Unit Essential Questions How can you represent quantities, patterns, and relationships? How are properties of real numbers related to algebra? Williams Math Lessons
TARGET RATING 3 VARIABLES AND EXPRESSIONS MACC.9.A-SSE.A.a: Interpret parts of an expression, such as terms, factors, and coefficients. LEARNING SCALE write algebraic expressions and apply them to real world situations or more challenging problems that I have never previously attempted write algebraic expressions write algebraic expressions with help understand that algebra uses symbols to represent quantities that are unknown or may vary WARM UP Which of these situations have a value that varies? a) The population of this school b) The number of classrooms in this school c) The time it takes you to get to your next class d) Your high school GPA KEY CONCEPTS AND VOCABULARY Addition Subtraction Multiplication Division sum plus added to more than increased by difference minus subtract less than decreased by less fewer than product times multiply multiplied by of double/ triple quotient divide shared equally divided by divided into anything that can be measured or counted a symbol, usually a letter, that represents the value of a variable quantity a mathematical phrase that includes one or more variables a mathematical phrase involving numbers and operation symbols, but no variables EXAMPLES EXAMPLE : REWRITING A WORD EXPRESSION (ADDITION OR SUBTRACTION) Write an algebraic expression for each word phrase. a) 3 more than f b) 0 less than c c) 5 decreased by p -8-
EXAMPLE : REWRITING A WORD EXPRESSION (MULTIPLICATION OR DIVISION) Write an algebraic expression for each word phrase. a) the quotient of 9 and k b) the product of 5 and y c) r divided by 5 d) twice a number s EXAMPLE 3: REWRITING A WORD EXPRESSION (WITH VARIOUS OPERATIONS) Write an algebraic expression for each word phrase. a) The sum of and twice y b) 7 less than the product of y and z c) 5 minus the quotient of x and y d) more than twice the number z EXAMPLE : REWRITING AN ALGEBRAIC EXPRESSION Write the word phrase for each algebraic expression. a) d + 5 b) p 3 c) x d) x/7 e) 00 + 6y f) c - 85 EXAMPLE 5: WRITING AN ALGEBRAIC EXPRESSION FOR REAL WORLD SITUATIONS A car salesman gets paid a weekly salary of $300. They are also paid $00 for each car that they sell during the week. Write a rule in words and as an algebraic expression to model the relationship in the table. Cars Sold Total Earned 0 $300 + (0 x $00) $300 + ( x $00) $300 + ( x $00) n RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3-9-
TARGET RATING 3 ORDER OF OPERATIONS MACC.9.A.SSE.A..b: Interpret complicated expressions by viewing one or more of their parts as a single entity LEARNING SCALE evaluate algebraic expressions and apply them to real world situations or more challenging problems that I have never previously attempted simplify expressions involving exponents use the order of operations to evaluate expressions simplify expressions involving exponents with help use the order of operations to evaluate expressions with help understand that you can use powers to shorten how you represent repeated multiplication WARM UP Find the greatest common factor of each pair of numbers. ) and 6 ) 9 and 5 3) 3 and 3 ) and 8 KEY CONCEPTS AND VOCABULARY has two parts, base and exponent a number that shows repeated multiplication a number that is multiplied repeatedly to replace an expression with its simplest form to substitute a given number for each variable, and then simplify ORDER OF OPERATIONS P PARENTHESIS perform any operations inside grouping symbols, such as parenthesis ( ), brackets [ ], and a fraction bar. E EXPONENTS simplify powers M D A S MULTIPLY AND DIVIDE from LEFT TO RIGHT (not multiplication before division) ADDITION AND SUBTRACTION from LEFT TO RIGHT (not addition before subtraction) EXAMPLES EXAMPLE : EVALUATING AN EXPRESSION What is the simplified form of each expression? a) b) 8 5 c) d) (0.3) 3-0-
EXAMPLE : ORDER OF OPERATIONS What is the simplified form of each expression? a) (5 3) b) + 6 c) 0 5 d) 3 5 EXAMPLE 3: EVALUATING AN ALGEBRAIC EXPRESSION What is the value of the expression for x = and y = 3? a) x + y b) 3xy + x c) y + y 8 x d) (x + y ) (y + ) EXAMPLE : WRITING AND EVALUATING AN EXPRESSION FOR REAL WORLD SITUATIONS You receive a weekly allowance. Every week you deposit ¼ of your allowance into a savings account. Evaluate the amount of spending money you have if your weekly wage is: a) $60 b) $ 00 c) $00 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3 --
TARGET REAL NUMBERS AND THEIR SUBSETS MACC.9.N-RN.B.3: Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. RATING 3 LEARNING SCALE provide counterexamples involving the sum and product of real number subsets classify real numbers into real number subsets understand the sum and product of real number subsets classify real numbers into real number subsets with help understand the sum and product of real number subsets with help understand that real numbers can be divided into subsets WARM UP Use order of operations to simplify. ) 3 + 6 ) 5[( + 5) 3] 3) 0 + 8 KEY CONCEPTS AND VOCABULARY SUBSETS OF REAL NUMBERS NAME DESCRIPTION EXAMPLES Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers EXAMPLES EXAMPLE : IDENTIFYING SUBSETS OF REAL NUMBERS Your math class is selling pies to raise money to go to a math competition. Which subset of real numbers best describes the number of pies p that your class sells? --
EXAMPLE : CLASSIFYING NUMBERS INTO SUBSETS OF REAL NUMBERS For each number, place a check in the column that the number belongs to. Remember the numbers may belong to more than one set. # Number Real Whole Natural Integer Rational Irrational a) 9 b) c) 8 d) e) 5 0 f) 0 g) h) 3π + EXAMPLE 3: OPERATIONS OF REAL NUMBERS Show each statement is false by providing a counterexample. a) The difference of two natural numbers is a natural number. b) The product of two irrational numbers is irrational. c) The product of a rational number and an irrational number is rational. d) The sum of a rational number and an irrational number is rational. RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3-3-
TARGET RATING 3 PROPERTIES OF REAL NUMBERS MACC.9.A.SSE.A..b: Interpret complicated expressions by viewing one or more of their parts as a single entity LEARNING SCALE use properties of real numbers to provide counterexamples or solve more challenging problems that I have never previously attempted identify and use properties of real numbers identify and use properties of real numbers with help understand that relationships that are always true for real numbers are called properties WARM UP Simplify each expression. ) + 5 ) 6 + 6 3) (3 + ) 8 ) 3 + 0 KEY CONCEPTS AND VOCABULARY Two algebraic expressions are if they have the same value for all values of the variables. COMMUTATIVE PROPERTY The order in which you add or multiply does not matter. For any numbers a and b, a + b = b + a and ab = ba ASSOCIATIVE PROPERTY The way three or more numbers are grouped when adding or multiplying does not matter. For any numbers a, b, and c, (a + b) + c = a + (b + c) and (ab)c = a(bc) ADDITIVE IDENTITY For any number a, the sum of a and 0 is a a + 0 = 0 + a = a or 7 + 0 = 0 + 7 = 7 ADDITIVE INVERSE A number and its opposite are additive inverses of each other, a + ( a) = 0 or 5 + ( 5) = 0 MULTIPLICATIVE IDENTITY For any number a, the product of a and is a. a = a = a or = = MULTIPLICATIVE INVERSE (RECIPROCALS) a For every number,, where a,b 0, there is exactly b one number b a such that the product is one. a b b a = or 3 3 = MULTIPLICATIVE PROPERTY OF ZERO Anything times zero is zero. a 0 = 0 a = 0 or 0 = 0 = 0 --
EXAMPLES EXAMPLE : IDENTIFYING PROPERTIES What property is illustrated? a) z 7 = 7z b) 9 9 = c) (f + 5) + 3 = f + (5 + 3) d) 3xyz + 0 = 3xyz EXAMPLE : USING PROPERTIES TO FIND UNKNOWN QUANTITIES Name the property then find the value of the unknown. a) n x = 0 b) 7 + (3 + z) = (7 + 3) + c) 0 + n = 8 d) 6h = 6 EXAMPLE 3: IDENTIFYING THE PROPERTY USED IN EACH STEP Name the property used in each step. (3+ 0) + 5 5 3 = (3) + 5 5 3 = 3+ 3 = 3+ 3 = Step ) Step ) Step 3) Step ) EXAMPLE : PROVIDING A COUNTEREXAMPLE Is the statement true or false? If it is false, give a counterexample. a) For all real numbers, a + b = ab b) For all real numbers, a() = a RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3-5-
TARGET RATING 3 WARM UP THE DISTRIBUTIVE PROPERTY MACC.9.A-SSE.A.a: Interpret parts of an expression, such as terms, factors, and coefficients LEARNING SCALE use the Distributive Property to rewrite fraction expressions or solve more challenging problems that I have never previously attempted use the Distributive Property to simplify expressions use the Distributive Property to simplify expressions with help understand that I can use the Distributive Property to simplify expressions Name the property that each statement illustrates. ) 8 + 0 = 8 ) ( ) = () 3) x + (y + 3) = (x + y) + 3 KEY CONCEPTS AND VOCABULARY THE DISTRIBUTIVE PROPERTY For any numbers a, b and c a (b + c) = ab + ac and (b + c)a = ab + ac and a(b c) = ab ac and (b c)a = ab ac is a number, a variable, or the product of a number and one or more variables is a term that has no variable is a numerical factor of a term have the same variable factors (same variables raised to the same power) An expressions is in when it contains no like terms or parenthesis EXAMPLES EXAMPLE : USING THE DISTRIBUTIVE PROPERTY OVER ADDITION AND SUBTRACTION Use the distributive property to write in simplified form. a) (y + 3) b) (xy + 8y 3) c) h(3h 7) -6-
EXAMPLE : WRITING EXPRESSIONS IN SIMPLEST FORM Simplify the following. a) (a 7) + 3a + a b) 9y 5 + 8 + y y c) 3h 5h + 8h d) x 3 x + x EXAMPLE 3: REWRITING FRACTION EXPRESSIONS Write each fraction as a sum or difference. x + a) b) 6 3x 9 EXAMPLE ; WRITING AND SIMPLIFYING EXPRESSIONS Use the expression twice the sum of x and y increased by six times the difference of x and 3y. a) Write an algebraic expression for the verbal expression. b) Simplify the expression. RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3-7-
TARGET RATING 3 WARM UP AN INTRODUCTION TO EQUATIONS MACC.9.A-CED.A.: Create equations and inequalities in one variable and use them to solve problems. LEARNING SCALE write and solve equations with one variable and apply them to real world situations or more challenging problems that I have never previously attempted write and solve equations with one variable write and solve equations with one variable with help understand that an equation is a mathematical sentence Identify and correct the error. ) 5(x + ) = 5(x ) 5() = 50x ) 0(a 3) = 0(a) + 0( 3) = 0a + 7 KEY CONCEPTS AND VOCABULARY A mathematical statement that contains algebraic expressions and symbols is an. An is a mathematical sentence that uses an equal sign (=). A of an equation containing a variable is a value that makes the equation true. - an equation that is true for every value of the variable. EXAMPLES EXAMPLE : IDENTIFYING SOLUTIONS OF AN EQUATION Determine if the given value is a solution to the equation. a) Is x = 7 a solution of the equation x + 0 = 3? b) Is x =0 a solution of the equation? x = 3-8-
EXAMPLE : APPLYING THE ORDER OF OPERATIONS Solve. a) 8 + ( ) 3 = x b) b = 7 ( 7) c) 3(x + ) 5 = 3x d) 6 3 c + 9 5 = ( ) c ( 5) EXAMPLE 3: USING MENTAL MATH TO FIND SOLUTIONS What is the solution of each equation? a) d + = 9 b) 3 m = 6 c) t 5 = 35 EXAMPLE : WRITING EQUATIONS Write an equation for each sentence. a) The sum of 3x and 5 is 3. b) The product of x and is 6. EXAMPLE 5: WRITING EQUATIONS FOR REAL WORLD SITUATIONS A grocery store cashier makes $.50 more per hour than a bagger. Write an equation that relates the amount x that a bagger earns each hour if a cashier makes $0.5 per hour. RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3-9-
RELATIONS MACC.9.A-REI.D.0: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line) RATING LEARNING SCALE represent a relation and interpret graphs of relations that apply real-world situations TARGET 3 represent a relation interpret graphs of relations represent a relation with help interpret graphs of relations with help understand how to read and plot points on a coordinate plane WARM UP Ticket prices for admission to a museum are $8 for adults, $5 for children, and $6 for seniors. a) What algebraic expression models the total number of dollars collected in ticket sales? b) If 0 adult tickets, 6 children s tickets, and 0 senior tickets are sold one morning, how much money is collected in all? KEY CONCEPTS AND VOCABULARY A is formed by the intersection of two number lines. - the horizontal axis - the vertical axis The is the point where the x and y axes intersect. - names the location of the point in the plane, usually written (x, y) y - axis Ordered Pair (,7) origin x - axis - a set of ordered pairs - the set of all inputs (x-coordinates) - the set of all outputs (y-coordinates) -0-
Ordered Pairs Mapping Diagram Table Graph (0, 0) (-, 3) (, 5) (-, -) (0, -7) EXAMPLES EXAMPLE : REPRESENTING A RELATION Express the relation as a table, a graph, and a mapping. Ordered Pairs Mapping Diagram Table Graph (5, 0) (-, 5) (, 3) (-6, ) (-, -) EXAMPLE : DETERMINING DOMAIN AND RANGE Determine the domain and range for each relation. a) {(, 3), (-,5), (-5, 5), (0, -7)} b) x y 0 3 3 - c) d) --
EXAMPLE 3: ANALYZING A GRAPH What are the variables? Describe what happens in the graph. a) The graph shows the volume of air in a balloon as Alyssa blows it up, until it pops. Volume Time b) Rocco rides his bike to the park. The graph represents the distance he travels. Time c) The graph represents the height of a basketball after Hadley dropped it from the top of a ladder. Height Distance Time RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3 --
FUNCTIONS MACC.9.F-IF.A.: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). MACC.9.F-IF.A.: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. RATING LEARNING SCALE use function values to solve more challenging problems that I have never previously attempted TARGET 3 WARM UP Simplify. determine whether a relation is a function find function values determine whether a relation is a function with help find function values with help understand that there are special relations called functions ) 3(h) ) ( ) + 3( 5 5 ) KEY CONCEPTS AND VOCABULARY A is a relationship that pairs each input value with exactly one output value. In a relationship between variables, the variable changes in response to the variable. - is a test to see if the graph represents a function. If a vertical line intersects the graph more than once, it fails the test and is not a function. A properly working vending machine is an example of a function. You put in a code (input B5) and it gives you exactly one item (output Mountain Dew). Equations that are functions can be written in a form called. It is used to find the element in the range that will correspond the element in the domain. Equation Function Notation y = x 0 f (x ) = x 0 Read: y equals four x minus 0 Read: f of x equals four x minus 0-3-
EXAMPLES EXAMPLE : IDENTIFYING A FUNCTION Determine whether each relation is a function. a) {(0, ), (, 0), (, ), (3, ), (, )} b) {(, 9), (, 3), (, 0), (, ), (, )} EXAMPLE : USING THE VERTICAL LINE TEST Use the vertical line test. Which graphs represent a function? a) b) c) EXAMPLE 3: EVALUATING FUNCTION VALUES Evaluate each function for the given value. a) f (x) = x + for f(5), f(-3), and [3 f(0)] b) f (x ) = x + 3x for f(), f(-), and [f(0) + f()] EXAMPLE : EVALUATING FUNCTION VALUES FOR REAL WORLD SITUATIONS Write a function rule to model the cost per month of a cell phone data plan. Then evaluate the function for given number of data. Monthly service fee: $.99 Rate per GB of data uses: $5 GB of data used: 3 RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3 --
TARGET INTERPRETING GRAPHS OF FUNCTIONS MACC.9.F-IF.B.: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship MACC.9.F-IF.B.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. RATING 3 LEARNING SCALE interpret intercepts and symmetry of graphs of functions and apply them to real world situations or more challenging problems that I have never previously attempted interpret intercepts and symmetry of graphs of functions identify the domain of a function interpret intercepts and symmetry of graphs of functions with help identify the domain of a function with help understand the definition of an intercept and symmetry WARM UP The cost of one scoop of ice cream is $3.50 and the cost of two scoops of ice cream is $5.75. Write and evaluate an expression to find the cost of 3 one-scoop ice creams and two-scoop ice creams. KEY CONCEPTS AND VOCABULARY EXAMPLES the point in which the graph intersects the x-axis the point in which the graph intersects the y-axis A function whose graph is a straight line is a A function whose graph is not a straight line is a A function has on some vertical line if each half of the graph on either side of the line matches exactly EXAMPLE : DETERMINING THE DOMAIN OF A FUNCTION GIVEN ITS GRAPH Identify the domain of the function. a) b) -5-
EXAMPLE : DETERMINING IF A GRAPH IS LINEAR VS. NON-LINEAR Identify the function as linear or non-linear. Explain. EXAMPLE 3: IDENTIFYING INTERCEPTS AND DETERMINING SYMMETRY Estimate the intercepts and determine if the graph has symmetry. a) b) c) d) RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson) Circle one: 3-6-