Student Resources. Problem-Solving Handbook

Transcription

1 Student Resources Student Resources Problem-Solving Handbook PS Draw a Diagram PS Make a Model PS3 Guess and Test PS Work Backward PS5 Find a Pattern PS6 Make a Table PS7 Solve a Simpler Problem PS8 Use Logical Reasoning PS9 Use a Venn Diagram PS10 Make an Organized List PS11 Skills Bank SB Place Value SB Rounding SB Factors and Multiples SB3 Divisibility Rules SB3 Prime and Composite Numbers SB Prime Factorization SB Greatest Common Factor (GCF) SB5 Least Common Multiple (LCM) SB5 55 Student Resources

2 Estimate by Rounding SB6 Compatible Numbers SB6 Estimate by Clustering SB7 Overestimates and Underestimates SB7 Multiply and Divide Decimals by Powers of SB8 Order of Operations SB8 Choose Appropriate Units of Measurement SB9 Points, Lines, and Planes SB10 Circle Graphs SB11 Inverse Variation SB1 Eperimental and Theoretical Probability SB13 Independent and Dependent Events SB1 Making Predictions SB15 Samples and Populations SB16 Transformations SB17 Student Resources Worked-Out Solutions WS Selected Answers SA Glossary Inde Table of Measures, Formulas, and Symbols G I1 inside back cover Student Resources 555

3 Draw a Diagram When problems involve objects, distances, or places, drawing a diagram can make the problem clearer. You can draw a diagram to help understand the problem and to solve the problem. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List June is moving her cat, dog, and goldfish to her new apartment. She can only take 1 pet with her on each trip. She cannot leave the cat and the dog or the cat and the goldfish alone together. How can she get all of her pets safely to her new apartment? Understand the Problem The answer will be the description of the trips to her new apartment. At no time can the cat be alone with the dog or the goldfish. Problem Solving Handbook Make a Plan Solve Draw a diagram to represent each trip to and from the apartment. In the beginning, the cat, dog, and goldfish are all at her old apartment. Old Apartment June, Cat, Dog, Fish June, Dog, Fish June, Dog, Fish June, Cat, Fish June, Cat June, Dog June June, Cat New Apartment June, Cat Cat June, Dog, Cat Dog Trip 1: She takes the cat and returns alone. Trip : She takes the dog and returns with the cat. June, Cat, Fish June, Cat June, Fish June June, Dog, Fish Dog, Fish Trip 3: She takes the fish and returns alone. June, Cat June, Cat June, Cat, Dog, Fish Trip : She takes the cat. Look Back Check to make sure that the cat is never alone with either the fish or the dog. PRACTICE 1. There are 8 flags evenly spaced around a circular track. It takes Ling 15 seconds to run from the first flag to the third flag. At this pace, how long will it take Ling to run around the track twice?. A frog is climbing a -foot tree. Every 5 minutes, it climbs up 3 feet, but slips back down 1 foot. How long will it take the frog to climb the tree? PS Problem Solving Handbook

4 Make a Model A problem that involves objects may be solved by making a model out of similar items. Make a model to help you understand the problem and find the solution. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List The volume of a rectangular prism can be found by using the formula V wh, where is the length, w is the width, and h is the height of the prism. Find all possible rectangular prisms with a volume of 16 cubic units and dimensions that are all whole numbers. Understand the Problem Make a Plan You need to find the different possible prisms. The length, width, and height will be whole numbers whose product is 16. You can use unit cubes to make a model of every possible rectangular prism. Work in a systematic way to find all possible answers. Solve Begin with a prism Keeping the height of the prism the same, eplore what happens to the length as you change the width. Then try a height of. Notice that an 8 1 prism is the same as an 8 1 prism turned on its side. Problem Solving Handbook 8 1 Not a rectangular prism 1 The possible dimensions are , 8 1, 1, and. Look Back The product of the length, width, and height must be 16. Look at the prime factorization of the volume: 16. Possible dimensions: 1 1 ( ) ( ) ( ) ( ) 1 ( ) PRACTICE 1. Four unit squares are arranged so that each square shares a side with another square. How many different arrangements are possible?. Four triangles are formed by cutting a rectangle along its diagonals. What possible shapes can be formed by arranging these triangles? Problem Solving Handbook PS3

5 Guess and Test When you think that guessing may help you solve a problem, you can use guess and test. Using clues to make guesses can narrow your choices for the solution. Test whether your guess solves the problem, and continue guessing until you find the solution. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List North Middle School is planning to raise $100 by sponsoring a car wash. They are going to charge $ for each car and $8 for each minivan. How many vehicles would have to be washed to raise $100 if they plan to wash twice as many cars as minivans? Problem Solving Handbook Understand the Problem Make a Plan Solve You must determine the number of cars and the number of minivans that need to be washed to make $100. You know the charge for each vehicle. You can guess and test to find the number of cars and minivans. Guess the number of cars, and then divide it by to find the number of minivans. You can organize your guesses in a table. First guess Second guess Third guess Cars Minivans Money Raised $ ( 00 ) $8 ( 100 ) $ $ ( 100 ) $8 ( 50 ) $ $ ( 150 ) $8 ( 75 ) $100 Too high Too low Look Back They should wash 150 cars and 75 minivans, or 5 vehicles. The total raised is $(150) $8(75) $100, and the number of cars is twice the number of minivans. The answer is reasonable. PRACTICE 1. At a baseball game, adult tickets cost $15 and children s tickets cost $8. Twice as many children attended as adults, and the total ticket sales were $80. How many people attended the game?. Angie is making friendship bracelets and pins. It takes her 6 minutes to make a bracelet and minutes to make a pin. If she wants to make three times as many pins as bracelets, how many pins and bracelets can she make in 3 hours? PS Problem Solving Handbook

6 Work Backward To solve a problem that asks for an initial value that precedes a series of steps, you may want to work backward. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Tyrone has two clocks and a watch. If the power goes off during the day, the following happens: Clock A stops and then continues when the power comes back on. Clock B stops and then resets to 1:00 A.M. when the power comes back on. When Tyrone gets home, his watch reads :7 P.M., clock B reads 5:1 A.M., and clock A reads 3:39 P.M. What time did the power go off, and for how long was it off? Understand the Problem Make a Plan Solve Look Back You need to find the time that the power went off and how long it was off. You know how each clock works. Work backward to the time that the power went off. Subtract from the correct time of :7 P.M., the time on Tyrone s watch. The difference between the correct time and the time on clock A is the length of time the power was off. :7 P.M. 3:39 P.M. 8 min The power was off for 8 min. Clock B reset to 1:00 A.M. when the power went on. Clock B reads 5:1 A.M. The power came on 5 h 1 min ago. Subtract 5 h 1 min from the correct time to find when the power came on. :7 P.M. 5 h 1 min 11:06 A.M. The power came on at 11:06 A.M. Subtract 8 min from 11:06 A.M. to find when the power went off. 11:06 A.M. 8 min 10:18 A.M. The power went off at 10:18 A.M. and was off for 8 minutes. If the power went off at about 10 A.M. for about an hour, it would come on at about 11 A.M., and each clock would run for about 5 1_ hours. PRACTICE 1. Jackie is years younger than Roger. Roger is 1_ years older than Jade. Jade is 1 years old. How old is Jackie?. Becca is directing a play that starts at 8:15 P.M. She wants the cast ready 10 minutes before the play starts. The cast needs 5 minutes to put on make-up, 15 minutes for a director s meeting, and then 35 minutes to get in costume. What time should the cast arrive? Problem Solving Handbook Problem Solving Handbook PS5

7 Find a Pattern If a problem involves numbers, shapes, or even codes, noticing a pattern can often help you solve it. To solve a problem that involves patterns, you need to use small steps that will help you find a pattern. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Gil is trying to decode the following sentence, which may have been encoded using a pattern. What does the coded sentence say? QEB NRFZH YOLTK CLU GRJMP LSBO QEB IXWV ALD. Understand the Problem You need to find whether there was a pattern used to encode the sentence and then etend the pattern to decode the sentence. Problem Solving Handbook Make a Plan Solve Find a pattern. Try to decode one of the words first. Notice that QEB appears twice in the sentence. Gil thinks that QEB is probably the word THE. If QEB stands for THE, a pattern emerges with respect to the letters and their position in the alphabet. Q: 17th letter T: 0th letter 3 letters E : 5th letter H: 8th letter 3 letters B: nd letter E: 5th letter 3 letters Continue the pattern. Although there is no 7th, 8th, or 9th letter of the alphabet, the remaining letters should be obvious (7 1 A, 8 B, and 9 3 C). Look Back QEB NRFZH YOLTK CLU GRJMP LSBO QEB IXWV ALD. THE QUICK BROWN FOX JUMPS OVER THE LAZY DOG. The sentence makes sense, so the pattern fits. PRACTICE Decode each sentence. 1. RFC DGTC ZMVGLE UGXYPBQ HSKN OSGAIJW. ( RFC THE ). U PYLS VUX KOUWE GCABN DCHR TCJJS ZIQFM. ( U A ) PS6 Problem Solving Handbook

8 Make a Table To solve a problem that involves a relationship between two sets of numbers, you can make a table. A table can be used to organize data so that you can look at relationships and find the solution. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Jill has 1 pieces of ft long decorative edging. She wants to use the edging to enclose a garden with the greatest possible area against the back of her house. What is the largest garden she can make? Understand the Problem You must determine the length and width of the edging. Make a Plan Solve Make a table of the possible widths and lengths. Begin with the least possible width and increase by multiples of ft. Remember that the width is the same on two sides. Use the table to solve. Width (ft) Length (ft) Garden Area ( ft ) w Problem Solving Handbook 10 0 Look Back The maimum area that the garden can be is 7 ft, with a width of 6 ft and a length of 1 ft. She can use 3 pieces of edging for the first side, 6 pieces for the second side, and another 3 pieces for the third side pieces 6 ft 1 ft 6 ft ft PRACTICE 1. Suppose Jill decided not to use the house as one side of the garden. What is the greatest area that she could enclose?. A store sells batteries in packs of 3 for $3.99 and for $.99. Barry got 1 batteries for $ How many of each package did he buy? Problem Solving Handbook PS7

9 Solve a Simpler Problem If a problem contains large numbers or requires many steps, try to solve a simpler problem first. Look for similarities between the problems, and use them to solve the original problem. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Noemi heard that 10 computers in her school would be connected to each other. She thought that there would be a cable connecting each computer to every other computer. How many cables would be needed if this were true? Problem Solving Handbook Understand the Problem Make a Plan Solve You know that there are 10 computers and that each computer would require a separate cable to connect to every other computer. You need to find the total number of cables. Start by solving a simpler problem with fewer computers. The simplest problem starts with computers. computers 3 computers computers 1 connection 3 connections 6 connections Organize the data in a table to help you find a pattern. Number of Computers Number of Connections Look Back So if a separate cable were needed to connect each of 10 computers to every other one, 5 cables would be required. Etend the number of computers to check that the pattern continues. PRACTICE A banquet table seats people on each side and 1 at each end. If 6 tables are placed end to end, how many seats can there be?. How many diagonals are there in a dodecagon (a 1-sided polygon)? PS8 Problem Solving Handbook

10 Use Logical Reasoning Sometimes a problem may provide clues and facts to help you find a solution. You can use logical reasoning to help solve this kind of problem. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Kim, Lily, and Suki take ballet, tap, and jazz classes (but not in that order). Kim is the sister of the person who takes ballet. Lily takes tap. Match each girl with the class she takes. Understand the Problem Make a Plan You want to determine which person is in which dance class. You know that there are three people and that each person takes only one dance class. Use logical reasoning to make a table of the facts from the problem. Solve List the types of dance and the people s names. Write Yes or No when you are sure of an answer. Lily takes tap. Ballet Tap Jazz Kim No Lily No Yes No Suki No The person taking ballet is Kim s sister, so Kim does not take ballet. Suki must be the one taking ballet. Problem Solving Handbook Ballet Tap Jazz Kim No No Lily No Yes No Suki Yes No No Kim must be the one taking jazz. Kim takes jazz, Lily takes tap, and Suki takes ballet. Look Back Make sure none of your conclusions conflict with the clues. PRACTICE 1. Patrick, John, and Vanessa have a snake, a cat and a rabbit. Patrick s pet does not have fur. Vanessa does not have a cat. Match the owners with their pets.. Isabella, Keifer, Dylan, and Chrissy are in the sith, seventh, eighth, and ninth grades. Isabella is not in seventh grade. The sith-grader has band with Dylan and lunch with Isabella. Chrissy is in the ninth grade. Match the students with their grades. Problem Solving Handbook PS9

11 Use a Venn Diagram You can use a Venn diagram to display relationships among sets in a problem. Use ovals, circles, or other shapes to represent individual sets. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List Patricia took a poll of 100 students. She wrote down that 3 play basketball and 5 run track. Of those students, 19 do both. Mrs. Thornton wants to know how many of the students polled only play basketball. Understand the Problem You know that 100 students were polled, 3 play basketball, 5 run track, and 19 play basketball and run track. The answer is the number of students who only play basketball. Make a Plan Use a Venn diagram to show the sets of students who play basketball, students who run track, and students who do both. Problem Solving Handbook Solve Draw and label two overlapping circles in a rectangle. Work from the inside out. Write 19 in the area where the two circles overlap. This represents the number of students who play basketball and run track. Use the information in the problem to complete the diagram. You know that 3 students play basketball, and 19 of those students run track. 19 So 13 students only play basketball. Look Back When your Venn diagram is complete, check it carefully against the information in the problem to make sure it agrees with the facts given ?? PRACTICE 1. How many of the students only run track?. How many of the students do not play basketball or run track? PS10 Problem Solving Handbook

12 Make an Organized List In some problems, you will need to find out eactly how many different ways an event can happen. When solving this kind of problem, it is often helpful to make an organized list. This will help you count all the possible outcomes. Problem Solving Strategies Draw a Diagram Make a Model Guess and Test Work Backward Find a Pattern Make a Table Solve a Simpler Problem Use Logical Reasoning Use a Venn Diagram Make an Organized List What is the great est amount of money you can have in coins (quarters, dimes, nickels, and pennies) without being able to make change for a dollar? Understand the Problem You are looking for an amount of money. You cannot have any combinations of coins that make a dollar, such as quarters or 3 quarters, dimes, and a nickel. Make a Plan Solve Make an organized list, starting with the maimum possible number of each type of coin. Consider all the ways you can add other types of coins without making eactly one dollar. List the maimum number of each kind of coin you can have. 3 quarters 75 9 dimes nickels pennies 99 Net, list all the possible combinations of two kinds of coins. 3 quarters and dimes dimes and 1 quarter quarters and nickels 95 9 dimes and 1 nickel 95 3 quarters and pennies 99 9 dimes and 9 pennies nickels and pennies 99 Problem Solving Handbook Look Back Look for any combinations from this list that you could add another kind of coin to without making eactly one dollar. 3 quarters, dimes, and pennies quarters, nickels, and pennies 99 9 dimes, 1 quarter, and pennies dimes, 1 nickel, and pennies 99 The largest amount you can have is 119, or $1.19. Try adding one of any type of coin to either combination that makes $1.19, and then see if you could make change for a dollar. PRACTICE 1. How can you arrange the numbers, 6, 7, and 1 with the symbols,, and to create the epression with the greatest value?. How many ways are there to arrange desks in 3 or more equal rows if each row must have at least desks? Problem Solving Handbook PS11

13 Skills Bank Place Value A place-value chart can help you read and write numbers. The number 35,01,678, (three hundred forty-five billion, twelve million, si hundred seventy-eight thousand, nine hundred twelve and five thousand seven hundred eighty-four ten-thousandths) is shown. Billions Millions Thousands Ones Tenths Hundredths Thousandths Ten-Thousandths 35, 01, 678, EXAMPLE Name the place value of the digit. A the 7 in the thousands column B the 0 in the millions column 7 ten thousands place 0 hundred millions place C the 5 in the billions column D the 8 to the right of the decimal point 5 one billion, or billions, place 8 thousandths PRACTICE Name the place value of the underlined digit ,56,789, ,56,789, ,56,789, ,56,789, ,56,789, ,56,789, Rounding Skills Bank To round to a certain place, follow these steps. 1. Locate the digit in that place, and consider the net digit to the right.. If the digit to the right is 5 or greater, round up. Otherwise, round down. 3. Change each digit to the right of the rounding place to zero. EXAMPLE A Round 15, to the nearest B Round 15, to the nearest thousand. tenth. 15, Locate digit. 15, Locate digit. The digit to the right is less than 5, The digit to the right is greater so round down. than 5, so round up. 15, ,000 15, ,539. PRACTICE Round 59,35.78 to the place indicated. 1. hundred thousand. ten thousand 3. thousand. hundred SB Skills Bank

14 Factors and Multiples When two whole numbers are multiplied to get a third, the two numbers are said to be factors of the third number. Multiples of a number can be found by multiplying the number by 1,, 3,, and so on. EXAMPLE A List all the factors of 8. B Find the first five multiples of , 8, , 1 8, and So the factors of 8 are 1,, 3,, 6, 8, 1, 16,, and , 3 6, 3 3 9, 3 1, and So the first five multiples of 3 are 3, 6, 9, 1, and 15. PRACTICE List all the factors of each number Write the first five multiples of each number Divisibility Rules A number is divisible by another number if the division results in a remainder of 0. Some divisibility rules are shown below. A number is divisible by... Divisible Not Divisible if the last digit is an even number. 11,99,175 3 if the sum of the digits is divisible by if the last two digits form a number divisible by. 1, if the last digit is 0 or 5. 15,195 10,007 6 if the number is even and divisible by 3. 1,33 8 if the last three digits form a number divisible by 8. 5,016 1,100 9 if the sum of the digits is divisible by Skills Bank 10 if the last digit is 0.,790 9,35 PRACTICE Determine which h of these numbers each number is divisible by:, 3,, 5, 6, 8, 9, , Skills Bank SB3

15 Prime and Composite Numbers A prime number is a whole number greater than 1 that has eactly two factors, 1 and the number itself. A composite number is a whole number greater than 1 that has more than two factors. Factors: 1 and ; prime 11 Factors: 1 and 11; prime 7 Factors: 1 and 7; prime EXAMPLE Factors: 1,, and ; composite 1 Factors: 1,, 3,, 6, and 1; composite 63 Factors: 1, 3, 7, 9, 1, and 63; composite Determine whether each number is prime or composite. A 17 B 16 C 51 Factors Factors Factors 1, 17 prime 1,,, 8, 16 composite 1, 3, 17, 51 composite PRACTICE Determine whether each number is prime or composite Prime Factorization A composite number can be epressed as a product of prime numbers. This is the prime factorization of the number. To find the prime factorization of a number, you can use a factor tree. Skills Bank EXAMPLE Find the prime factorization of The prime factorization of is 3, or 3 3. PRACTICE Find the prime factorization of each number SB Skills Bank

16 Greatest Common Factor (GCF) The greatest common factor (GCF) of two whole numbers is the greatest factor the numbers have in common. EXAMPLE Find the GCF of and 60. Method 1: List all the factors of both numbers. Find all the common factors. : 1,, 3,, 6, 8, 1, 60: 1,, 3,, 5, 6, 10, 1, 15, 0, 30, 60 The common factors are 1,, 3,, 6, and 1. So the GCF is 1. Method : Find the prime factorizations. : : 3 5 Find the least power of each common prime factor: and 3. The product of these is the GCF. So the GCF is 3 1. PRACTICE Find the GCF of each pair of numbers by either method. 1. 9, 15. 5, , 30., , , , , , , , , 8 Least Common Multiple (LCM) The least common multiple (LCM) of two whole numbers is the least multiple the numbers share. EXAMPLE Find the least common multiple of 8 and 10. Method 1: List multiples of both numbers. 8: 8, 16,, 3, 0, 8, 56, 6, 7, 80 10: 10, 0, 30, 0, 50, 60, 70, 80, 90 The smallest common multiple is 0. So the LCM is 0. Method : Find the prime factorizations. 8: 3 10: 5 Find the greatest power of each prime factor: 3 and 5. The product of these is the LCM. Skills Bank PRACTICE So the LCM is Find the LCM of each pair of numbers by either method. 1.,. 3, , 5. 10, , , , , 1 9., , , 1. 8, 36 Skills Bank SB5

17 Estimate by Rounding To estimate a sum or difference, find the least number and identify its first nonzero digit (from the left). Round all the numbers to this place value. Then add or subtract the rounded numbers. To estimate a product, round each number to its greatest place value. Then multiply the rounded numbers. EXAMPLE Use rounding to estimate. A B is least. Round each number to the nearest ,000 Round each number to its greatest place value. PRACTICE Use rounding to estimate Compatible Numbers Compatible numbers are numbers that you can calculate with mentally and that are close to the numbers in a problem. You can use compatible numbers to estimate quotients. EXAMPLE Skills Bank Use compatible numbers to estimate each quotient. A B Think: Think: Compatible Estimate Compatible Estimate numbers numbers PRACTICE Estimate the quotient by using compatible numbers , , , SB6 Skills Bank

18 Estimate by Clustering Sometimes all the numbers in an addition problem are close to the same number. When this happens, you can estimate the sum by clustering. To use clustering, round the numbers to the same value. EXAMPLE Estimate by clustering The addends cluster around 0, so round each to 0. 0(5) 00 Use multiplication instead of repeated addition. PRACTICE Estimate by clustering Overestimates and Underestimates An overestimate is an estimate that is greater than the actual answer. An underestimate is an estimate that is less than the actual answer. EXAMPLE Gina has $3.. Her brother repays her $8.50, and her mother loans her $5.50. Gina wants to know if she can buy a portable music player that costs $100. Find an overestimate and an underestimate of the total amount Gina has. Then determine which is more appropriate for the situation. $3. $8.50 $5.50 $50 $30 $50 $130 To overestimate, round each number up. $0 $0 $0 $100 To underestimate, round each number down. Gina should use the underestimate. Her actual amount of money will be greater than her underestimate. Her underestimate shows that she has more than $100, so she has more than enough money for the music player. PRACTICE Skills Bank Find an overestimate and an underestimate for each situation. Then determine which is more appropriate. 1. Rasheed has $5 to buy art supplies. He wants a tube of paint for $5.5, a brush for $8.95, and a sketch pad for $9.75. He needs to determine if he has enough money.. Lionel received a paycheck for $156. and earned $39.50 in tips. He also has $6.50 in his savings account. He wants to buy a new game system for $0. He needs to determine if he has enough money. Skills Bank SB7

19 Multiply and Divide Decimals by Powers of 10 Notice the pattern below. Notice the pattern below , , , Think: When multiplying decimals by powers of 10, move the decimal point one place to the right for each power of 10, or for each zero. Think: When dividing decimals by powers of 10, move the decimal point one place to the left for each power of 10, or for each zero. PRACTICE Find each product or quotient , Order of Operations When simplifying epressions, follow the order of operations. 1. Simplify within parentheses. 3. Multiply and divide from left to right.. Evaluate eponents and roots.. Add and subtract from left to right. EXAMPLE Skills Bank Simplify the epression 3 ( 11 ). 3 ( 11 ) 3 7 Simplify within parentheses. 9 7 Evaluate the eponent. 63 Multiply. PRACTICE Simplify each epression ( 15 8 ) ( 6 8 ) ( 3 6 ) ( 8 ) SB8 Skills Bank

20 Choose Appropriate Units of Measurement Use the following benchmarks to help you choose appropriate units of measurement and to estimate measurements. Customary Unit Benchmark Length Inch (in.) Length of a small paper clip Foot (ft) Length of a standard sheet of paper Yard (yd) Width of a doorway Weight Ounce (oz) Weight of a slice of bread Pound (lb) Weight of 3 apples Capacity Fluid ounce (fl oz) Amount of water in two tablespoons Cup (c) Capacity of a standard measuring cup Gallon (gal) Capacity of a large milk jug Metric Unit Benchmark Length Millimeter (mm) Thickness of a dime Centimeter (cm) Width of a large paper clip Meter (m) Width of a doorway Mass Gram (g) Mass of a small paper clip Kilogram (kg) Mass of a tetbook Capacity Milliliter (ml) Amount of water in an eyedropper Liter (L) Amount of water in a large water bottle EXAMPLE A Choose the most appropriate customary unit to measure the length of a city bus. Justify your answer. Yards; the length of a city bus is similar to the width of several doorways. B Choose the most reasonable estimate of the mass of a rocking chair. A g B 1 g C kg D 1 kg The most reasonable estimate is 1 kg (a mass of about 1 tetbooks). PRACTICE Skills Bank Choose the most appropriate unit for each measurement. Justify your answer. 1. capacity of a soup bowl (customary). weight of a bo of tissue (customary) 3. mass of a carrot stick (metric). length of pencil (metric) 5. width of a butterfly (customary) 6. capacity of a bathtub (metric) Choose the most reasonable estimate. 7. capacity of a shampoo bottle 8. width of a cell phone A 5 ml B 0.5 L C 5 L A in. B ft C yd Skills Bank SB9

21 Points, Lines, and Planes Points, lines, and planes are the building blocks of geometry. Segments, rays, and angles are defined in terms of these basic figures. A point names a location. A point A A line is perfectly straight and etends forever in both directions. B C line, or BC A plane is a perfectly flat surface that etends forever in all directions. P D E F plane P, or plane DEF A segment, or line segment, is the part of a line between two points. G H GH A ray is a part of a line that starts at one point and etends forever in one direction. J K KJ EXAMPLE Use the diagram to name each figure. A four points Q, R, S, T Z Q R S T B a line Possible answers: QR, QS, or RS Use any two points on the line. C a plane Skills Bank D E Possible answers: plane Z or plane QRT four segments Possible answers: QR, RS, RT, QS five rays RQ, RS, RT, SQ, QS Use any three points in a plane that form a triangle to name the plane. Write the two endpoints in any order. Write the endpoint first. PRACTICE Use the diagram to name each figure. 1. three points. a line 3. a plane. three segments 5. three rays X Y Z A SB10 Skills Bank

22 Circle Graphs A circle graph shows how a set of data is divided into parts. The entire circle represents 100% of the data. Each sector, or slice of the circle, represents one part of the data set. To make a circle graph, determine the percent for each part of the data. Use the percents to determine the angle measure of each sector. EXAMPLE Skunks are legal as pets in some states and illegal in others. Use the first two columns of the table to make a circle graph of the data. Step 1: Find the percent of states with each category of legality. Step : Use the percents to determine the angle measure of each sector of the graph. Step 3: Use a compass to draw a circle. Use a ruler to draw a vertical radius. Legality Step : Use a protractor to draw each angle clockwise from the radius. Step 5: Label the graph and each sector. Color the sectors. Number of States Legal (no restrictions) Legal with permit Percent of States Legal in some areas 50 % 100 Illegal Other conditions % Legal in some areas % Angle of Sector 1% % % Legal (no restrictions) 1% Legal with permit % PRACTICE 1. Suppose you made a circle graph in which all the states where it is legal to own a pet skunk were grouped together in one category. What would be the angle measure of the sector for this category?. How many states would need to legalize skunks as pets for the largest sector to measure 180? Make a circle graph for each set of data. 3. Land Use in Minnesota. Type of Land Number of Acres (millions) Crops.7 Forest 1. Water, marshes 8.9 Other 8.0 Illegal 5% Government Spending in Florida Category Other conditions 6% Amount (billions of dollars) Health care 6.9 Education 6.1 Transportation 15.0 Police, firefighting 0.1 Other 66.1 Skills Bank Skills Bank SB11

23 Inverse Variation An Inverse variation is a relationship between two variables, and y, that can be written in the form y _ k or y k, where k is a nonzero constant and 0. In an inverse variation, as one quantity increases, the other quantity decreases. The product of and y is constant. You can use this fact to distinguish inverse variations from other relationships. EXAMPLE 1 Tell whether each relationship is an inverse variation. Eplain. A 3 6 Find the product of and y. y (16) 8, (1) 8, 6(8) 8 The product y is constant, so the relationship is an inverse variation with k 8. B y 8 6 Find the product of and y. 0(8) 160, 5(6) 150, 50() 100 The product y is not constant, so the relationship is not an inverse variation. EXAMPLE As shown in the table, when the volume of a gas increases, the pressure in atmospheres decreases. Find the inverse variation equation, and use it to find the pressure of the gas when the volume is 8 liters. Volume (L) Pressure (atm) The relationship is an inverse variation with y 0. The inverse variation equation is y 0 0. When the volume is 8 L, the pressure is y 8 5 atm. PRACTICE Skills Bank Tell whether each relationship is an inverse variation. Eplain y y Ohm s Law relates the current in a circuit to the resistance. Find the inverse variation equation, and use it to find the current in a 1-volt circuit with 16 ohms of resistance. Current (amps) Resistance (ohms) Given that a triangle has an area of 7 c m, the relationship between the height h and the length of the base b is an inverse variation. If b 8 cm when h 3 cm, find b when h 1 cm. SB1 Skills Bank

24 Eperimental and Theoretical Probability A trial is one repetition or observation of an eperiment, such as rolling a number cube. Each result is an outcome, and a set of outcomes is an event. The eperimental probability of an event is the ratio of the number of times the event occurs to the total number of trials. EXAMPLE 1 A marble is randomly drawn from a bag and then replaced. The table shows the results after 100 draws. Find the eperimental probability of drawing a yellow marble. Outcome Green Red Yellow Blue White Frequency eperimental probability The eperimental probability is 0.1 or 1%. number of times a yellow marble was drawn total number of draws You can find the theoretical probability of an event when all of the outcomes for an eperiment are equally likely. The theoretical probability is the ratio of the number of ways the event can occur to the total number of equally likely outcomes. EXAMPLE An eperiment consists of rolling a number cube. Find the theoretical probability of rolling an even number. There are 6 equally likely outcomes: 1,, 3,, 5, 6. There are 3 possible even numbers:,, 6. number of outcomes that are even numbers P(even) 3 total number of equally likely outcomes 6 1 PRACTICE Larissa spins a spinner 36 times. The table shows the results of her spins. Use the table for Find the eperimental probability that the spinner lands on green. Outcome Red Blue Green Frequency Skills Bank. Find the eperimental probability that the spinner does not land on red. 3. Find the eperimental probability that the spinner lands on red or blue. Kendrick has a set of 0 cards numbered 1 through 0. He chooses a card without looking. Find the theoretical probability of each event.. choosing 7 or 1 5. choosing an odd number 6. choosing a number less than 5 7. not choosing choosing a prime number 9. choosing a multiple of 3 Skills Bank SB13

25 Independent and Dependent Events Two events are independent events if the occurrence of one event does not effect the probability of the other event. To find the probability that two independent events will happen, multiply the probabilities of the two events. That is, if A and B are independent events, then EXAMPLE 1 P(A and B) P(A) P(B) Find the probability of rolling a 1 on a number cube and then flipping a coin and getting tails. The outcome of rolling the number cube does not affect the result of flipping the coin, so the events are independent. P(1 and tails) P(1) P(tails) There are 6 outcomes for the number cube and outcomes for the coin. Multiply. Events are dependent events if the occurrence of one event affects the probability of the other event. To find the probability of dependent events, multiply the probability of the first event by the probability of the second event given that the first event has occurred. That is, if A and B are dependent events, then EXAMPLE P(A and B) P(A) P(B after A) Skills Bank A jar contains 16 quarters and 10 nickels. If coins are chosen at random, what is the probability of getting two quarters? Because the first coin that is chosen is not replaced, the sample space is different when choosing the second coin, so the events are dependent. P(first coin is a quater) There are 16 quarters out of 6 coins. 13 P(second coin is a quater) 15 5 _ 3 There are 15 quarters remaining out of 5 coins. 5 P(two quaters) P(A) P(B after A) PRACTICE 1. Elena rolls a number cube three times. What is the probability that she rolls a 5 on all three rolls? A bag contains 6 red marbles, blue marbles, and 8 yellow marbles. Tell whether the events are dependent or independent, and find the probability.. Jennifer chooses a marble, puts it back in the bag, and then chooses another marble. What is the probability that she chooses a yellow marble then a red marble? 3. Zack chooses a marble, puts it aside, and then chooses another marble. What is the probability that he chooses a yellow marble then a red marble? SB1 Skills Bank

26 Making Predictions A prediction is something you can reasonably epect to happen in the future. You can use probability to help you make a prediction. EXAMPLE 1 At a carnival, a spinner is used to determine a player s prize. If the spinner lands on red, the player gets a stuffed animal. The manager of the carnival epects 160 players to spin the spinner each day. What is the best prediction of the number of stuffed animals that will be given away each day? First find the probability that the spinner lands on red. total number of red outcomes 1 total number of outcomes n Set up a proportion n Find the cross products. 0 n Divide both sides by 8 About 0 stuffed animals will be given away each day. You can also use probability to determine whether a game is fair. A game of chance is fair if each player is equally likely to win. EXAMPLE A bag contains 100 tiles with letters on them. The table shows the number of tiles with vowels. Lisa chooses a tile from the bag. If the tile is a vowel, she gets a point. If the tile is a consonant, Brent gets a point. Decide whether the game is fair. Find the probability that each player gets a point. P(vowel) There are vowels. 50 P(consonant) There are consonants. 50 Since , the game is not fair. 50 PRACTICE Number Letter of Tiles A 9 E 1 I 9 O 8 U Skills Bank 1. Jorge rolls a number cube 8 times. Predict how many times he will roll a 5.. Brian has a 0-sided number cube that is numbered 1 through 0. He rolls the cube 100 times. Predict the number of times he will roll a number less than 7. Decide whether each game is fair. Eplain. 3. A spinner is divided evenly into 8 sections. There are blue sections, red, 1 green, and 1 yellow. Player A wins if the spinner lands on blue. Otherwise, Player B wins.. Alicia rolls two number cubes and adds the two numbers. She wins the game if the sum is 6 or less. Otherwise, Kendra wins the game. Skills Bank SB15

27 Samples and Populations A population is an entire group being considered for a survey. Researchers often study a part of the population called a sample. In a random sample, each member of the population has an equal chance of being selected. A random sample is more likely to be representative of the population than samples chosen in other ways. EXAMPLE 1 Which sampling method better represents the population? Justify your answer. Student Survey: New School Uniforms Sampling Method Results of Survey Kurt surveys 30 students by randomly choosing 57% favor new uniforms. names from the school enrollment list. Sabine surveys 15 of her friends at school. 73% favor new uniforms. Kurt s sample is a random sample. His results are probably more representative of the entire student population than Sabine s results. EXAMPLE Skills Bank A factory produces 600 MP3 players each day. The factory s manager claims that fewer than 50 defective players are produced each day. A random sample of 00 MP3 players shows that of them are defective. Evaluate the manager s claim. The random sample is likely to be representative of all the MP3 players. Use a proportion to make a prediction about the entire population of MP3 players. defective MP3 players in sample size of sample PRACTICE 00 defective MP3 players in population size of population Find the cross products. 1,00 00 Multiply. 31 Divide both sides by 00. Based on the random sample, there are likely to be 31 defective MP3 players in the entire population. The manager s claim is likely to be accurate. 1. Which sampling method better represents the population? Justify your answer. Consumer Survey: Breakfast Foods Sampling Method Results of Survey Ale surveys 60 customers at a bakery. 6% eat bagels for breakfast. Tenisha surveys 50 people chosen at 36% eat bagels for breakfast. random from the phone book.. A novel has 800 pages. Juan claims that more than 60 pages of the novel contain a typographic error, or typo. A random sample of 50 pages from the novel shows that 3 pages have typos. Evaluate Juan s claim. SB16 Skills Bank

28 Transformations A transformation changes the position or orientation of a figure. A translation slides a figure along a straight line without turning. A reflection flips a figure across a line to create a mirror image. A rotation turns a figure around a fied point. EXAMPLE 1 Identify each type of transformation. A B The figure flips across a line. The transformation is a reflection. The figure slides along a straight line. The transformation is a translation. EXAMPLE Graph the result of each transformation. A Reflect ABC across the y-ais. B A C O -coordinates y-coordinates y C A B opposites same B Rotate DEF 90 counterclockwise about the origin. F y F E D O D E Corresponding sides DE and D E form a right angle, as do DF and D F. PRACTICE Identify each type of transformation. 1.. Graph the result of each transformation. 3. Translate JKL 3 units right and. Reflect ABCD across the -ais. 1 unit up. J O y K A L y C B D O Skills Bank Skills Bank SB17

29 Worked-Out Solutions Worked-Out Solutions Chapter (pp. 6 9) 9. Evaluate y for y Evaluate ( z ) 5 for z ( 3 ) (pp ) 9. twice the quotient of m and 35 ( m_ 35) ( m 5 ) times the sum of m and 5 (pp. 1 17) 15. ; Identity Property of Multiplication 5. ( c d ) (pp. 18 1) ( 3 ) (pp. 5) 1. Evaluate b ( 6 ) for b. 1-6 ( 6 ) ( 11 ) (pp. 6 9) 5. Evaluate 16 for. 16 ( 16 ) _ (1) _ 1 8 (pp ) 35. Evaluate y _ 11 for y (pp. 36 0) 35. t t t 7 7 C g ; g 58, 5, or g So 58 is not the solution. 3 7 g 3 7 ( 5 ) 3 So 5 is not the solution. 3 7 g So 59 is the solution (pp. 1 5) _ 1_ n_ 5 3 n_ n_ n_ n 35 WS Worked-Out Solutions

30 Chapter -1 (pp. 6 65) _ _ GCF ; 1_ ; No, the fraction cannot be 3 further simplified because the numerator and denominator are relatively prime. - (pp ) 7. 5_ 7 _ _ 70 6_ 10 7_ 7 _ 70 50_ 70 _ 5_, so _ _ _, so (pp. 7 76) ( ) , or 16 8 The inside diameter is 5 8 in _ The rates of erosion differ by meter. - (pp ) 1. 5 ( ) 5 ( 19 11) 5 ( 19 ) ( 5 9) 3 6 ( 7 9) 3 ( 7 ) 6 ( 9 ) (pp ) She can have 11 more glasses. 9. Let equal the height of the largest doll in The height of the largest doll is in. (pp ) ,87, or carats j ( 7. ) j j 1.6 Chapter (pp ) 19. y 3; ( 5, 8 ) 8? 5 3 8? 8 ( 5, 8 ) is a solution. 1. y 3 5; (, )? 3 ( ) 5? 6 5? 11 (, ) is not a solution. Worked-Out Solutions Worked-Out Solutions WS3

31 3- (pp ) ; 1 units 1. a. y 3-3 Derrick O Monica b. 6 () 10 10; They are 10 kilometers apart from each other at the end of their rides. (pp ) 13. Discrete graph 17. Possible answer: The airplane takes off and climbs quickly. Then it levels off and stays near the same altitude for minutes before it lands. 3-5 b. Any nonnegative number of hours ( 0 ) c. The independent variable represents the number of hours the bulb is lit. The dependent variable represents the cost in dollars of using the bulb. Possible answer: For each etra hour the bulb is lit, the cost of using the bulb increases by $ d. y The bulb was used for 550 hours. 7. d 0g (pp. 1 17) g d Worked-Out Solutions Height (ft) Time (min) (pp ) 15. y ( 1 ) ( 1 ) y ( 1 ( )) 10 1 ( 1 ( 1 )) 6 0 ( 1 ( 0 )) 1 ( 1 ( 1 )) ( 1 ( )) 6 1. a. First 8 hours a day multiplied a week, 7 days, equals 56 hours. y y ( 56 ) y It costs $0.0 to use a 60-watt bulb 8 hours a day for a week d O 9. y.5 g y Chapter -1 (pp. 1 17) 39. ( 5 ) ( 5 16 ) ( 80 ) Evaluate m ( p n q ) for m, n 6, p 3, and q 3. ( ) - ( 3 16 ) ( 13 ) 6 (pp ) 33. Evaluate ( ) for.1. (.1.1 ) (.1.1 ) ( 6.51 ) 13.0 WS Worked-Out Solutions

32 ( ) (pp ) million 1,000, A grape-size nucleus weighs kg , , (pp ) 39. Let be the number of times as many There are 676 times as many ways. 5. w w 3 w 3 w (pp ) (pp ) ft 9. a. r 1.88d r ,000 r b. Evaluate d rt for d 3000 and r ,000. t ,000 t 7.8 It would take approimately 7.8 hours. -7 (pp ) Since 15 is closer to 16 than to 9, 15 So t t is closer to than 1, so 3 t about.5 seconds (pp ) (pp ) 1. a 7 60 a,09 3,600,09,09 a 1,391 a 1,391 a b 6 b b 51 b 51 b.6 The ladder will safely reach about.6 feet. -10 (pp ) No h h 8 h The altitude of the triangle is 8 units. 3 Worked-Out Solutions Worked-Out Solutions WS5

33 Worked-Out Solutions Chapter (pp. 0 05) No; Carmen s ratio of pancake mi to servings is not equal to the ratio the pancake recipe calls for. She needs 3 1 cups of pancake mi to make 1 servings ; yes (pp ) 6 beats 13. measures 6 beats measures 16 beats 1 measure 16 beats per measure 50 heartbeats heartbeats 6 min 50 heartbeats 5 min 1 min approimately 50 beats per minute (pp ) , 1 1 c 15 computers 1. 1 student 5- computers 5 students c ( c 15 ) ( 5 )( 15 ) c 1 computers (pp. 16 0) 7. ABC and DEF have corresponding angles that are congruent. 15. AB DE BC EF CA FD 18 3 The ratios of the corresponding sides of ABC and DEF are equal. similar in. 5-5 (pp. 1 ) ft : 1 in. enlarges ft 1 in. 1 in. 1 1 in. 5-6 (pp. 5 9) 13. boes 60 s 1 s 1 min 10 boes 1 min 10 boes per minute cereal boes 3. 0 knots 0 mi 1 h 1,85 m 1 h 1 mi 3,600 s 37,00 m 3,600 s 10.3 m 10.3 m/s 1 s 5-7 (pp ) The shadow etends 7. m past the storage shed The ramp is 85 feet long. Chapter (pp ) 3. Yellow: 0. 0% Blue: % Purple: 0. 0% Red: % 5. Red: % 0 Blue: % Purple: % 10 Yellow: % 6 - (pp. 5 58) % of 0 100% % of cars 6-3 (pp ) % of WS6 Worked-Out Solutions