1.1 Numbers & Number Operations

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Gertrude Newton
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1 1.1 Numbers & Number Operations Objectives: Learn how to represent numbers and number operations Learn how to use grouping symbols Numbers Whole 0, 1, 2, 3, 4,... Counting 1, 2, 3, 4,... Fractions 1, 2, 5 (part or whole plus part) Rational can be expressed as the quotient of two numbers. Even though some fractions convert to a repeating decimal, they are still rational. 1 = 0.3 is rational 3 Irrational A number possessing a non-repeating AND non-terminating decimal (π = ; 2 are irrational) Too many others to list here! How a number is written determines what it is called. Example: what do we call the number two divided by three Fraction: 2 3 Decimal: 0.6 Percent: % Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

2 1.1 Numbers & Number Operations Numbers can be approximated to different degrees of accuracy: The symbol means is approximately equal to. Degrees of Accuracy and Purpose: Purpose Symbol Degree of Accuracy Example Money $ 2 decimal places $2.17 Batting Average % 3 decimal places.321 Operations: There are 4 Basic Operators: +,,, +, Addition, sum is what the answer is called, Subtraction, difference is what the answer is called, Multiplication, product is what the answer is called Warning! Do not use the symbol!, Division, quotient is what the answer is called Parenthesis Brackets Grouping Symbols: ( ), [ ] Are used to show calculations that need to be done preferentially to the Basic Operators (+,,, ). Remember! Work from the inside out [(35 7) i 2] ( ) (7 i 2) 12 + [5 i 2] = 3 = Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

3 1.2 Variables in Algebra Objectives: Learn how to evaluate variable expressions Learn how to represent real-life quantities using variable expressions Terms: Variables: Values of variables: A letter used to represent one or more numbers the number the variable represents Algebraic expression: a collection of numbers, variables, operations and grouping symbols Value of expression: result when the variable is given a number and operations are complete Algebraic Expression 5x 2a c 2x + 3 Meaning 5 times x 2 times a, divided by c 2 times x, plus 3 Rules for evaluating algebraic expressions: 1. Write (or re-write) the algebraic expression 2. Substitute numerical values for variables ( plug-in what we know) 3. Simplify the expression 4. Give the answer! Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

4 1.2 Variables in Algebra Example: 5x (2x+4) when x = 2, 5 5x ( 2x+ 4) 52 ( ) ( ) 5x ( 2x+ 4) 55 ( ) ( ) 10 ( 4 + 4) 25 ( ) 10 8 = = 11 Real-world Problems: Area: A = bh or A = lw Interest: Temperature: Distance: I = p r t (where p = principal, r = rate, and t = time) 9 F = C d = r t Substitute ( plug-in ) variables you know to find the answers! Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

5 1.3 Exponents and Powers Objectives: Learn how to evaluate expressions with exponents Learn how to use exponents in real-life problems Terms: Exponents: Used to represent repeated multiplication Base: Power: The number being multiplied by itself The number of times (or level) a base is multiplied by itself Read 27 as: two to the seventh power Special Powers: Power Special Significance 3 Referred to as cubed for cubic units (in.3) 2 Referred to as squared for square units (in.2) 1 anything to the first power is written as the base number itself 1 5 = = 2 0 anything to the zero power is ALWAYS 1 (more on that later!) Exponents, Variables and Grouping Symbols: Be careful when evaluating expressions containing powers, variables and grouping symbols! Remember the correct order of operations... Parenthesis (Grouping Symbols) need to be evaluated BEFORE exponents. Does 4x 2 = (4x) 2 Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

6 1.3 Exponents and Powers Example: Does 4 x = (4 x ) 2 2 One way to check is to pick any one possible value for x, and substitute that value into each expression. Let s say x = 4 4x 2 step 1: re-write the expression (4x) 2 4(4)2 step 2: plug-in what we know (4 4) step 3: Simplify the expression (16) 2 = 64 step 4: give the answer = 256 So 4x 2 (4x) 2 Real-life Problems: Surface area of a cube: S = 6s 2 A = s 2 why area of a square Since there are 6 sides on a cube, the total surface area of a cube is 6 times the area of a single side. Surface area of a sphere: SA = 4πr 2 Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

7 1.3 Exponents and Powers Practice: What is the base and power of the following Base Power 1. 2x 2 x 2 2. (2x) 2 2x x 16 x Write the mathematical expression for each of the following: 4. Three cubed: y to the ninth: y 9 6. x x x x = x 4 Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach

8 1.4 Order of Operations Objectives: Learn to use order of operations to evaluate algebraic expressions P E M D A S P - Parenthesis ( ), [ ], { } (Grouping Symbols). Start from the inside and work outward E - Exponents x 2 These operators have equal priority, perform which ever occurs first in an expression M - Multiply ( )( ), or -- NOT -- D - Divide A - Add + S - Subtract Examples: 1) (12 8) + 3 First, evaluate the grouping symbols and then the exponents Next, perform the multiplication and division from left to right Then, perform the remaining addition and subtraction from left to right Last, give the answer! Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

9 1.4 Order of Operations 2) 23 [(45 15) 3 9] 2 First, evaluate the innermost pair of grouping symbols 23 [(3) 3 9] 2 Next, evaluate the next pair of grouping symbols. Remember, exponents before division! 23 [27 9] 2 23 [3] 2 After the grouping symbols have been finished, continue according to the order of operations... Exponents before subtraction Last, give the answer! = 14 3) {8 + [(84 12) 4] 2 7} + 3 {8 + [7 4] 2 7} + 3 {8 + [3] 2 7} + 3 { } = 13 Real-world: To determine the class average of the following grades: 87, 92, 85, and 81 Average = sum of all grades divided by the number of grades ( ) 4 = What is the tax on two sweaters when tax is 7.25%, sweater #1 is $19.95 and sweater #2 is $24.99 ($ $24.99) (0.0725) = $3.26 Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

10 1.5 Equations and Inequalities Objectives: Learn how to solve and check equations Learn how to solve and check inequalities Terms: Equation: Two mathematical expressions separated by an equal sign ( = ) examples: 2a = 5 t = = 3 Open sentence: an equation (or inequality) that contains at least one variable Solution: A number that produces a true statement when it is substituted for the variable in an open sentence. Example: 2a = 5 when a = 2 ½ and when a = 3 2(2 ½) = 5 2(3) = 5 5 = 5 true 6 5 false Inequality: Two mathematical expressions separated by an inequality sign. symbols: > greater than < less than greater than or equal to less than or equal to not equal to Example: If t = 7, does t = 65 (7) = = false Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

11 1.5 Equations and Inequalities More Examples: Is 6 a solution to these inequalities 3x + 1 < 8 3(6) +1 < < 8 19 < 8 No 14 x > > 9 8 > 9 No 2x (6) Yes Equations with No Solutions, One Solution, or Many Solutions: 2y = 2(y + 4) 2y = 2y + 8 No Solutions 2y = y + 4 y = 4 One Solution 2y = 2(y + 4) 8 2y = 2y y = 2y Many Solutions Inequalities with No Solutions, or Many Solutions: x + 1 < x x + 3 > 4 No Solutions Many Solutions Any # greater than 1 Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

12 1.6 Verbal & Algebraic Models Objectives: Learn how to translate verbal phrases into algebraic expressions Learn how to translate verbal sentences into algebraic equations and inequalities Words often have more than one meaning. An extreme example is bad, which may mean evil or rotten, or is slang for good. Can you identify the different ways the following words can be used Plus: Product: add or benefit the result after multiplying or something produced or manufactured Difference: the result after subtracting or the condition of being unlike something else The Language of Addition: Verbal Phrase Algebraic Expression(s) The sum of 5 and a number 5 + n n more than a number n n A number plus 3 n n Addition is commutative, (meaning that the order in which numbers are added does not matter) The Language of Subtraction: Verbal Phrase The difference of 5 and a number Algebraic Expression(s) 5 n Four less than a number n 4 Eight minus a number 8 n Subtraction is NOT commutative, (meaning that the order in which numbers are subtracted matters!) Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

13 1.6 Verbal & Algebraic Models The Language of Multiplication: Verbal Phrase The product of 6 and a number Five times a number Algebraic Expression(s) A number multiplied by 4 4n n4 Multiplication is commutative, (However, we usually write the coefficient BEFORE the variable) 6n 5n The Language of Division: Verbal Phrase The quotient of a number and 8 Fifteen divided by a number Division is NOT commutative Algebraic Expression(s) n 8 15 x Translating Verbal Phrases into Algebra: Read each verbal phrase and write the corresponding algebraic expression: 1. The sum of 6 and a number is 17: 6 + n = twelve more than a number is 21: n + 12 = The quotient of 51 and 17 is equal to a number minus 4: = x 4 Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

14 1.6 Verbal & Algebraic Models Writing an equation as an Algebraic Model: The cost for a theme park is $50.00 for adults and $35.00 for children. Write an equation when 3 adults take several children to the theme park and the total cost was $ Verbal Model: Adult Admission Cost Children Admission Cost + = Total Admission Cost Labels: Adult Admission cost = $50 Adults going to theme park = 3 Children Admission cost = $35 Children going to theme park = x Equation: 50(3) + 35x = 255 Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach

15 1.7 A Problem-Solving Plan Objectives: Learn how to use algebra to solve simple, real-life problems Learn how to make an algebraic model for a real-life problem Problem Solving Strategies: Guess & Check Make a Table Look for a Pattern Draw a Diagram Using Algebra: 1. Set-up a verbal model 2. Label what you know and what you don t know 3. Write an equation 4. Solve Example 1: Janet decided to purchase a computer that cost $1500. If she made a $575 down payment and borrowed the rest from her aunt with the promise to repay her in 20 equal monthly payments, how much will each monthly payment be Step 1: Verbal Model monthly payment ($) = ($ amount borrowed) 20 months Where the amount borrowed = cost of computer down payment Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

16 1.7 A Problem-Solving Plan Step 2: Apply Labels Cost of computer = $1, Down payment = $ Number of Monthly payments = 20 Amount of Monthly payment = x Step 3: Write an Equation x = ( ) 20 Step 4: Solve ( ) 925 x = = = The monthly payment is $46.25 How can we check this answer 20(46.25) = = = 1500 Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

17 1.7 A Problem-Solving Plan Example 2: If an airplane flies from Phoenix to St. Louis at a speed of 500 miles per hour (mph) and flies over Tulsa, Oklahoma, 400 miles from St. Louis and is told it needs to wait 3 hours before landing, should the pilot reduce his speed Step 1: Verbal Model We need to apply the Distance formula: D = r t Speed of Airplane Flight time = Distance to travel Step 2: Apply Labels Speed of airplane = x Flight time = 3 Distance to travel = 400 Step 3: Write an Equation 3x = 400 Step 4: Solve 3x = 400 x = mph Does this answer seem reasonable Commercial aircraft must fly at a speed of at least 350 mph to avoid stalling, so, no, this answer is not reasonable. The pilot would need to do something else besides just slowing down, perhaps circling in a holding pattern. Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach

18 1.8 Exploring Data: Tables & Graphs Objectives: Learn how to use tables to organize data Learn how to use graphs to model data visually Terms: Data - (plural) facts or numbers that describe something Organizing Data: Often, data is just numbers on a page. Without some form of organization, data is meaningless, however, once it is organized, it becomes a key building block of information and knowledge. Organizing Data with Tables: Arranging data in a table often reveals patterns or trends. There is no best way to organize data, but there are reliable techniques. Increasing order Decreasing order Groups Frequency Table: Using natural groups or ranges can be useful for data organization. Example: Student Birthdates Birthdays by month Boys Girls Frequency Boys Frequency Girls Jan I II Feb III March II I April IIII III Total Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach

19 1.8 Exploring Data: Tables & Graphs Visualizing Data with Graphs: Graphs are a common way to display data as a visual model. Common Types of Graphs: Bar Line Circle/Pie Stem-and-Leaf plot Box-and-Whisker plot Example: Bar Graphs Average Grade by Class Period 1 Period 2 Period 3 0 1st Qtr 2nd Qtr 3rd Qtr 4th Qtr Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS

UNIT 4 NOTES: PROPERTIES & EXPRESSIONS Vocabulary Mathematics: (from Greek mathema, knowledge, study, learning ) Is the study of quantity, structure, space, and change. Algebra: Is the branch of mathematics