Integers and Rational Numbers

Transcription

1 1 Skills Intervention: Integers The opposite, or additive inverse, of a number is the number that is the same distance from zero on a number line as the given number. The integers are the set of whole numbers and their opposites. Graphing Integers and Their Opposites on a Number Line Graph 2 and its opposite on a number line. Vocabulary opposite additive inverse integer absolute value How many units from 0 is 2? Plot a point on 2 on the number line What number is also 2 units from zero but on the other side of zero? Where should you plot the other point? Ordering Integers Using a Number Line Graph the integers on a number line, and then write them in order from least to greatest: 3, 6, 1, 0, 2, 4. Graph the numbers that are not already graphed on the number line List the integers in order from least to greatest. 3,,, 1,, What direction on the number line shows numbers from least to greatest? The absolute value of a number is the distance from zero to the number on a number line and is represented by horizontal lines on either side of the number. Finding Absolute Value Use a number line to find 4 and 2. Graph the integer 4 on the number line How many units are between 0 and 4? So, 4 Graph the integer 2 on the number line. How many units are between 0 and 2? So, 2 12

2 2 There are two commonly used methods of adding integers; using a number line and using absolute values. To add integers using a number line, start at the origin. If the number is positive, move right. If the number is negative, move left. Modeling Integer Addition Add. A. 3 ( 2) Skills Intervention: Adding Integers Start at 0. Move right Then move 3 ( 2) B. 1 ( 3) 2 units. units. Start at. Move 1 unit. Then move left units. 1 ( 3) To add integers, first determine the sign of each number. If the signs are the same, find the sum of the absolute values and keep the sign. If the signs are different, find the difference of the absolute values of the numbers and keep the sign of the number with the greatest absolute value Adding Integers Using Absolute Value Add. A. 1 ( 6) The signs are both. The absolute value of 1 is. The absolute value of 6 is. 1 6 Find the of the absolute values. Is the answer positive or negative? B. 5 3 The signs are. The absolute value of 5 is. The absolute value of 3 is. 5 3 Find the of the absolute values. Is the answer positive or negative? 13

3 3 There are two commonly used methods of subtracting integers; using a number line and adding the opposite. To subtract a positive integer, move left on the number line. To subtract a negative integer, move right. Modeling Integer Subtraction Subtract. A. 3 4 Start at 0. Move right units To subtract 4, move to the 3 4 B. 2 ( 4) Start at 0. Move left Skills Intervention: Subtracting Integers units To subtract 4, move to the 4 units. 4 units ( 4) To subtract an integer, add its opposite and follow the rules for adding integers Subtracting Integers by Adding the Opposite Subtract The opposite of 10 is. 6 Rewrite the expression as a sum. Evaluating Expressions with Integers Evaluate c d for c 8, d 5. c d Since the signs are different, find the absolute values. Subtract. Is the answer positive or negative? 5 Substitute for c and d. 8 Add the opposite of 5. 8 ( 5) Since the signs are the same, find the. Add. Is the answer positive or negative? of the 14

4 3 Problem Solving Intervention: Subtracting Integers You can subtract integers to find horizontal and vertical distances on a coordinate plane. In a town with square blocks, the zoo is 8 blocks east and 13 blocks south of City Hall. The stadium is 17 blocks west and 13 blocks south of City Hall. The mall is 17 blocks west and 5 blocks north of City Hall. How far is the total distance from the zoo to the stadium to the mall? y Understand the Problem 1. Think of the town as a coordinate plane. Which axis runs east-west? Which runs north-south? 2. On the coordinate plane to the right, mark the approximate location of City Hall, the zoo, stadium, and mall. x Make a Plan 3. Why does it make sense to use City Hall as the origin? 4. What coordinates will you use for the zoo? The stadium? The mall? Solve 5. What integers can you subtract to find the distance between the zoo and the stadium? 6. What integers can you subtract to find the distance between the stadium and the mall? 7. How many blocks is it from the zoo to the stadium? From the stadium to the mall? From the zoo to the mall by way of the stadium? Look Back 8. Do the distances you calculated reasonably match the distances on your sketch? 15

5 4 Skills Intervention: Multiplying and Dividing Integers You can multiply integers using repeated addition or by multiplying the absolute values. The product or quotient of integers with the same sign is positive. The product or quotient of integers with different signs is negative. Multiplying Integers Using Repeated Addition Find each product. Use a number line to model repeated addition. 2 ( 3) Start at 0. Add two times. 3 3 Move to the 2 ( 3) two times Multiplying Integers Find the product. 3 ( 2) 3 = 2 = = Multiply. Find the absolute values. 3 ( 2) = Both signs are, so the product is. To divide integers, find the quotient of the absolute values. Then, use the rules above to determine the sign. Dividing Integers Find the quotient. 144 ( 3) First, divide 144 by and 3 are integers. 144 ( 3) The signs are the same, so the quotient is. Consumer Math Application The price of a couch is discounted by the same amount each month it does not get sold. In 5 months, it is discounted $100. What is the price change per month? = First, divide 100 by 5. The price change is. The signs are, so the quotient is. 16

6 4 Problem Solving Intervention:Multiplying and Dividing Integers a and b are two integers. If you subtract their product from the absolute value of their product, you get 60. If you subtract their quotient from the absolute value of their quotient, you get 15. Find the values of a and b. Understand the Problem 1. Complete these equations to match the clues in the problem. ab ; a b Make a Plan 60 ab 0 ab 2. ab and ab are opposites. Picture them on a number line, 60 units apart. How far is ab from 0? What is the product of a b? (Remember, a and b cannot have the same sign.) 3. How can you use guess and check to solve the problem? Solve 4. What pairs of integers have a product of 30? Hint: There are 6 different pairs. 5. Now look at the quotient clue. Since a b a b equals 15, then a b must equal a b Look back at the pairs of integers you listed in Exercise 5. Which of those pairs have a quotient of 7 1 2? 0 a b Look Back 7. Show that your answer fits all the conditions stated in the problem. 17

7 5 Skills Intervention: Solving Equations Containing Integers To solve an equation, use the opposite operation to isolate the variable. Addition and subtraction are opposite operations, as are multiplication and division. Solving Addition and Subtraction Equations Solve. Check your answers. A. x 6 3 What is the variable? What number should you add to both sides of the equation? x What does x equal? Check: x 6? 3 6? 3 What do you substitute into the original equation to check? 3 Subtract. Does the solution check? B. a 5 13 What do you subtract from both sides of the equation? a What does a equal? Check: a 5? 13 5? 13 What do you substitute into the original equation to check? 13 Add. Does the solution check? Solving Multiplication and Division Equations Solve. Check your answer. g 6 5 What is the variable? g 6 ( 5) Multiply both sides of the equation by to isolate g. g Check: g 6? 5 What does g equal? 6? 5 5 What do you substitute into the original equation to check? Divide. Does the solution check? 18

8 5 Problem Solving Intervention: Solving Equations Containing Integers You can use equations with integers to model situations and solve problems. The table shows how the number of passengers on a bus changes for five stops. The number for Ash Street is missing. If you know that the mean change for the five stops is 3, then at Ash Street, did more people get on the bus or off the bus? How many more? Stop Elm St. Oak St. Ash St. Maple St. Cedar St. Change Understand the Problem 1. Can you tell how many people are on the bus after Elm St? 2. Can you tell how many people got on and how many got off at Elm St? What can you tell about Elm St? Make a Plan 3. Let n stand for the total change in the number of passengers for the 5 stops. If you know that the mean change is 3, what equation can you use to find n? 4. Let x stand for the missing value, the change at Ash St. If you know n, the sum of the 5 changes, what equation can you use to find x? Solve 5. To find n, the sum of the changes, solve the equation you wrote for Exercise Find the missing value x by solving the equation you wrote for Exercise 4. Hint: Use the value of n you found in Exercise 5. Look Back 7. Use the value you found for x and see if the mean is 3. 19

9 SECTION 2A SECTION A: Quiz for Lessons 1 Through 5 1 Integers Compare the integers. Use or Graph the integers 7, 7, 1, 3, 16, 11, and 8 on a number line and then list them from least to greatest Use a number line to find each absolute value Adding Integers Find each sum ( 12) ( 2) ( 3) ( 6) ( 22) Evaluate g h for the given values. 24. g 13, h g 6, h g 24, h g 7, h g 10, h g 8, h 5 20

10 SECTION 2A SECTION A: Quiz for Lessons 1 Through 5 continued 3 Subtracting Integers Find each difference ( 6) ( 11) Evaluate c d for the given values. 33. c 3, d c 9, d c 11, d 10 4 Multiplying and Dividing Integers Find each product or quotient ( 8) ( 7) ( 3) ( 2) ( 6) ( 5) ( 16) 5 Solving Equations Containing Integers Solve each equation. Check your answer x y k k x y When the Chinook winds came to Calgary one spring, the temperature rose from 10 C to 18 C in 24 hours. How much did the temperature rise in 24 hours? On average, how much did the temperature rise each hour? 21

11 SECTION 2A SECTION A Enrichment: Gains and Losses Gains and losses in business can be represented by integers. Gains are called profits and are represented by positive integers. Losses are represented by negative integers. A dividend is part of a profit that is paid to the investors. If you want to find out how well a business is doing, whether they have made or lost money, you can perform operations involving integers. Look at the example below. Video game maker VGM is a public company with 8,750,000 shares of stock owned by investors. Last year the company showed a loss of $15 million. This year the company expects to show a profit of $35 million, half of which will be paid to investors as a dividend. One share of stock currently costs $80. Use the information to answer the questions below. 1. Corporations need to make a profit in order to attract investors and to grow. What is the total profit for VGM after 2 years? 2. What is the equation for the loss ( ) per share last year? 3. What is the loss ( ) per share last year? 4. What is the equation for profit (p) per share this year? 5. What is the profit per share this year? 6. But remember the dividend will only be half the profit per share. What is the equation for dividend per share? What is the dividend per share, d, for this year? 7. Mr. Collins bought 65 shares of stock last year. What was the loss on his shares? 8. Ms. Crenshaw bought 40 shares of stock this year. What will the profit on her shares be? 22

12 6 Decimals that can be written as fractions are either terminating decimals or repeating decimals. Terminating decimals are decimals that come to an end. Repeating decimals are decimals that have a series of numbers that repeat forever. Writing Fractions as Decimals Write each fraction has a decimal. Round to the nearest thousandth, if necessary. A Set up the division. Divide into Skills Intervention: Equivalent Fractions and Decimals Vocabulary terminating decimal repeating decimal Place the decimal point in the quotient directly above the decimal point in the dividend. Add as many zeros as necessary Continue to divide until the decimal terminates or starts to repeat. Is there a remainder? B Multiply to get a power of ten in the denominator. Simplify. Put the numerator in the decimal place indicated by the denominator. Writing Decimals as Fractions Convert 0.52 to a fraction in simplest form can be written as fifty-two Write the decimal as a fraction with a denominator of Divide the numerator and denominator by their 00 greatest common factor Simplify. 23

13 6 Problem Solving Intervention: Equivalent Fractions and Decimals Knowing how to convert between fractions and decimals can help you solve some problems. So far this season, your baseball team has won 29 games and lost 24. How many of the remaining 7 games must you win in order to finish with a winning rate of 0.550? Understand the Problem 1. How is the winning rate calculated? 2. How many games has your team played so far? How many will your team have played by the end of the season? Make a Plan 3. What are the 8 possible win-loss records your team could end up with? 4. Which of the records are obviously not a winning rate? Explain. 5. How can you tell which of the other records have a winning rate of or greater? Solve 6. What is the winning rate for a record of 31 wins and 29 losses? Does your team need more than a total of 31 wins? 7. Use guess and check to find the total number of wins your team needs in order to have a winning rate of How many of the remaining games must your team win? Look Back 9. Is your answer reasonable? Explain. 24

14 7 Skills Intervention: Comparing and Ordering Rational Numbers Rational numbers include integers, fractions, terminating decimals, and repeating decimals. To compare and order fractions, be sure all fractions have the same denominator. Comparing Fractions Compare the fractions using or and What is the LCM of 8 and 10? , so Write each fraction with denominators Since the are the same, compare the numerators. Which is greater? To compare and order decimals, line up the decimal points in each number and compare digits from left to right. Comparing Decimals Compare and using or Line up the points. Vocabulary rational number The tenths and are the same. Compare the. 4 So, To order fractions and decimals, convert all fractions to decimals. Ordering Fractions and Decimals Place the following numbers in order from least to greatest. 1 4, 0.2, 0.28 Convert each number to a decimal with the same number of decimal places Since the tenths place is the same in each number, compare the Which number is least? Which number is greatest? place. Place the numbers in order. 25

15 7 You can use number sense or convert to compare rational numbers. The table shows some records of 3 basketball players during the 2001 NBA playoffs. Which of the 3 players had the highest rate of making field goals? Who had the lowest? Understand the Problem 1. If a player tries 12 field goals and makes 7 of them, what is his rate of making field goals? Make a Plan Problem Solving Intervention: Comparing and Ordering Rational Numbers 2001 NBA Playoffs Player Field Goals Made Field Goals Tried Dikembe Mutombo Tim Duncan Shaquille O Neal When rates are written as fractions, do you have to convert them to decimals to compare? Explain with an example. 3. Write fractions to show the 3 rates you re comparing. Solve 4. Which of the rates is greater than 1? How can that help you? 2 5. Convert 1 02 and 1 20 to decimals. Which is greater? Which of the 3 players had the highest rate? Who had the lowest? Look Back 7. Are the decimals you found in Exercise 7 reasonable? Explain. 26

16 SECTION 2B SECTION B: Quiz for Lessons 6 Through 7 6 Equivalent Fractions and Decimals Write each fraction as a decimal. Round to the nearest hundredth, if necessary Write each decimal as a fraction in simplest form The weatherperson predicted 1 2 inch of rain. Since, 1 inch 25.4 millimeters, how many millimeters of rain is expected? 18. Brad Ausmus had a batting average of in the 2005 World Series. Out of 16 at bats, how many hits did he get? 27

17 SECTION 2B SECTION B: Quiz for Lessons 6 Through 7 continued 7 Comparing and Ordering Rational Numbers Compare the fractions. Write,, or Compare the decimals. Write,, or Order the numbers from least to greatest ; 1 6 ; , ; ; 1.8; ; 2.69; Density is used to separate and sort the minerals in a crushed rock sample. Arrange the approximate mineral densities in the table from least to greatest. Mineral quartz feldspar hornblende apatite pyrite Density (g/cc)

18 SECTION 2B The set of rational numbers includes all terminating and repeating decimals. Terminating decimals are easily converted to fractions, but repeating decimals are more difficult. Convert the repeating decimal to a fraction. Let x Then 1000x So 1000x x x 864 x Convert the repeating decimal 0.16 to a fraction. Let Then So SECTION B Enrichment: Rational Numbers x x x x x 1.5 x Write each repeating decimal as a fraction or mixed number Why can t irrational numbers such as be represented as fractions? 10. What is the exact product of ? 29