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1 Table of Contents Topic 1: Variables and Expressions 1-1: Numerical Expressions (pg 3-5) 1-2: Algebraic Expressions (pg 6-7) 1-3: Writing Algebraic Expressions (pg 7-8) 1-4: Evaluating Algebraic Expressions (pg 8-10) 1-5: Expressions with Exponents (pg 10-11) Topic 9: Rational Numbers 9-1: Rational Numbers and the Number Line (pg 71) 9-2: Comparing Rational Numbers (pg 72) 9-3: Ordering Rational Numbers (pg 73) 9-4: rational #s & the Coordinate Plane (p 74-75) 9-5: Polygons in the Coordinate Plane (pg 75-78) Topic 2: Equivalent Expressions 2-1: The Identity and Zero Properties (pg 12) 2-2: The Commutative Properties (pg 13) 2-3: The Associative Properties (pg 14) Special: Divisibility Rules & Prime Factorization (p 15-6) 2-4: Greatest Common Factor (pg 17-18) 2-5: The Distributive Property (pg 19-20) 2-6: Least Common Multiple (pg 21-22) Topic 10: Ratios 10-1: Ratios (pg 79-80) 10-2 to 10-4: Equivalent Ratios (pg 80-81) 10-5: Ratios as Decimals (pg 82) 10-6: Problem Solving (pg 83) Topic 11: Rates 11-1/11-2: Unit Rates and Unit Price (pg 84-86) 11-3: Constant Speed (pg 87-88) 11-4: Measurements and Ratios (pg 89-90) Topic 3: Equations and Inequalities 3-1: Expressions to Equations (pg 23 ) 3-3: Solving Addition & Subtraction Equations (pg 24 ) 3-4: Solving Multiplication & Division Equations (pg 25) 3-5: Equations to Inequalities (pg 26-27) 3-6: Solving Inequalities (pg 27-28) Topic 12: Ratio Reasoning 12-1: Plotting Ratios and Rates (pg 91) 12-2: Recognizing proportionality (pg 92) 12-3: Introducing Percents (pg 93) Topic 4: Two-Variable Relationships 4-1: 2 Variables to Represent a Relationship (pg 29-30) Special: Graphing Linear Functions (pg 31-32) 4-2: Analyzing Patterns with Tables & Graphs (pg 33-4) 4-3: Relating Tables & Graphs to Equations (pg 35-36) Topic 13: Area 13-1/13-3:Rectangles, Squares, & Parallelograms (p 94) 13-2/13-4: Triangles (pg 95) 13-5: Polygons (pg 96) 13-6: Problem Solving (pg 97) Special: Interior Angles of Triangles & Quads(p 98) Special: Adding and Subtracting Fractions Special: Simplifying and Converting Fractions (pg 37 ) Special: Adding and Subtracting Fractions (pg 38-39) Topic 14: Surface Area and Volume 14-1/14-2: Analyzing 3-D Figures & Nets (pg 99-101) 14-3: Surface Areas of prisms (pg 102-103) 14-4: Surface Areas of Pyramids (pg 104) 14-5: Volumes of Rectangular Prisms (pg 105) Topic 5: Multiplying Fractions 5-1: Multiplying Fractions and Whole Numbers (pg 40 ) 5-2: Multiplying Two Fractions (pg 41-42) 5-3/5-4: Multiplying Mixed Numbers (pg 42-43) Topic 15: Data Displays 15-1: Statistical Questions (pg 106) 15-2: Dot Plots (pg 107-108) 15-3: Histograms (pg 109-110) 15-4: Box Plots (pg 111-112) 15-5: Choosing an Appropriate Display (pg 113-114) Topic 6: Dividing Fractions 6-1: Dividing Fractions and Whole Numbers (pg 44-45) 6-2/6-3: Dividing Fractions (pg 46-47) 6-4: Dividing Mixed Numbers (pg 48 ) Special: Solving Equations Fractions & Mixed #s (p 49) Topic 16: Measures of Center and Variation 16-1/16-2: Median and Mean (pg 115-116) 16-3: Variability (pg 117) 16-4: Interquartile Range (pg 118-119) 16-5: Mean Absolute Deviation (pg 120-121) Topic 7: Fluency with Decimals 7-1: Adding and Subtracting Decimals (pg 50) Special: Place Value and Rounding (pg 51) 7-2: Multiplying Decimals (pg 52-53) 7-3: Dividing Multi-Digit Numbers (pg 54) 7-4: Dividing Decimals (pg 55-56) 7-5: Decimals and Fractions (pg 56-58) 7-6: Compare/Order Decimals & Fractions (p 58-9) Topic 8: Integers 8-1: Integers and the Number Line (pg 60) 8-2: Comparing and Ordering Integers (pg 61) 8-3: Absolute Value (pg 62) 8-4: Integers and the Coordinate Plane (pg 63-64) 8-5: Distance (pg 65-66) 8-6: Problem Solving (pg 67) Special: Adding Integers (pg 68) Special: Subtracting Integers (pg 69) Special: Multiplying and Dividing Integers (pg 70) 2 Topic 1: Variables and Expressions 1-1 Numerical Expressions _____________________________________________________________________________________ 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_____________________________________________________________________________________ 3 Topic 1: Variables and Expressions 1-1 Numerical Expressions _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 5 1-2 Algebraic Expressions Algebraic Expression – an expression with numbers, variables, and operations. Variable – a symbol usually a letter, that takes the place of a number. Term – a number, variable, or the product of a number and one or more variables. Ex – Constant – a term that only has a number part Ex – Coefficient – the number part of a term with a variable *if there is no number in front of the variable, then the coefficient is _____ Ex – Variable Quantity – a quantity that varies or changes. Ex 1 Classify each expression as a numerical expression or an algebraic expression Ex 2 6 1-2 Algebraic Expressions Ex 3 Identify the terms, constants, and coefficients. 1.) 3x + 4 + x 2.) 10 + 4y + 5y + 8 3.) 7k + 9k + 2k + k Terms Constants Coefficients 1-3 Writing Algebraic Expressions Ex 1 Let Statement - Ex 2 A state park has 3 lakes. In the spring, each lake was stocked with the same number of fish. Write an algebraic expression to represent the total number of fish in the lake. Ex. 3 Spring Lake is 3 miles shorter than Grand lake. Write an expression that represents the length of Spring Lake where l is the length of Grand Lake. 7 1-3 Writing Algebraic Expressions Ex 4 Ex 5 A national park charges $25 per bus and $12 per person for an organized tour group. Write an algebraic expression to represent the total cost for a one-day tour with one bus and n people? A national park charges $26 per adult and $16 per child for rafting down one of their two rivers. Write an algebraic expression that can be used to represent the total cost for a adults and c children to raft down the Wild River? 1-4 Evaluating Algebraic Expressions _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 8 1-4 Evaluating Algebraic Expressions _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 9 1-4 Evaluating Algebraic Expressions Ex. 3 A dog walker charges $10 to walk a large dog and $6 to walk a small dog. Write an expression to represent how much the dog walker earns for walking b large dogs and s small dogs. How much will he earn for 8 large dogs and 3 small dogs? 1-5 Expressions with Exponents Power – a way to write repeated multiplication Base – the repeated factor Exponent – the number of times a factor is repeated Special Powers Squared – Cubed – 10 1-5 Expressions with Exponents Ex. 1 Ex. 2 Evaluate the following expressions. a.) 43 + 6 b.) 2x2 – y2 when x = 12 and y = 8 Example 3 11 c.) 52 – 3(4 + 2) Topic 2: Equivalent Expressions 2-1 The Identity and Zero Properties Identity Property of Addition: _____________________________________________________________________________________ _____________________________________________________________________________________ Identity Property of Multiplication: _____________________________________________________________________________________ _____________________________________________________________________________________ Zero Property of Multiplication: _____________________________________________________________________________________ _____________________________________________________________________________________ Ex 1 Tell which property each statement represents. 1×a=a _____________________________ 1 × 99 = 99 _____________________________ 1+0=1 _____________________________ 3×0=0 _____________________________ x+0=x _____________________________ w×0=0 _____________________________ Ex 2 At a carnival, you earn 1 point for each balloon you pop with a dart. You pop b balloons. Write two equivalent algebraic expressions that show how many points you earned. Ex 3 Hockey teams earn 3 points for a win, 0 points for a loss, and 1 point for a tie. Use the expression below to find the total number of points Team A and Team B have earned. Name the properties used to simplify the expression. 12 2-2 The Commutative Property Commutative Property of Addition: _____________________________________________________________________________________ Commutative Property of Multiplication: _____________________________________________________________________________________ Ex 1 Write an equivalent expression to the given expression using the commutative property a.) b.) c.) d.) Ex 2 In the first 10 months of the year, California experienced 555 earthquakes. The remaining two months of the year are predicted to have e earthquakes. Write 2 equivalent expressions to show the total number of earthquakes that California will have in that year. Ex 3 While planning dinners, Sam, Carl, Jack, and Mary wrote their initials beneath the meals that they would make for the family. Use the commutative property of multiplication to write 2 equivalent expressions that show the total number of meals that will be prepared each week. 13 2-3 The Associative Property Associative Property of Addition: _____________________________________________________________________________________ _____________________________________________________________________________________ Associative Property of Multiplication: _____________________________________________________________________________________ _____________________________________________________________________________________ Ex 1 Use the associative property to write an equivalent expression for the given expression. a.) b.) c.) d.) Ex 2 A pilot’s schedule for the month of May is 63 hours in-flight, 75 hours at work but not flying, and 25 hours on-call. Use the associative property of addition to write 2 equivalent expressions to find the total number of hours the pilot worked during the month of May. Ex 3 An airline ships 3,000 lb. of salmon each day. How many pounds of salmon are shipped in w weeks? Use the associative property of multiplication to write 2 equivalent expressions. 14 Special: Divisibility Rules/Prime Factorization Divisibility – the ability to divide a number into another number evenly (NO REMAINDERS!) Divisibility Rules A number is divisible by: 4,212 2 if the number is even 3 if the sum of the digits is divisible by 3 4 if the last two digits of the number are divisible by 4 5 if the number ends in five or zero 6 the number is divisible by 2 and 3 9 if the sum of the digits is divisible by 9 10 if the number ends in zero Ex 1 YES or NO…are these numbers divisible by….. 204 99 6, 060 2: 2: 2: 3: 3: 3: 4: 4: 4: 5: 5: 5: 6: 6: 6: 9: 9: 9: 10: 10: 10: 15 Special: Divisibility Rules/Prime Factorization factor – a number that divides into another number without a remainder prime number: a number larger than 1 that only has one and itself as a factor 2, 3, 5, 7, 11, 13, 17, 19, 23, ….. composite number: a number that has more than one and itself as factors 4, 6, 8, 9, 10, 12, 14 15,…….. Ex 2 Write whether each number below is prime or composite and list all the factors. 17 46 91 51 Factor trees are a way to write the prime factorization of a number. Prime Factorization – showing a number as the product of prime numbers Ex 3 Let’s make some factor trees for the numbers below. 28 36 44 16 100 2-4 Greatest Common Factor Common Factor: a factor that 2 or more numbers share Ex – Greatest Common Factor (GCF): the largest of the common factors There are 2 ways you can find the GCF of a set of numbers: 1.) Listing all the factors of each 2.) Using Prime factorization Ex. 1 Find the GCF of 36 and 54 by listing the factors. Prime Factorization Method 1.) Make a factor tree for each number 2.) Write the prime factorization of each number (do not use exponents) 3.) Circle sets of common factors 4.) Multiply one from each set Ex. 2 30 and 75 Ex. 3 24 and 15 17 Ex. 4 20 and 49 2-4 Greatest Common Factor Ex. 5 60 and 88 Ex. 6 30, 25, and 90 Ex. 7 I am handing out treats to my classes. I have 150 blue jolly ranchers and 200 pink jolly ranchers. Every class will get the same number of each color to split. What is the greatest number of each color that I can give to each class? 18 2-5 The Distributive Property Distributive Property – each term inside a set of parentheses can be multiplied by a factor outside the parentheses to produce an equivalent expression. Ex – Ex 1 Which rocks show the distributive property? Cross out the ones that do not. Ex 2 Use the Distributive property and common factors to write an equivalent expression. a.) 15 + 35 b.) 36 – 27 Ex 3 19 2-5 The Distributive Property Ex 4 Use the distributive property to write an equivalent expression. a.) 5(3x + 10) b.) 7(4y – 5) Ex 5 Use the distributive property and common factors to write an equivalent expression. a.) 20y + 15 b.) 12x – 16y c.) 17x + 12x Ex. 6 A family makes a budget for a camping trip to Spring Lake. Write two equivalent expressions that shows the cost for camping and groceries for d days. Ex. 7 A camper canoes for 2 hours and hikes for 3 hours each day. The number of hours she hikes and canoes in d days is represented by the expression d(2 + 3). Use the distributive property to write an equivalent expression that also shows the number of hours the camper canoes and hikes in d days. 20 2-6 Least Common Multiple multiple: the number multiplied by any other number other than zero Ex - LCM Least Common Multiple: the least multiple shared by the numbers There are 2 ways to find LCM: 1.) Listing the Multiples 2.) Using Prime Factorization Ex. 1 Find the LCM of 80 and 30 by listing the multiples. Prime Factorization Method 1. Make factor trees for each number. 2. List the prime factorizations in exponent form. 3. Circle the largest power of a # and cross out any others with the same base 4. Multiply the circled numbers. Ex. 2 Ex. 3 40 and 60 42 and 72 Ex. 4 49 and 20 21 2-6 Least Common Multiple Ex. 5 22 Topic 3: Equations and Inequalities 3-1 Expressions to Equations Equation – a mathematical sentence that has an equal sign comparing two expressions. Ex.1 Tell which are expressions and which are equations. Expressions Equations True Equation – has equivalent expressions on each side of the equal sign. False Equation – has expressions that are not equivalent on each side of the equal sign. Open Sentence – an equation that includes one or more variables. Solution – a value of a variable that makes the equation true. Ex. 2 Which equations have a solution of 12? a.) 17 – m = 5 b.) 3x = 24 c.) 2g – 4 = 20 23 3-3 Solving Addition and Subtraction Equations Inverse Operations – Operations that undo each other. Ex Solving 1-step Addition/Subtraction Equations Addition Subtraction Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex 7 How many comic books do you need to add to the 17 comic books you already own to have a total of 30 comic books? Write an equation to model the situation along with a let statement identifying your variable. Ex 8 You have a deck of trading cards. You gave 19 cards to a friend and have 5 left for yourself. How many cards were in the original deck? Write an equation to model the situation along with a let statement identifying your variable. 24 3-4 Solving Multiplication and Division Equations Steps for Solving Multiplication Equations Division Equations Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex. 7 An octopus’s suction cups are evenly divided among its 8 arms. There are 240 suction cups in all. Write and solve a multiplication equation with a let statement to find the number of suction cups on each arm. Ex. 8 A band director wants to make 3 groups of 10 students. Write and solve a division equation with a let statement to find the total number of students in the band. 25 3-5 Equations to Inequalities Differences Between Equations and Inequalities Equations – Inequalities – Symbol Meaning Graphing Inequalities Open Points Algebraic Numerical Closed Points Sentence Example Examples When shading your number line: ___________________________________________________________________________________ Ex. 1 Write a let statement and inequality for the situation described. A.) My age is greater than 36. B.) The candy left in the bag is 66 pieces or less. 26 3-5 Equations to Inequalities Ex. 2 Decide whether each situation could be represented by an equation or an inequality. Next, write the equation or inequality. Finally, graph it. A.) John rode his scooter 2 miles. B.) The speed limit is 45 miles per hour. C.) The voting age is 18 and older. D.) To go on the kiddie rides, you must be 36 inches or less E.) The movie ticket costs $6. F.) My mom gave me less than $10 to spend on snacks. 3-6 Solving Inequalities _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 27 3-6 Solving Inequalities _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 28 Topic 4: Two-Variable Relationships 4-1 Using 2 Variables to Represent a Relationship Related Unknown Quantities – Quantities that when a change happens in one, it affects the other. Ex 1 A fruit basket business takes 103 orders in one month. Orders can be placed online or over the phone. Identify the related and unknown quantities. Ex 2 The fruit basket business takes 312 orders in one month. Orders can be shipped overnight or standard delivery. Identify the related and unknown quantities. 29 4-1 Using 2 Variables to Represent a Relationship Ex 3 After-school practice is 90 minutes long. Some of that time is spent doing warm-ups and the rest is spent doing drills. Write an equation to represent the situation. Ex 4 The fruit basket company offers gift wrapping for an additional charge. Some customers choose to have their order gift-wrapped and others do not. One month, 145 people do not want their order giftwrapped. Write an equation to represent the situation. Dependent Variable – changes in response to another variable. Independent Variable – affects change on the dependent variable. 30 Special: Graphing Linear Functions using a table Function – a pairing of each number in one set with a number in a second set. Linear Function – a function whose graph is a straight line. Input – a number in the first set (the independent variable, the x-value) Output – the number in the second set that pairs with exactly one input (the dependent variable, the yvalue). Function Rule – an equation that relates the input to the output (how the independent variable affects the dependent variable, how the x-value affects the y-value). How to graph using an input/output table: 1.) Write the values of x in the first column 2.) Use the function rule and plug in the values of x to find the y (SHOW WORK IN THE CENTER COLUMN) 3.) Write the values found for y in the third column 4.) Write the ordered pairs you want to graph 5.) Graph your ordered pairs (x-tells you how much to move right, y-tells you how much you move up) Make an input-output table using the function rule and the input values x = 0, 4, 6, and 8, then graph the function. Ex. 1 y = x + 2 x y 31 Special: Graphing Linear Functions using a table Ex. 2 y = 8 – x x y Ex. 3 𝒚 = 𝒙 𝟐 x Ex. 4 x y 𝒚= 𝟏 𝒙+ 𝟐 𝟑 y 32 4-2 Analyzing Patterns Using Tables and Graphs Ex. 1 Ex 2 33 4-2 Analyzing Patterns Using Tables and Graphs Ex. 3 Ex 4 Acres # of Trees 0 0 5 3100 10 6200 15 20 25 Ex 5 Year Height (ft.) 0 0 1 12 2 24 3 4 5 34 4-3 Relating Tables and Graphs to Equations Ex. 1 Ex. 2 *look for a pattern! (It should be what you do to x to get to a value for y). Ex. 2 Use the graph to complete the table, then write an equation that represents a relationship between x and y. 35 4-3 Relating Tables and Graphs to Equations Ex. 3 Write an equation that represents the relationship between x and y. Use the graph to help you come to your conclusion. x 1 2 3 5 9 y 10 9 8 6 2 Ex. 3 Complete the table and then write an equation that shows the relationship between the number of games bowled and the total cost. (It costs $3 to bowl a game and $5 to rent shoes). Ex. 4 Write an equivalent equation to the given equation. Original: x + y = 19 Equivalent Equations: 1.) Can be found by subtracting x from both sides. 2.)Can be found by subtracting y from both sides. 3.) Can be found by using the commutative property. 36 Special: Simplifying Fractions and Converting Mixed Numbers and Improper Fractions Simplifying Fractions – finding a common factor of the numerator and denominator and dividing it out until they have a denominator of 1 (If you are stuck, refer back to page 13 for divisibility rules). Ex. 1 Ex. 2 Ex. 3 Converting Mixed Numbers to Improper Fractions 1.) Multiply the denominator by the whole number 2.) Add the numerator to the resulting product 3.) Place that over the original denominator Ex. 4 Ex. 5 Ex. 6 Converting Improper Fractions to Mixed Numbers 1.) Divide the numerator by the denominator until you get a remainder 2.) The quotient becomes the whole number 3.) The remainder becomes the new numerator 4.) The denominator stays the same Ex. 7 Ex. 8 37 Ex. 9 Special: Adding and Subtracting Fractions Adding/Subtracting Fractions 1.) Rename the fractions so they have the same denominators if necessary 2.) Add/Subtract the fractions 3.) Add/Subtract the whole numbers 4.) Simplify Ex. 1 Ex. 2 Ex. 3 Ex. 4 Ex. 5 Ex. 6 Ex. 7 Ex. 8 38 Special: Adding and Subtracting Fractions Steps for Subtracting Fractions if borrowing is necessary 1.) Rewrite fractions to have common denominators 2.) Borrow (look at common denominator and add that to the numerator of the top fraction) 3.) Subtract 4.) Simplify Ex. 9 Ex. 10 39 Ex. 11 Topic 5: Multiplying Fractions 5-1 Multiplying Fractions with Whole Numbers Ex 1 Ex 2 Multiplying Fractions and Whole Numbers 1.) 2.) 3.) 4.) Change the Whole number to a fraction (place it over one) Multiply the numerators Multiply the denominators Simplify (as a fraction in simplest form or as a mixed number) Ex 3 Ex 4 40 5-2 Multiplying Fractions 2 Fractions How to show a model of multiplying a fraction by a fraction: 1.) Draw a rectangle 2.) Draw the number of rows as the number in the denominator of the first fraction (shade the fraction that is shown) 3.) Draw the number of columns as the number in the denominator of the second fraction (shade the fraction that is shown) 4.) Count how many total boxes within the rectangle that is the denominator of the product 5.) Count the number of shaded boxes that overlap that is the numerator of the product 6.) Simplify Ex 1 Multiplying Fractions 1.) Simplify numbers in the numerators with numbers in the denominators (if possible… this will make your multiplication easier) 2.) Multiply the numerators 3.) Multiply the denominators 4.) Make sure your answer is in simplest form Ex 2 Ex 3 41 Ex 4 5-2 Multiplying Fractions 2 Fractions Ex 5 5-3/5-4 Multiplying Mixed Numbers Ex. 1 Ex. 2 1 1 Use the Area Model to model 2 3 × 3 4 42 5-3/5-4 Multiplying Mixed Numbers Multiplying Fractions (the final List) 1.) Change any mixed number to an improper fraction. 2.) Simplify early if possible 3.) Multiply Numerators 4.) Multiply Denominators 5.) Make sure your answer is in simplest form (change any improper fractions back to a mixed number) Ex 3 Ex. 4 Ex. 5 Ex. 6 1 3 An Ipod has 7 2 gigabytes of memory available. If you use 5 of that memory for music, how many gigabytes are used for music? Ex. 7 An art club has made a mural for the wall of their school. Using the diagram of the mural, what is its area? Ex. 8 The table below shows the trails in Diablo State Park in Walnut Creek, CA. A jogger has decided to run all 1 4 trails 3 2 times. How many miles did the jogger run? 43 6-1 Dividing Fractions and Whole Numbers Ex 1 Ex 2 44 6-1 Dividing Fractions and Whole Numbers Reciprocals – If two numbers, when multiplied together, produce a product of 1. Ex 3 Ex 4 Ex 5 Dividing Fractions, Whole Numbers, and Mixed Numbers 1.) 2.) 3.) 4.) 5.) 6.) Change any mixed number or whole number to an improper fraction Change the division sign to a multiplication sign and the SECOND fraction to its reciprocal Simplify if possible Multiply the numerators Multiply the denominators Make Sure your answer is in simplest form Ex 6 Ex 7 45 6-2/6-3 Dividing Fractions Ex 1 Ex 2 46 6-2/6-3 Dividing Fractions Ex 3 Part 2 Dividing Fractions, Whole Numbers, and Mixed Numbers 1.) 2.) 3.) 4.) 5.) 6.) Change any mixed number or whole number to an improper fraction Change the division sign to a multiplication sign and the SECOND fraction to its reciprocal Simplify if possible Multiply the numerators Multiply the denominators Make Sure your answer is in simplest form Ex 4 Ex 5 Ex 6 47 6-4 Dividing Mixed Numbers Dividing Fractions, Whole Numbers, and Mixed Numbers 1.) 2.) 3.) 4.) 5.) 6.) Ex 1 Change any mixed number or whole number to an improper fraction Change the division sign to a multiplication sign and the SECOND fraction to its reciprocal Simplify if possible Multiply the numerators Multiply the denominators Make Sure your answer is in simplest form Ex 2 Ex 4 Ex 5 48 Ex 3 Special: Solving Equations with Fractions and Mixed Numbers _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 49 Topic 7: Fluency with Decimals 7-1 Adding and Subtracting Decimals Steps for Adding and Subtracting Decimals 1.) 2.) 3.) 4.) Line up the decimal points Add zero place holders when necessary Add/Subtract as normal Place decimal point in the answer directly in line with the decimal point in the problem Ex. 1 Ex. 2 Ex 3 Ex 4 Ex 5 50 Special: Place Value and Rounding Ex. 1 State the place value of the given digit. a.) b.) c.) Rounding Rules: 1.) Underline the place value you want to round to 2.) Look at the digit behind it: 4 or below leave it alone 5 or above add one 3.) Drop all digits after the underlined digit Ex. 2 Round to the indicated place value a.) b.) d.) e.) c.) 51 7-2 Multiplying Decimals Steps for Multiplying Decimals 1.) 2.) 3.) 4.) Place the number with the most nonzero digits on top DO NOT line up the decimal points Multiply as normal Count the number of decimal points in the problem and place the same amount in the answer If you do not have enough digits in your answer for the decimal places, place additional zeros IN FRONT of the digits you got from multiplying Ex 1 Ex 2 Ex 3 Ex 4 0.7 × 0.08 4.29(5.03) 4.8 × 3.235 0.066 × 0.05 Ex 5 Find the area of the mini-tv shown below. 52 7-2 Multiplying Decimals Ex 6 * Rounding to the nearest cent means rounding to the nearest hundredth (because you can’t have less than a whole penny) Ex 7 53 7-3 Dividing Multi-Digit Numbers 1.) 2.) 3.) 4.) Divide Multiply Subtract Bring down Remember IOU (Inner divided by the outer gives U (you) the answer) Ex. 1 Ex. 2 Ex. 3 Ex. 5 54 Ex. 4 7-4 Dividing Decimals Remember your IOU! -Be sure to place your decimal point in your answer directly in line with the one in the dividend (inner number). Ex 1 Ex 2 Steps for Dividing a Decimal by a Decimal 1.) Move the decimal point in the divisor (second number) so it becomes a whole number 2.) Move the decimal point in the dividend (first number) the same number of times 3.) Divide as normal (IOU) adding zeros to bring down when necessary until you get a remainder of 0, the digits begin to repeat, or the problem asks you to round to a certain place value. 4.) Place decimal point in answer directly in line with the decimal point in the dividend (inner number) Ex 3 Ex 4 55 Ex 5 7-4 Dividing Decimals Ex 6 Last week, a 6th grade class received $10.32 for aluminum cans they recycled. The scrap yard paid them $0.48 for each pound. How many pounds of cans did the class recycle? Ex 7 A shelf used to store DVDs is 60.96 cm long. If each DVD is 1.5 cm wide, what is the maximum number of DVDs that can be stored on the shelf? 7-5 Decimals and Fractions Common Fractions and Decimal Equivalence Fraction Decimal Fraction 𝟐 𝟓 𝟑 𝟓 𝟒 𝟓 𝟏 𝟑 𝟐 𝟑 𝟏 𝟒 𝟏 𝟐 𝟑 𝟒 𝟏 𝟐 𝟑 , , , 𝒆𝒕𝒄. 𝟏𝟎 𝟏𝟎 𝟏𝟎 𝟏 𝟓 56 Decimal 7-5 Decimals and Fractions Decimals Fractions 1.) The denominator will be the same as the place value of the last digit of the decimal (ex if the last digit is in the tenths place the denominator is a 10, hundredths place the denominator is 100). 2.) Place anything left of the decimal point as a whole number and anything to the right of the decimal in the numerator. 3.) Simplify Ex. 1 Ex. 2 Fractions Decimal 𝐼 1.) Divide the numerator by the denominator (your IOU looks like this 𝑂 = 𝑈) If you see a number begin to repeat, round to the indicated place value (this means divide to one place PAST the indicated place value, and then round.) Ex. 3 Ex. 4 Ex. 5 Round to the nearest hundredth Ex. 6 Round to the nearest tenth 57 7-5 Decimals and Fractions Changing a fraction to a decimal without dividing Factors of 100 – 1, 2, 4, 5, 10, 20, 25, 50, 100 - If you have these numbers as a denominator, you can find the decimal equivalence by using equivalent fractions. Ex. 7 a.) b.) 7-6 Comparing and Ordering Decimals and Fractions -When comparing decimals you must compare the digits from left to right (once the digits stop being the same compare to see which is greater in that place value) Ex 1 * BE CAREFUL TO SEE IF YOU ARE ORDERING FROM LEAST TO GREATEST OR GREATEST TO LEAST! Ex 2 Compare 58 7-6 Comparing and Ordering Decimals and Fractions In order to compare fractions and decimals, you need to change the numbers so that they are all fractions or all decimals (I don’t care which you choose) For Examples 3 – 5 Compare the numbers Ex 3 Ex 4 Ex 6 59 Ex 5 Topic 8: Integers 8-1 Integers and the Number Line Integers: _____________________________________________________________________________________ _____________________________________________________________________________________ Positive Numbers: _____________________________________________________________________________________ Negative Numbers: _____________________________________________________________________________________ Opposites: _____________________________________________________________________________________ Draw a number line from -10 to 10. Label the area where there are positive numbers, negative numbers, and show one set of opposite numbers. Ex. 1 What is the opposite of 9? Ex. 2 What is the opposite of -3? Ex. 3 Ex. 4 The highest point as an integer: ___________ The lowest point as an integer: ___________ 60 8-2 Comparing and Ordering Integers On a number line, the more right a number is when you graph it, the larger the number is. Ex Ex Ex 1 Ex 2 61 8-3 Absolute Value Absolute Value – The distance a number is from zero on a number line. (Always positive) Symbol Ex 1 Find the absolute value. a.) b.) c.) Ex 2 Ex 3 Depth will always be measured as an absolute value. The larger the absolute value the deeper a diver is. 62 8-4 The Coordinate Plane Graphing Ordered Pairs (points on a coordinate plane) Ordered Pair (x, y) (1st # right (+) or left (-), 2nd # up (+) or down (-)) Ex 1 Graph the following ordered pairs and give its location A (2, -3) _____________________ B (0, -2) _____________________ C (-1, 4) _____________________ D (3, 0) _____________________ 63 8-4 The Coordinate Plane Ex 2 State the ordered pairs for the locations on the graph. Hospital: ________________ School: ________________ Grocery Store: ________________ Library: ________________ Transformation – when a figure changes size, position, or shape on a coordinate plane. Reflection – a transformation that “flips” over a line called the line of reflection to produce a mirror image. Reflection on the x-axis (x, y) (x, -y) Reflection on the y-axis (x, y) (-x, y) Ex 3 a.) b.) c.) 64 8-5 Distance To find distance between 2 numbers on a number line: 1.) Same Signs (both positive or both negative) – Subtract the smaller absolute value from the larger absolute value. 2.) Different Signs ( one positive and one negative) – Add the absolute values Ex 1 Ex 2 Distance on a coordinate plane Vertical Distance – add/sub the absolute values of the y-coordinate depending on the sign. Ex 3 Ex 4 65 8-5 Distance Horizontal Distance – add/subtract the absolute values of the x-coordinate depending on the sign. Ex 5 *To double check you can always count the spaces between the points on a number line. Ex6 Ex 7 66 8-6 Problem Solving Ex. 1 Ex 2 Ex 3 Ex.4 67 Special: Adding Integers _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 68 Special: Subtracting Integers _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 69 Special: Multiplying and Dividing Integers _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 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_____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ _____________________________________________________________________________________ 70 Topic 9: Rational Numbers 9-1 Rational Numbers and the Number Line Rational Numbers – Numbers that can be written in the form 𝑎 𝑏 𝑎 𝑏 𝑜𝑟 − where a is a whole number and b is a positive whole number. Ex 1 Prove that the number is a rational number. a.) b.) c.) Ex 2 Ex 3 Explain what the number 0 represents in each situation. Then write a rational number to represent the situation. 2 The pant leg shrunk 3 inch after washing. _______________________________________ Dad gained 1 pounds during vacation. 1 4 _______________________________________ It is 10 seconds to blast off. _______________________________________ 71 9-2 Comparing Rational Numbers *In order to compare rational numbers, they must be the same type of numbers – all fractions or all decimals. Ex 1 <, >, or = Ex 2 <, >, or = Ex 3 72 9-3 Ordering Rational Numbers Ex 1: Order the numbers from least to greatest Ex 2: Order the numbers from greatest to least Ex 3: Order the animals depths from the least depth to the greatest depth Ex 4: Accountants use parentheses as negative signs. (Example ($209) means -209 dollars). Whenever you see an accounting number in parentheses, you can write is as a negative number, showing a ______________ or __________________ balance. **** Outside of accounting, however, you should NOT interpret a number in parentheses to be a negative number**** Use rational numbers to write an expression that compares an income of ($41.16) to an income of ($14.61), interpreted as size of loss. 73 9-4 Rational Numbers and the Coordinate Plane Ex 1 Give the ordered pair of each point on the coordinate plane then tell which quadrant the point is located in. Ex 2 74 9-4 Rational Numbers and the Coordinate Plane Ex 3 *Remember: Reflection on the x-axis (x, y) (x, -y) Reflection on the y-axis (x, y) (-x, y) 9-5 Polygons and the Coordinate Plane Polygon – a closed figure formed by 3 or more line segments that do not cross. Regular Polygon – a polygon whose sides and angles are all equal in measure. Examples Non-examples Vertex – a point where any 2 sides of a polygon meet. Ex – 75 9-5 Polygons and the Coordinate Plane Types of Polygons Name Number of Sides What it looks like 76 9-5 Polygons and the Coordinate Plane Ex 1 Ex 2 Find the length of segments AC and BC. AC = BC = 77 9-5 Polygons and the Coordinate Plane Ex 3 Plot the following points and connect them to complete a polygon. (-4, 5) (-6,2) (-2,2) Now, reflect the original polygon over the y-axis. The new coordinates are: Finally, reflect the original polygon over the x-axis. The new coordinates are: Ex 4 78 Topic 10: Ratios 10-1 Ratios Ratio: _____________________________________________________________________________________ Terms of a Ratio: _____________________________________________________________________________________ Ex 1 Ex 2 Write 2 different ratios to compare the number of headphones to the number of MP3 players. Ex 3 You toss a coin 12 times and get heads 7 times. Write the ratio of the number of heads to the number of tails. 79 10-1 Ratios Ex 3 Use the table below to state whether the ratios are true or false. If the statement is false, correct the statement. a. The ratio of votes for Jordyn to total votes is 10:35 b. The ratio of votes for Devin to votes for Jordyn is 10 to 20. c. The ratio of votes for Micah to total votes is 5:30. d. The ratio of votes for Devin to votes for Micah is 20:5. e. The ratio of total votes to votes for Devin is 35 to 20. 10-2 to 10-4 Equivalent Ratios Equivalent Ratios – ratios that express the same relationship Lesser Terms – terms that are smaller than the original terms Greater Terms – terms that are larger than the original terms Ex 1 80 10-2 to 10-4 Equivalent Ratios Ex 2 Write one ratio using lesser terms and one ratio using greater terms for the given ratio. Ex 3 In one class, 3 out of 8 students have braces. There are 32 students in the class. How many students have braces? Ex 4 A volunteer at an animal shelter recorded the ratio of the number of dogs adopted to the number of cats adopted each month. She finds that all ratios are equivalent. Complete the table. Ex 5 a.) the number of students who play the saxophone to the number of students who play the drums. b.)The number of students who play the flute to the total number of students Ex 6 Write 2 equivalent ratios to 10 . 25 Ex 7 Simplify the ratio 12 to 40 as a fraction. 81 10-5 Ratios as Decimals Steps for writing the ratio as a decimal 1.) Write the ratio as a fraction 2.) Divide the numerator by the denominator to change that fraction to a decimal Ex 1 Ratio Fraction Decimal 3:8 Ex 2 Write the ratio as a fraction in simplest form. 0.45 Changing a Decimal Ratio to a fraction 1.) Count the number of decimal places in the number 2.) Place a 1 plus 0s for the number of decimal places in the problem in the denominator (2 decimal places 100, 3 decimal places 1000, etc.) 3.) Place everything else in the numerator Ex 3 A batter’s average is 0.350 in 60 at bats. How many hits did the batter have? 1.) Change the decimal to a fraction 2.) Simplify 3.) Set up 2 equivalent ratios to find the missing piece. 82 10-6 Problem Solving Ex. 1 Fill in the table using equivalent ratios. Ex 2 *** Make a table to help you solve this. 83 Topic 11: Rates 11-1/11-2 Unit Rate and Unit Price Ex. 3 1.) Set up the ratios for each class (or item you are looking for in other types of problems) 2.) Write equivalent ratios to find the different combinations of fish that add up to the total number of fish. Ex. 4 A gym teacher has 15 soccer balls and 16 footballs. The teacher wants to put twice as many footballs as soccer balls in one bin. The teacher wants to put 5 soccer balls for every 3 footballs in another bin. How many soccer balls and footballs should the teacher put in each bin? 84 11-1/11-2 Unit Rate and Unit Price Rate – A ratio that compares 2 quantities that are measured in different units. Ex – Unit Rate – A rate for one unit of a given quantity. When a unit rate is written as a fraction the denominator is 1. Unit Price – The price of an item per unit. Ex – Unit Rate Unit Price Ex. 1 A box of crackers contains 84 crackers and has a total of 7 servings. How many crackers are there per serving? Write the rate Divide both the numerator and the denominator by the denominator to find the unit rate. Ex. 2 In 16 years the trunk of a tree grew approximately 4 inches. How much did the tree trunk grow per year? Ex. 3 Your dishwasher uses 11 gallons of water to wash 2 loads of dishes. How many gallons of water will your dishwasher use to wash 7 loads of dishes? 85 11-1/11-2 Unit Rate and Unit Price Ex. 4 A 5-minute shower uses approximately 12 gallons of water. Each minute the shower is running, the rate of water used is the same. Use this rate to complete the table. *Make a unit rate first! Ex. 5 86 11-3 Constant Speed Constant Speed – Comparing something’s distance traveled to time traveled. Ex – Equation d = rt where d = distance, r = rate of travel (constant speed), and t = time Steps: 1.) Write the formula 2.) Substitute in for what you know 3.) Solve the resulting equation (show balance if necessary) When finding distance this means – multiplying the rate and time When finding rate this means – dividing both sides of the equation by the time When finding time this means – multiply both sides of the equation by the reciprocal of the rate Ex 1 Your aunt drives at a constant speed of 45 miles per hour. How far will your aunt travel in 20 minutes? *watch for time not being in the same unit! Ex 2 Lisa bikes at a constant speed of 8 miles per hour. If she bikes for 30 minutes, how far does she travel? 87 11-3 Constant Speed Ex 3 Ex. 4 A bus travels 70 miles in 2 hours. What is the speed of the bus in miles per hour? Ex. 5 Marie and her brother are driving from city A to city B. The two cities are 318 miles apart. Marie drives 53 miles per hour. How long does it take them to make the trip? Ex. 6 On a busy road, cars can travel only 5 miles in 10 minutes. a.) At this speed, how far will a car travel in 15 minutes? 88 *Find the constant speed first! b.) At this speed, how long will it take the car to travel 20 miles? 11-4 Measurements and Ratios Conversion Factor – A relationship between 2 equivalent measurements of different units. It will always equal 1. Ex – 1 pound = 16 ounces 1 ton = 2000 pounds 1 foot = 12 inches 1 yard = 3 feet = 36 inches 1 mile = 1760 yards = 5280 feet 1 cup = 8 fluid ounces 1 pint = 2 cups 1 quart = 2 pints 4 quarts = 1 gallon Steps for converting units 1.) Write the unit you have as a fraction 2.) Multiply that by a conversion factor from the chart (a second fraction that is 𝑢𝑛𝑖𝑡 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡 𝑡𝑜 𝑐ℎ𝑎𝑛𝑔𝑒 𝑡𝑜 ) 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑢𝑛𝑖𝑡 3.) Simplify your answer making sure you label the new unit Ex 1 Ex 2 Ex 3 4 pints to cups 35 oz to pounds Ex 4 89 3 5 2 𝑇 𝑡𝑜 𝑝𝑜𝑢𝑛𝑑𝑠 11-4 Measurements and Ratios Is a rider 3 feet 10 inches tall allowed to ride this roller coaster? Use 1 in. = 2.54 cm. Ex. 5 A bread recipe calls for 500 grams of flour. About how many pounds of flour do you need? Use 1 oz = 28.4 g Ex. 6 You have 2 gallons of bubble solution. About how many liters of bubble solution do you have? Use 1 qt. = 0.95 L 90 Topic 12: Ratio Reasoning 12-1 Plotting Ratios and Rates Ex 1 𝑦 Ratios are always plotted as 𝑥 where y is the y-coordinate and x is the x-coordinate. This is always 2 constant. You will use this fact to write equivalent ratios (Ex 3 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠 Ex 2 Ex 3 91 4 6 ) 12-2 Understanding Proportionality Equivalent ratios form a proportional relationship. Ex. 1 Tell whether the ratios form a proportional relationship. a.) b.) c.) d.) Proportional Relationships can be shown with a table, graph, or equation. Table Graph Ex. 2 Does each table show a proportional relationship? Equation Ex. 3 Do the equations show proportional relationship? 4 a.) 𝑦 = 3 𝑥 4 b.) 𝑦 = 3 𝑥 + 3 92 12-3 Introducing Percent Percent – a ratio that compares a number to 100. Ex. 1 Ex. 2 Complete the table Ex. 3 2 out of 5 students have braces. What percentage of students have braces? 93 Topic 13: Area 13-1 and 13-3 Rectangles, Squares, and Parallelograms Area – the number of square units a figure encloses Area Formula Rectangle A=lw Area = length × width Square A=s2 Area = (side length)2 Parallelogram A=bh Area = base × height *When labeling area you use units2 (ft2, mm2, in2, ect). Steps for solving for all 2D and 3D figures 1.) Write the appropriate formula 2.) Substitute in the values you know 3.) Solve the resulting equation (showing balance if necessary) Find the area. Ex. 1 Ex. 2 Ex. 3 Find the missing dimension. Ex 4 A rectangle with an area of 72 mm2 and a length of 9 mm. Find the width. Decomposing – Breaking up a shape into other shapes. 94 Ex 5 A parallelogram with an area of 15 in2 1 and a base of 2 in. Find the height. 2 13-2 and 13-4 Area of Triangles Area of a Triangle Formula 𝟏 𝑨 = 𝟐 𝒃𝒉 𝒐𝒓 𝑨 = 𝟎. 𝟓𝒃𝒉 Formula comes from Composed Rectangle and Parallelogram Find the area. Ex. 1 Ex. 2 Ex 4. Ex. 3 Ex. 5 You have a triangle with an area of 34 cm2 and a base of 4 cm. What is the height of the triangle? 95 13-5 Polygons Formulas Trapezoid A = ½h(b1 + b2) Area = ½height(base1 + base2) Hexagon A= 6(½bh) A= 6(area of 1 triangle) Octagon A=8(½bh) A= 8(area of 1 triangle) Ex 1 Ex 2 Ex 3 Ex 4 96 13-6 Problem Solving Find the area of the figure. Ex. 1 Ex. 2 Find the area of the shaded region. Ex. 3 Ex. 4 The area of the rectangle is 48 m2. Find the value of x then the area of the trapezoid. 97 Special: Interior Angles of Triangles and Quadrilaterals Classifying Triangles by Sides Equilateral Isosceles Scalene Obtuse Right Acute Angle – Obtuse Angle – Right Angle – Classifying Triangles by Angles Acute Find the value of x. Ex. 1 Ex. 2 Are the angle measures that of a triangle. Justify your answer. If yes, classify the triangle by angles. Ex. 3 Ex. 4 98 Topic 14: Surface Area and Volume 14-1/14-2 Analyzing 3D Figures and Nets Face – a flat surface of a 3D figure shaped like a polygon. Edge – a segment formed by the intersection of 2 faces. Vertex – a point where 3 or more edges intersect on a 3D figure. Ex. 1 Prism – a 3D figure with 2 parallel polygonal faces that are the same size and shape. 99 14-1/14-2 Analyzing 3D Figures and Nets Ex 2 Pyramid – a 3D figure with a base that is a polygon and triangular faces that meet at a vertex. Ex 3 100 14-1/14-2 Analyzing 3D Figures and Nets Net – a 2D pattern that you can fold to form a 3D figure. Ex. 4 Match the figure with its net. Ex. 5 Which net is the net of a cube? Ex. 6 Ex. 7 Draw the net for the given triangular prism. 101 14-3 Surface Area of Right Prisms Surface Area – the sum of the areas of a 3D figures faces. To find surface area: 1.) Make a net of the solid 2.) Find the area of each polygon 3.) Add all the areas together Rectangular Prism Formula Cube Formula Ex. 1 102 14-3 Surface Area of Right Prisms Ex. 2 Ex. 3 Ex. 4 103 14-4 Surface Area of Right Pyramids Ex. 1 Ex. 2 Ex. 3 104 14-5 Volume of Rectangular Prisms Rectangular Prism Formula Cube Formula Labeled as __________ Find the volume of the prism or cube. Ex. 1 Ex. 2 Ex. 3 105 Topic 15: Data Displays 15-1 Statistical Questions Statistical Question – a question that investigates an aspect of the world and can have more than one possible response. Ex. 1 Ex. 2 Which questions are statistical questions and which questions are not statistical questions? Data – are numbers or other pieces of information collected by asking questions, measuring, or making observations about the real world. Ex. 3 106 15-2 Dot Plots Dot Plots show the shape of a corresponding value on a number line. by representing each point as a dot over its Frequency describes how often a _________________ occurs. Ex 1 Your friends hold a basket-ball shooting contest. The person who makes the most baskets in one minute wins. Use the dot plot to answer the questions. a. How many people made 8 baskets? b. How many baskets did the most people make? c. What is the least number of baskets that a person made? Ex 2 The data show the age of the dancers on a dance team. Make a dot plot of the data to find out which age is most common on the team. 107 15-2 Dot Plots Distribution: The way data is over all values. Cluster: Area where dots are stacked Gaps: Area where there are significantly number of dots. Values that stray: Dots that are located from the main set of data. Ex 3 The dot plot shows the heights of plants in a research laboratory. Identify the clusters, the gaps, and any data values that stray. What do they tell you about the heights of the plants? 108 15-3 Histograms Histograms show the shape of a values on a number line. with above intervals of Ex 1 You are helping a new social networking website company analyze data. Use a histogram to answer the questions. How many users were surveyed? of the users surveyed have 150 friends or more How many of the users surveyed have 0 and 49 friends ____ 50 is in interval __________ 109 15-3 Histograms Ex 2 The participants in a ski race are divided into four groups of six skiers each. The table shows the results of the races. Make a histogram to show how the race times are distributed among all the participants. Use the intervals 86 – 87.9, 88 – 89.9, 90 – 91.9, 92 – 93.9, 96 – 97.9, 98 – 99.9, 100 – 101.9 Ex 3 Three wind turbines were constructed in your town. Each is expected to generate 3,000 kilowatts (kW) per day. During the testing phase, engineers recorded the daily amount of energy produced by each turbine. What does the histogram show about the test results? 110 15-4 Box Plots A box plot shows five boundary values. To find these values you MUST put the data values in numerical order. Minimum – the least value in a data set Maximum – the greatest value in a data set Middle of the Data (Median) – the middle value in an ordered data set. If there are 2 middle values, you add the numbers and divide by 2. Middle of the Lower Half (Lower Quartile or 1st Quartile) – the middle value of the lower half of the ordered data set. If there are 2 middle values, you add the numbers and divide by 2. Middle of the Upper Half (Upper Quartile or 3rd Quartile) – the middle value of the upper half of the ordered data set. If there are 2 middle values, you add the numbers and divide by 2. Ex. 1 Order the data set. Then identify the 5 boundary values. 30, 16, 68, 35, 57, 5, 27, 76, 21, 91, 44 Box Plot 111 15-4 Box Plots Ex 2 Ex 3 112 15-5 Choosing Appropriate Data Displays Dot Plots are helpful to see: A A A A Histograms are helpful to see: A A A Box Plots are helpful to see: A A Ex. 1 Ex. 2 113 15-5 Choosing Appropriate Data Displays Ex. 3 Ex. 4 You need to be in the top 25% to move on to the second round in your town’s golf tournament. Which data display would you use if you wanted to see what the lowest possible score is so that you can move on? Explain. 114 Topic 16: Measures of Center and Variation 16-1/16-2 Median and Mean Measures of Center- A measure that is a value that represents the middle of a data set. There may be more than one measure of center for a data set (The ______________________ and the ____________________ are both measures of center). To find the median you must FIRST __________________________________________________________ Odd number of data values Even number of data values Mean – the sum of all the data values divided by the total number of data values in the data set. Also can be called the “average.” Ex 1 Find the median of each data set a.) 7, 8, 9, 12, 14, 16, 17, 19, 20, 50 b.) -132, -105, -19, 16, 17, 17, 22, 25 Ex. 2 115 16-1/16-2 Median and Mean Ex. 3 Find the mean. 24, 27, 30, 31, 33, 35 Ex. 4 Find the value of x. 16.5, 17.5, x 1.) Set up the problem like in example 3 2.) Add the known data values 3.) Multiply each side by the denominator 4.) Subtract that sum from both sides to Find the value of x. 116 16-3 Measure of Variability Variability – how spread out or not spread out data is 3 Types of Variability No Variability High Variability Low Variability Measures of Variability – a value that describes the amount of variability in a data set. Range = Ex. 1 Find the range and identify any stray value(s). 33, 10, 34, 33, 35 Ex. 2 For a project, the teacher asked a group of seven students to each count the number of pedestrians that cross the street where they live on a certain day. The data collected has a range of 41. Six of the seven data values are shown below. Find the missing data value. 29, 20, 25, 11, 34, 20 117 16-4 Interquartile Range Minimum: the LEAST value Range: maximum - minimum Maximum: the GREATEST value First Quartile (lower quartile): the LOWER MIDDLE value Median: the MIDDLE value of the data Interquartile Range (IQR): the middle 50% Third Quartile (upper quartile): the UPPER MIDDLE value Ex 1 118 16-4 Interquartile Range Ex 2 Find the range and the IQR of the data set. 110, 123, 3, 8, 64, 2, 35, 45, 12, 3, 33, 23, 91, 64, 12, 8, 150, 42, 40, 76 Ex 3 119 16-5 Mean Absolute Deviation Deviation of Data Values from the Mean – how far away a number is from the mean. If the number is less than the mean the deviation is negative. If the number is greater than the mean the deviation is positive. Ex. Ex. 1 Absolute Deviation - the distance a data value is away from the mean of the data set (it is the absolute value of the deviation). Mean Absolute Deviation (MAD) – the mean of the absolute deviations. 120 13-5 Polygons To find MAD 1.) Find the mean of the data set (if not given) 2.) Find the Deviation of each of the data points (the mean’s deviation is 0) 3.) Change them to their absolute deviations by finding the absolute value 4.) Add all the absolute deviations and divide by the total number of numbers in the data set. Ex. 2 Ex. 3 Find the MAD of the data set. 1, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10 121