12.2 Adding Rational Numbers

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1 Rational Numbers. Rational Numbers. Adding Rational Numbers. Subtracting Rational Numbers. Multiplying and Dividing Rational Numbers On the count of, I m going to give you half of my dog biscuits.,,,,,,,... 7 three-digit range.

$What You Learned Before (MACC..NF..6) ess than pet s number... You go first. Example Write 0.7 as a fraction. Example Write as a decimal. 0.7 = 7 00 = = 0 = 0.$

2 What You Learned Before (MACC..NF..6) ess than pet s number... You go first. Example Write 0.7 as a fraction. Example Write as a decimal. 0.7 = 7 00 = = 0 = 0. Write the decimal as a fraction or the fraction as a decimal (MACC..NF..) Example Find +. Example Find 9. + = + 9 = = + = = = 6 (MACC..NF.., MACC.6.NS..) Example Find 6 = 6 = 6. Example 6 Find = 0 9 = 0 9 = 0 7 Multiply by the reciprocal of the divisor. Evaluate the expression

$. Rational Numbers rational numbers? How can you use a number line to order The word rational comes from the word ratio. Recall that you can write a ratio using fraction notation.$

3 . Rational Numbers rational numbers? How can you use a number line to order The word rational comes from the word ratio. Recall that you can write a ratio using fraction notation. If you sleep for hours in a day, then the ratio of your sleeping time to the total hours in a day can be written as Rational h h. A rational number is a number that can be written as the ratio of two integers. = = = 0. = ACTIVITY: Ordering Rational Numbers Work in groups of five. Order the numbers from least to greatest. Use masking tape and a marker to make a number line on the floor similar to the one shown Write the numbers on pieces of paper. Then each person should choose one. COMMON CORE Rational Numbers In this lesson, you will understand that a rational number is an integer divided by an integer. convert rational numbers to decimals. Learning Standards MACC.7.NS..b MACC.7.NS..d Stand on the location of your number on the number line. Use your positions to order the numbers from least to greatest. a. 0.,.,, 0., b. 7,.,, 0,. c..,, 9,, 0.9 d., 0.7,, 0.,. Chapter Rational Numbers

ACTIVITY: The Game of Math Card War Math Practice Consider Similar Problems What are some ways to determine which number is greater? Preparation: Cut index cards to make 0 playing cards.

4 ACTIVITY: The Game of Math Card War Math Practice Consider Similar Problems What are some ways to determine which number is greater? Preparation: Cut index cards to make 0 playing cards. Write each number in the table on a card. To Play: Play with a partner. Deal 0 cards to each player facedown. Each player turns one card faceup. The player with the greater number wins. The winner collects both cards and places them at the bottom of his or her cards. Suppose there is a tie. Each player lays three cards facedown, then a new card faceup. The player with the greater of these new cards wins. The winner collects all ten cards and places them at the bottom of his or her cards. Continue playing until one player has all the cards. This player wins the game IN YOUR OWN WORDS How can you use a number line to order rational numbers? Give an example. The numbers are in order from least to greatest. Fill in the blank spaces with rational numbers..,,,, 7,.,,.9,,, 6.,, 0.,,, 7..,,.,,., Use what you learned about ordering rational numbers to complete Exercises 0 on page. Section. Rational Numbers 9

5 . Lesson Lesson Tutorials Key Vocabulary rational number, p. 0 terminating decimal, p. 0 repeating decimal, p. 0 Rational Numbers A rational number is a number that can be written as a where a and b are b integers and b 0. Rational Numbers. Integers 0 0 Whole Numbers. Because you can divide any integer by any nonzero integer, you can use long division to write fractions and mixed numbers as decimals. These decimals are also rational numbers and will either terminate or repeat. A terminating decimal is a decimal that ends.., 0., 0.6 A repeating decimal is a decimal that has a pattern that repeats..... = = 0. Use bar notation to show which of the digits repeat. EXAMPLE Writing Rational Numbers as Decimals a. Write as a decimal. b. Write as a decimal. Notice that = 9. Divide by. 0. ).0000 Divide 9 by.. ) The remainder is 0. So, it 0 The remainder repeats. So, is a terminating decimal. 0 it is a repeating decimal. So, =.. So, = 0.. Exercises Write the rational number as a decimal Chapter Rational Numbers

6 EXAMPLE Writing a Decimal as a Fraction Write 0.6 as a fraction in simplest form. Study Tip If p and q are integers, then p q = p q = p q. 0.6 = 6 00 = 0 Write the digits after the decimal point in the numerator. The last digit is in the hundredths place. So, use 00 in the denominator. Simplify. Exercises 0 7 Write the decimal as a fraction or a mixed number in simplest form EXAMPLE Ordering Rational Numbers Creature Elevation (kilometers) The table shows the elevations of four sea creatures relative to sea level. Which of the sea creatures are deeper than the whale? Explain. Anglerfish 0 Squid Shark Whale 0. Write each rational number as a decimal. 0 =. =. = 0. Then graph each decimal on a number line.. Squid..0.6 Anglerfish. Whale 0. Shark Elevation (kilometers) Both. and. are less than 0.. So, the squid and the anglerfish are deeper than the whale. Exercises 9. WHAT IF? The elevation of a dolphin is kilometer. Which of the 0 sea creatures in Example are deeper than the dolphin? Explain. Section. Rational Numbers

7 . Exercises Help with Homework. VOCABULARY Is the quotient of two integers always a rational number? Explain.. WRITING Are all terminating and repeating decimals rational numbers? Explain. Tell whether the number belongs to each of the following number sets: rational numbers, integers, whole numbers Tell whether the decimal is terminating or repeating (-6)= +(-)= +(-9)= 9+(-)= Write the rational number as a decimal ERROR ANALYSIS Describe and correct the error in writing the rational number as a decimal. 7 = 0.6 Write the decimal as a fraction or a mixed number in simplest form Order the numbers from least to greatest.., 0.,, 7,. 9. 9,.,.,, ,, 0.6, 0.9,.., 6 0, 9, 0.7,. 7,.,,,..,.,.6, 0,.. COINS You lose one quarter, two dimes, and two nickels. a. Write the amount as a decimal. b. Write the amount as a fraction in simplest form.. HIBERNATION A box turtle hibernates in sand at feet. A spotted turtle hibernates at 6 feet. Which turtle is deeper? Chapter Rational Numbers

8 Copy and complete the statement using <, >, or = OPEN-ENDED Find one terminating decimal and one repeating decimal between and. Player Hits At Bats Eva 90 Michelle 0. SOFTBALL In softball, a batting average is the number of hits divided by the number of times at bat. Does Eva or Michelle have the higher batting average?. PROBLEM SOLVING You miss out of 0 questions on a science quiz and out of questions on a math quiz. Which quiz has a higher percent of correct answers?. SKATING Is the half pipe deeper than the skating pool? Explain. Skating pool Lip Base 0 ft Half pipe Lip 9 ft 6 Base 6. ENVIRONMENT The table shows the changes from the average water level of a pond over several weeks. Order the numbers from least to greatest. Week Change (inches) Given: a and b are integers. a. When is a positive? b. When is ab positive? Add or subtract. (Section.6 and Section.) MULTIPLE CHOICE In one year, a company has a profit of $ million. In the next year, the company has a profit of $7 million. How much more profit did the company make the second year? (Section.) A $ million B $ million C $7 million D $9 million Section. Rational Numbers

9 . Adding Rational Numbers How can you use what you know about adding integers to add rational numbers? ACTIVITY: Adding Rational Numbers Work with a partner. Use a number line to find the sum. a..7 + (.) Start at 0. Move.7 units to the right. Add...7 Then move. units left to end at So,.7 + (.) =. b.. + (.) c d. + e. 0 ( + 0) ACTIVITY: Adding Rational Numbers COMMON CORE Rational Numbers In this lesson, you will add rational numbers. solve real-life problems. Learning Standards MACC.7.NS..a MACC.7.NS..b MACC.7.NS..d MACC.7.NS.. Work with a partner. Use a number line to find the sum. a. ( + ) Then move unit left to end at. Add. So, ( + ) =. Start at 0. Move units to the left. 0 b. 7 0 ( + 7 0) c. ( + d (.9) e.. + ( 0.6) ) Chapter Rational Numbers

10 ACTIVITY: Writing Expressions Math Practice Use Operations What operation is represented in each number line? How does this help you write an expression? Work with a partner. Write the addition expression shown. Then find the sum. a. b. 7 Then move units 0 right to end at. Start at 0. Move. units to the right. 0 7 Add. 0 Add... Then move. units left to end at Start at 0. Move 0 units to the left. 0 c. Then move 0.7 unit left to end at. Add Start at 0. Move. units to the left. 0. IN YOUR OWN WORDS How can you use what you know about adding integers to add rational numbers? PUZZLE Find a path through the table so that the numbers add up to the sum. You can move horizontally or vertically.. Sum: 6. Sum: 0.07 Start 7 Start End End Use what you learned about adding rational numbers to complete Exercises 6 on page. Section. Adding Rational Numbers

11 . Lesson Lesson Tutorials Adding Rational Numbers Words To add rational numbers, use the same rules for signs as you used for integers. Numbers + 6 = = + 6 = 6 = 6 Study Tip EXAMPLE In Example, notice how is written as = = 6 6. Adding Rational Numbers Find +. Estimate + = = = = 6 = 6 Rewrite using the LCD (least common denominator). Write the sum of the numerators over the common denominator. Add. Write the improper fraction as a mixed number. The sum is 6. Reasonable? 6 EXAMPLE Adding Rational Numbers Find = >.0. So, subtract.0 from 7.6. Use the sign of 7.6. The sum is.7. Exercises Add ( 7 ) (.) (.7) 6 Chapter Rational Numbers

12 EXAMPLE Evaluating Expressions Evaluate x + y when x = and y =. x + y = ( ) ( + ) Substitute for x and for y. = ( + ) Multiply. = + ( ) = Simplify. Write the sum of the numerators over the common denominator. EXAMPLE Year Profit (billions of dollars) Real-Life Application The table shows the annual profits (in billions of dollars) of a financial company from 00 to 0. Positive numbers represent gains, and negative numbers represent losses. Which statement describes the profit over the five-year period? A gain of $0. billion B gain of $0 million C loss of $ million D loss of $00 million To determine whether there was a gain or a loss, find the sum of the profits. five-year profit =.7 + (.7) Write the sum. = (.7) Comm. Prop. of Add. = 0 + (.7) Additive Inv. Prop. = Add. Prop. of Zero = Add.7 and 0.. = 0. Add.9 and.6. The five-year profit is $0. billion. So, the company has a five-year loss of $0. billion, or $00 million. The correct answer is D. Exercises 7 Evaluate the expression when a = and b =. 7. b + a. a + b 9. WHAT IF? In Example, the 0 profit is $.07 billion. State the company s gain or loss over the six-year period in millions of dollars. Section. Adding Rational Numbers 7

$. Exercises Help with Homework. WRITING Explain how to find the sum.6 +... OPEN-ENDED Write an addition expression using fractions that equals.. DIFFERENT WORDS, SAME QUESTION Which is different?$

13 . Exercises Help with Homework. WRITING Explain how to find the sum OPEN-ENDED Write an addition expression using fractions that equals.. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Add. and.. What is the distance between. and.? What is. increased by.? Find the sum of. and.. 9+(-6)= +(-)= +(-9)= 9+(-)= Add. Write fractions in simplest form.. ( + ) 7. ( ) ( ) 9. ( + ) ( 0.).. + ( 0.66) (.7) ERROR ANALYSIS Describe and correct the error in finding the sum ( 0.) + = + = 6 = 0.6 Evaluate the expression when x = and y = 7.. x + y 6. x + y 7. x + y. BANKING Your bank account balance is $0.. You deposit $.0. What is your new balance? 9. HOT DOGS You eat of a pack of hot dogs. 0 Your friend eats of the pack of hot dogs. What fraction of the pack of hot dogs do you and your friend eat? Chapter Rational Numbers

$Add. Write fractions in simplest form. 0. 6 + ( ) ( + ). + ( + 7 ). 0.9 + (.6) +.. NUMBER SENSE When is the sum of two negative mixed numbers an integer?$

14 Add. Write fractions in simplest form ( ) ( + ). + ( + 7 ) (.6) +.. NUMBER SENSE When is the sum of two negative mixed numbers an integer?. WRITING You are adding two rational numbers with different signs. How can you tell if the sum will be positive, negative, or zero? June July August 9 6. RESERVOIR The table at the left shows the water level (in inches) of a reservoir for three months compared to the yearly average. Is the water level for the three-month period greater than or less than the yearly average? Explain. 6. BREAK EVEN The table at the right shows the annual profits (in thousands of dollars) of a county fair from 00 to 0. What must the 0 profit be (in hundreds of dollars) to break even over the five-year period? 7. REASONING Is a + b = a + b for all rational numbers a and b? Explain. Year Profit (thousands of dollars) ?. Repeated Reasoning Evaluate the expression. 9 0 ( + 0 ) ( ) ( 0 ) + 0 ( + 0 ) + 0 Identify the property. Then simplify. (Section.) 9. + ( ) + = + + ( ) 0. (. 9) = (.) 9. + ( + ) = ( + ) = 7 7. MULTIPLE CHOICE The regular price of a photo album is $. You have a coupon for % off. How much is the discount? (Section.6) A $.70 B $ C $ D $.0 Section. Adding Rational Numbers 9

15 Study Help Graphic Organizer You can use a process diagram to show the steps involved in a procedure. Here is an example of a process diagram for adding rational numbers. Adding rational numbers with the same sign Add the absolute values of the rational numbers. Write the sum by using the common sign. Example. + ( 6.9) Because the numbers have the same sign, add. and ( 6.9) =. Use the common sign. with different signs Subtract the lesser absolute value from the greater absolute value. Write the sum by using the sign of the rational number with the greater absolute value. Example Because the numbers have different signs, subtract. from =. Use the sign of 6.9. Make a process diagram with examples to help you study the topic.. writing rational numbers as decimals After you complete this chapter, make process diagrams with examples for the following topics.. subtracting rational numbers. multiplying rational numbers. dividing rational numbers Does this process diagram accurately show how a cat claws furniture? 0 Chapter Rational Numbers

$.. Quiz Progress Check Write the rational number as a decimal. (Section.). 0. 6 Write the decimal as a fraction or a mixed number in simplest form. (Section.). 0... Order the numbers from least to greatest.$ $Your friend 7 drinks of the pitcher. What fraction of the pitcher do you and your friend drink? (Section.) 7. FOOTBALL The table shows the statistics of a running back in a football game.$

16 .. Quiz Progress Check Write the rational number as a decimal. (Section.) Write the decimal as a fraction or a mixed number in simplest form. (Section.) Order the numbers from least to greatest. (Section.)., 0.,, 0.,. 6.,., 0., 9, 0. Add. Write fractions in simplest form. (Section.) ) 7. + ( (.6) Evaluate the expression when x = and y =. (Section.). x + y. x + y. x + y. x + y. STOCK The value of Stock A changes $.6, and the value of Stock B changes $.7. Which stock has the greater loss? Explain. (Section.) 6. LEMONADE You drink of a pitcher of lemonade. Your friend 7 drinks of the pitcher. What fraction of the pitcher do you and your friend drink? (Section.) 7. FOOTBALL The table shows the statistics of a running back in a football game. Did he gain more than 0 yards total? Explain. (Section.) Quarter Total Yards? Sections.. Quiz

17 . Subtracting Rational Numbers How can you use what you know about subtracting integers to subtract rational numbers? ACTIVITY: Subtracting Rational Numbers Work with a partner. Use a number line to find the difference. a. Then move unit left to end at. Subtract. Start at 0. Move units to the left. 0 b. So, = c. d e ACTIVITY: Finding Distances on a Number Line Work with a partner. a. Plot and on the number line. Then find and ( ). What do you notice about your results? COMMON CORE Rational Numbers In this lesson, you will subtract rational numbers. solve real-life problems. Learning Standards MACC.7.NS..c MACC.7.NS..d MACC.7.NS b. Plot and on the number line. Then find and. What do you notice about your results? 0 c. Choose any two points a and b on a number line. Find the values of a b and b a. What do the absolute values of these differences represent? Is this true for any pair of rational numbers? Explain. Chapter Rational Numbers

18 ACTIVITY: Financial Literacy Work with a partner. The table shows the balance in a checkbook. Black numbers are amounts added to the account. Red numbers are amounts taken from the account. Math Practice Interpret Results What does your answer represent? Does your answer make sense? Date Check # Transaction Amount Balance Previous balance /0/0 Groceries.7 /07/0 Check deposit 7.0 //0 ATM withdrawal 0.00 //0 Electric company 7. /7/0 Music store 0. //0 6 Shoes 7. //0 Check deposit.00 //0 Interest. //0 7 Cell phone 9.99 /6/0 Clothes 6. /0/0 9 Cable company 7.00 You can find the balance in the second row two different ways = 6. Subtract.7 from (.7) = 6. Add.7 to a. Copy the table. Then complete the balance column. b. How did you find the balance in the twelfth row? c. Use a different way to find the balance in part (b).. IN YOUR OWN WORDS How can you use what you know about subtracting integers to subtract rational numbers?. Give two real-life examples of subtracting rational numbers that are not integers. Use what you learned about subtracting rational numbers to complete Exercises on page 6. Section. Subtracting Rational Numbers

19 . Lesson Lesson Tutorials Subtracting Rational Numbers Words To subtract rational numbers, use the same rules for signs as you used for integers. Numbers ( ) = + = + = EXAMPLE Subtracting Rational Numbers Find 7 ( 6 7 ). Estimate ( ) = 7 ( 6 7) = = = = 7 = 7 Add the opposite of 6 7. Write the mixed number as an improper fraction. Write the sum of the numerators over the common denominator. Add. Write the improper fraction as a mixed number. EXAMPLE The difference is 7. Reasonable? 7 Subtracting Rational Numbers Find =. + (.6) Add the opposite of.6. =..6 >.. So, subtract. from.6. The difference is.. Use the sign of.6. Exercises. ( ) ( 0.) ( 0.07) Chapter Rational Numbers

20 The distance between any two numbers on a number line is the absolute value of the difference of the numbers. EXAMPLE Finding Distances Between Numbers on a Number Line 0 Find the distance between the two numbers on the number line. To find the distance between the numbers, first find the difference of the numbers. ) Add the opposite of = ( +. = ( + 7 Write the mixed numbers as improper fractions. ) = Add. = Simplify. Because =, the distance between and is. EXAMPLE Clearance: ft in. Real-Life Application In the water, the bottom of a boat is. feet below the surface, and the top of the boat is.7 feet above it. Towed on a trailer, the bottom of the boat is. feet above the ground. Can the boat and trailer pass under the bridge? Step : Find the height h of the boat. h =.7 (.) Subtract the lowest point from the highest point. =.7 +. Add the opposite of.. = 0. Add. Step : Find the height t of the boat and trailer. t = =. Add. Add the trailer height to the boat height. Because. feet is greater than feet inches, the boat and trailer cannot pass under the bridge. Exercises 7. Find the distance between 7. and. on a number line.. WHAT IF? In Example, the clearance is feet inch. Can the boat and trailer pass under the bridge? Section. Subtracting Rational Numbers

21 . Exercises Help with Homework. WRITING Explain how to find the difference.. WHICH ONE DOESN T BELONG? Which expression does not belong with the other three? Explain your reasoning. + ( + ) 9+(-6)= +(-)= +(-9)= 9+(-)= Subtract. Write fractions in simplest form.. ( 7 ) ( 9) (.) (.9). ERROR ANALYSIS Describe and correct the error in finding the difference. 9 = 9 = 6 = Find the distance between the two numbers on a number line..,..,.. 7, 6. SPORTS DRINK Your sports drink bottle is full. After practice, the bottle is 6 full. Write the difference of the amounts after practice and before practice. 7. SUBMARINE The figure shows the depths of a submarine. a. Find the vertical distance traveled by the submarine. b. Find the mean hourly vertical distance traveled by the submarine ft (now) 7.6 ft ( hours ago) Evaluate.. 6 ( ) ( + 7 9) ( 7.) (.) ( ) 6 Chapter Rational Numbers

22 . REASONING When is the difference of two decimals an integer? Explain.. RECIPE A cook has cups of flour. A recipe calls for cups of flour. Does the cook have enough flour? If not, how much more flour is needed? mi Springville mi 6 new road Uniontown. ROADWAY A new road that connects Uniontown to Springville is miles long. What is the change in distance when using the new road instead of the dirt roads? RAINFALL In Exercises 6, the bar graph shows the differences in a city s rainfall from the historical average.. What is the difference in Monthly Rainfall rainfall between the wettest and the driest months?.0.0 Historical Average.6. Find the sum of the differences for the year. 6. What does the sum in Exercise tell you about the rainfall for the year? Rainfall (inches) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month 7. OPEN-ENDED Write two different pairs of negative decimals, x and y, that make the statement x y = 0.6 true. REASONING Tell whether the difference between the two numbers is always, sometimes, or never positive. Explain your reasoning.. two negative fractions 9. a positive decimal and a negative decimal 0. Fill in the blanks to make the solution correct.. (. ) =.6 Evaluate. (Section., Section., Section., and Section.6) MULTIPLE CHOICE A sports store has 6 soccer balls. Over 6 months, it sells soccer balls per month. How many soccer balls are in inventory at the end of the 6 months? (Section. and Section.) A B C 6 D 0 Section. Subtracting Rational Numbers 7

23 . Multiplying and Dividing Rational Numbers Why is the product of two negative rational numbers positive? In Section., you used a table to see that the product of two negative integers is a positive integer. In this activity, you will find that same result another way. ACTIVITY: Showing ( )( ) = Work with a partner. How can you show that ( )( ) =? To begin, assume that ( )( ) = is a true statement. From the Additive Inverse Property, you know that + ( ) = 0. So, substitute ( )( ) for to get ( )( ) + ( ) = 0. If you can show that ( )( ) + ( ) = 0 is true, then you have shown that ( )( ) =. Justify each step. ( )( ) + ( ) = ( )( ) + ( ) So, ( )( ) =. = ( )[( ) + ] = ( )0 = 0 ACTIVITY: Multiplying by Work with a partner. a. Graph each number below on three different number lines. Then multiply each number by and graph the product on the appropriate number line. COMMON CORE Rational Numbers In this lesson, you will multiply and divide rational numbers. solve real-life problems. Learning Standards MACC.7.NS..a MACC.7.NS..b MACC.7.NS..c MACC.7.NS.. b. How does multiplying by change the location of the points in part (a)? What is the relationship between the number and the product? c. Graph each number below on three different number lines. Where do you think the points will be after multiplying by? Plot the points. Explain your reasoning.. d. What is the relationship between a rational number a and the product (a)? Explain your reasoning. Chapter Rational Numbers

24 ACTIVITY: Understanding the Product of Rational Numbers Work with a partner. Let a and b be positive rational numbers. a. Because a and b are positive, what do you know about a and b? b. Justify each step. ( a)( b) = ( )(a)( )(b) = ( )( )(a)(b) = ()(a)(b) = ab c. Because a and b are positive, what do you know about the product ab? d. What does this tell you about products of rational numbers? Explain. ACTIVITY: Writing a Story Math Practice Specify Units What units are in your story? Work with a partner. Write a story that uses addition, subtraction, multiplication, or division of rational numbers. At least one of the numbers in the story has to be negative and not an integer. Draw pictures to help illustrate what is happening in the story. Include the solution of the problem in the story. If you are having trouble thinking of a story, here are some common uses of negative numbers: A profit of $ is a loss of $. An elevation of 00 feet is a depth of 00 feet below sea level. A gain of yards in football is a loss of yards. A score of in golf is strokes under par.. IN YOUR OWN WORDS Why is the product of two negative rational numbers positive? 6. PRECISION Show that ( )( ) = How can you show that the product of a negative rational number and a positive rational number is negative? Use what you learned about multiplying rational numbers to complete Exercises 7 9 on page. Section. Multiplying and Dividing Rational Numbers 9

25 . Lesson Lesson Tutorials Remember The reciprocal of a b is b a. Multiplying and Dividing Rational Numbers Words To multiply or divide rational numbers, use the same rules for signs as you used for integers. Numbers 7 = 7 = = 9 = 9 = 9 = 9 = 9 EXAMPLE Dividing Rational Numbers Find. Estimate = = 6 7 = 6 7 = 6 7 = 7, or Write mixed numbers as improper fractions. Multiply by the reciprocal of 7. Multiply the numerators and the denominators. Simplify. The quotient is. Reasonable? EXAMPLE Multiplying Rational Numbers Find The decimals have different signs. The product is negative. The product is 9. 0 Chapter Rational Numbers

26 EXAMPLE Multiplying More Than Two Rational Numbers Find 7 [ ] ( 7). You can use properties of multiplication to make the product easier to find. 7 [ ] ( 7) = 7 ( 7 Commutative Property ) of Multiplication = 7 ( 7) = = Associative Property of Multiplication Multiplicative Inverse Property Multiplication Property of One The product is. Exercises 0 0 Multiply or divide. Write fractions in simplest form.. 6 ( ). ) (. 6.( 0.6) (.) ( 00) EXAMPLE Real-Life Application Account Positions Stock Original Value Current Value Change A B C An investor owns Stocks A, B, and C. What is the mean change in the value of the stocks? ( 0.) mean = =.6 =. The mean change in the value of the stocks is $.. 7. WHAT IF? The change in the value of Stock D is $6.. What is the mean change in the value of the four stocks? Section. Multiplying and Dividing Rational Numbers

$Tell whether the expression is positive or negative without evaluating.. 0 ( ). ( ). 6.. 6..6.7 9+(-6)= +(-)= +(-9)= 9+(-)= Multiply. 7. ( ) (. ) Divide. Write fractions in simplest form. 9. 0.( ) 0.$

27 . Exercises Help with Homework. WRITING How is multiplying and dividing rational numbers similar to multiplying and dividing integers?. NUMBER SENSE Find the reciprocal of. Tell whether the expression is positive or negative without evaluating.. 0 ( ). ( ) (-6)= +(-)= +(-9)= 9+(-)= Multiply. 7. ( ) (. ) Divide. Write fractions in simplest form ( ) (. ( 7). 0 7 ( ) ). 9 ( 9) ( ) (.6).. (.) Multiply. Write fractions in simplest form.. ( ). 6( ) (. ) ( 0.0) ( 0.) 0) (. (0.09)( 0.) 9. 6 ( ERROR ANALYSIS Describe and correct the error....7 =. ) ( ) ( 0. ). = = = 6. HOUR HAND The hour hand of a clock moves 0 every hour. How many degrees does it move in hours?. SUNFLOWER SEEDS How many 0.7-pound packages can you make with 6 pounds of sunflower seeds? 0 Chapter Rational Numbers

28 Evaluate (.9) ( ) ( + ) OPEN-ENDED Write two fractions whose product is. 0 yd 0 yd 9 0. ( ) ( ). FENCING A farmer needs to enclose two adjacent rectangular pastures. How much fencing does the farmer need?. GASOLINE A.-gallon gasoline tank is full. How many gallons will it take to fill the tank?. PRECISION A section of a boardwalk is made using boards. Each board is 9 inches wide. The total width of the section is inches. The spacing between each board is equal. What is the width of the spacing between each board?. RUNNING The table shows the changes in the times (in seconds) of four teammates. What is the mean change? 6. The daily changes in the barometric pressure for four days are 0.0, 0.09, 0.0, and 0.0 inches. a. What is the mean change? b. The mean change after five days is 0.0 inch. What is the change on the fifth day? Explain. Teammate Change Add or subtract. (Section. and Section.) (.7) 9. 9 ( 7 0) 0. 6 ( +. MULTIPLE CHOICE What are the coordinates of the point in Quadrant IV? (Section 6.) A (, ) B (, ) C C (0, ) D (, ) y A D O x B F E 9) Section. Multiplying and Dividing Rational Numbers

29 .. Quiz Progress Check Subtract. Write fractions in simplest form. (Section.). 7 ( 6 7). 7 ( ( 9.) 9) Find the distance between the two numbers on the number line. (Section.) Divide. Write fractions in simplest form. (Section.) 7. ( 6). 9 ( (.) Multiply. Write fractions in simplest form. (Section.). ( ).. 9. (.7). 00( 0.6)(0.0). PARASAILING A parasail is at 00.6 feet above the water. After minutes, the parasail is at 0. feet above the water. What is the change in height of the parasail? (Section.) 6. TEMPERATURE Use the thermometer shown. How much did the temperature drop from :00 p.m. to 0:00 p.m.? (Section.) 7. LATE FEES You were overcharged $. on your cell phone bill months in a row. The cell phone company says that it will add $. to your next bill for each month you were overcharged. On the next bill, you see an adjustment of.. Is this amount correct? Explain. (Section.) 7) 0 :00 P.M. 7. F 0:00 P.M F. CASHEWS How many -pound packages can you make with 7 pounds of cashews? (Section.) 0 Chapter Rational Numbers

30 Chapter Review Review Key Vocabulary rational number, p. 0 terminating decimal, p. 0 repeating decimal, p. 0 Vocabulary Help Review Examples and Exercises. Rational Numbers (pp. ) a. Write as a decimal. Notice that =. Divide by. The remainder is 0. So, it is a terminating decimal So, =.6. b. Write 0. as a fraction in simplest form. 0. = 00 = 7 0 Write the digits after the decimal point in the numerator. The last digit is in the hundredths place. So, use 00 in the denominator. Simplify. Write the rational number as a decimal Write the decimal as a fraction or a mixed number in simplest form Chapter Review

31 . Adding Rational Numbers (pp. 9) Find = + = + = 9 = Rewrite using the LCD (least common denominator). Write the sum of the numerators over the common denominator. Add. Write the improper fraction as a mixed number. The sum is. Add. Write fractions in simplest form ( + ) (.). Subtracting Rational Numbers (pp. 7) Find ( ( ). ) = + = + = + = 9, or Add the opposite of. Write the mixed number as an improper fraction. Write the sum of the numerators over the common denominator. Simplify. The difference is. Subtract. Write fractions in simplest form ( 7.). TURTLE A turtle is 0 inches below the surface of a pond. It dives to a 6 depth of inches. What is the change in the turtle s position? 6 Chapter Rational Numbers

$. Multiplying and Dividing Rational Numbers (pp. ) a. Find 6. 6 = 6 = 6 = 6 =, or Write mixed numbers as improper fractions. Multiply by the reciprocal of.$

32 . Multiplying and Dividing Rational Numbers (pp. ) a. Find 6. 6 = 6 = 6 = 6 =, or Write mixed numbers as improper fractions. Multiply by the reciprocal of. Multiply the numerators and the denominators. Simplify. The quotient is. b. Find The decimals have different signs The product is negative. The product is.. Divide. Write fractions in simplest form ( 6 ) (.) 9.. (.) Multiply. Write fractions in simplest form ( 7 9). ( )..9( 9.7)..(.6). ( ) ( )..6 (0.) ( 0) 6. SUNKEN SHIP The elevation of a sunken ship is 0 feet. Your elevation is of the ship s elevation. What is your elevation? Chapter Review 7

$Chapter Test Write the rational number as a decimal.. 7 0. 9. 6 Test Practice. 6 Write the decimal as a fraction or a mixed number in simplest form.. 0. 6. 0. 7... 7.09 Add or subtract.$

33 Chapter Test Write the rational number as a decimal Test Practice. 6 Write the decimal as a fraction or a mixed number in simplest form Add or subtract. Write fractions in simplest form ( ) ( (.)..6. Multiply or divide. Write fractions in simplest form ( ) ( 6.0) 6.. ) 7. ( 7) ( ). 6 ( 0.0) ( 0.) 9. ALMONDS How many.-pound containers can you make with.7 pounds of almonds? 0. FISH The elevation of a fish is 7 feet. a. The fish decreases its elevation by feet, and then increases its elevation by feet. What is its new elevation? b. Your elevation is of the fish s new elevation. What is your elevation?. RAINFALL The table shows the rainfall (in inches) for three months compared to the yearly average. Is the total rainfall for the three-month period greater than or less than the yearly average? Explain. November December January BANK ACCOUNTS Bank Account A has $70.9, and Bank Account B has $67.. Account A changes by $6., and Account B changes by $6.9. Which account has the greater balance? Explain. Chapter Rational Numbers

34 Standards Assessment. When José and Sean were each years old, José was inches taller than Sean. José grew at an average rate of inches per year from the time that he was years old until the time he was years old. José was 6 inches tall when he was years old. How tall was Sean when he was years old? (MACC.7.NS..) Test-Taking Strategy Estimate the Answer A. 9 in. C. in. B. in. D. 7 in.. Which expression represents a positive integer? (MACC.7.NS..a) Using estimation you can see that there are about 0 tabbies. So about 0 are not tabbies. F. 6 H. ( ) G. ( ) I.. What is the missing number in the sequence below? (MACC.7.NS..a) 9 6, 9, 9, 9, 9,. What is the value of the expression below? (MACC.7.NS..c) (.) A.. C. 0. B. 0. D... What is the distance between the two numbers on the number line? (MACC.7.NS..c) 7 0 F. G. H. I. Standards Assessment 9

What is the value of the expression below when q =, r =, and s =? (MACC.7.NS..) q r s F. H. G. I.

35 6. Sandra was evaluating an expression in the box below. = 9 = 9 = 9 = 0 09 What should Sandra do to correct the error that she made? (MACC.7.NS..) A. Rewrite 9 as 9 and multiply by. B. Rewrite as and multiply by 9. C. Rewrite as D. Rewrite and multiply by 9. as and multiply by. 7. What is the value of the expression below when q =, r =, and s =? (MACC.7.NS..) q r s F. H. G. I.. You are stacking wooden blocks with the dimensions shown below. How many blocks do you need to stack to build a block tower that is 7 inches tall? (MACC.7.NS..) in. in. in. 0 Chapter Rational Numbers

36 9. What is the area of a triangle with a base length of inches and a height of inches? (MACC.7.NS..c) A. in. C. in. B. in. D. in. 0. What is the value of the expression below? (MACC.7.NS..) ( ) F. 6 H. G. I. 6. Four points are graphed on the number line below. (MACC.7.NS..) R S T U 0 Part A Part B Part C Part D Choose the two points whose values have the greatest sum. Approximate this sum. Explain your reasoning. Choose the two points whose values have the greatest difference. Approximate this difference. Explain your reasoning. Choose the two points whose values have the greatest product. Approximate this product. Explain your reasoning. Choose the two points whose values have the greatest quotient. Approximate this quotient. Explain your reasoning.. What number belongs in the box to make the equation true? (MACC.7.NS..) =. A. C. 0. B. 0. D. Standards Assessment

Work with a partner. How can you show that ( 1)( 1) = 1?

Work with a partner. How can you show that ( 1)( 1) = 1? . Multiplying and Dividing Rational Numbers numbers positive? Why is the product of two negative rational In Section., you used a table to see that the product of two negative integers is a positive integer.