Operations (Cont’d) Page 173 Order of Operations (Cont’d) Please Excuse My Dear Aunt Sally. P: Parentheses & fraction bars E: Exponents & Roots M: Multiplication D: Division A: Addition S: Subtraction Left to right Examples – page 174 – practice 12: Even Group Work – page 174 – practice 12: odd

Algebraic Expressions Page 174 Algebraic Expressions Variable = a letter or symbol that represents an unknown number. Letters such as p, l, x or y are used to represent variables. A variable can be used in addition, subtraction, multiplication &/or division problems. Some variables have coefficients like the –8 in –8s, where the coefficient or number –8 is multiplied by the unknown number s. If a variable appears by itself as s, or xy, it is understood to have a coefficient of 1 because s = 1s & xy = 1xy

Algebraic Expressions: Cont. Page 174 Algebraic Expressions: Cont. Expressions Expression = a mathematical statement that may use numbers and/or variables. Algebraic expression = an expression that contains one or more variables. To evaluate an expression at some number means we replace or substitute a variable in an expression with the number, and simplify the expression. Examples – page 175-178 – practice 13-17: even Group Work – page 175-178 – practice 13-17: odd

Algebraic Expressions: Cont. Page 178 Algebraic Expressions: Cont. Writing Expressions from Word Problems The most important use of writing expression is in real life situations. Careful reading of the problem will help you ensure that you use the correct mathematical operation. = + - x ÷ is add subtract multiply divide as sum difference product quotient equals plus minus times split equal to total remainder "of" per   more than less than increased by decreased by Examples – page 179 – practice 18: Evens Group Work – page 179 – practice 18: odds

Algebraic Expressions: Cont. Page 179 Algebraic Expressions: Cont. Solving Equations To solve an equation, get the variable by itself on one side of the equal to sign. Use inverse operations to do this: Addition is the inverse of subtraction and vice versa Multiplication is the inverse of division and vice versa Examples – page 179 – practice 19: Evens Group Work – page 179 – practice 19: odds

Algebraic Expressions: Cont. Pages 180 – 181 Algebraic Expressions: Cont. Writing Equations from Word Problems Algebraic equation = an equation that contains one or more variables. There will also be algebraic expressions on both sides of the equation. So an equation is a mathematical sentence with an equal sign that illustrates that two expressions represent the same number. To understand word problems, you must be able to translate word sentences or other data into an equation. Examples – page 181-182 – practice 20: evens Group Work – page 181-182 – practice 20: odds