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

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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. In this activity, you will find that same result another way. ACTIVITY: Showing ( )( ) = Work with a partner. How can you show that ( )( ) =? From the Additive Inverse Property, you know that + ( ) = 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. Rational Numbers In this lesson, you will multiply and divide rational numbers. solve real-life problems. 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. 6 Chapter Rational Numbers

2 3 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 ( )( 3) = 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 6. Section. Multiplying and Dividing Rational Numbers 6

3 . 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 3 = 7 3 = = 9 = 9 = 9 = 9 = 9 EXAMPLE Dividing Rational Numbers Find 3. Estimate = 3 = = = = 7 3, or 3 Write mixed numbers as improper fractions. Multiply by the reciprocal of 7 3. Multiply the numerators and the denominators. Simplify. The quotient is 3. Reasonable? 3 EXAMPLE Multiplying Rational Numbers Find The decimals have different signs. The product is negative. The product is Chapter Rational Numbers

4 EXAMPLE 3 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 30 Multiply or divide. Write fractions in simplest form.. 6 ( ). 3) 3 (. 6.3( 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 = 3 = =. 3 The mean change in the value of the stocks is $.. 7. WHAT IF? The change in the value of Stock D is $6.3. What is the mean change in the value of the four stocks? Section. Multiplying and Dividing Rational Numbers 67

5 . 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 ( ). ( ) (-6)=3 3+(-3)= +(-9)= 9+(-)= Multiply. 7. ( ) (. 3 ) Divide. Write fractions in simplest form ( ) ( 3. ( 7). 0 7 ( ) ). 9 ( 9) ( ) ( 3.6).. (.) 3 Multiply. Write fractions in simplest form.. ( 3) 3. 6( ) (. ) ( 0.03) ( 0.) 0) 3 (. (0.09)( 0.) 9. 6 ( ERROR ANALYSIS Describe and correct the error =. ) ( ) ( 30. 3) = 3 = = HOUR HAND The hour hand of a clock moves 30 every hour. How many degrees does it move in hours? 3. SUNFLOWER SEEDS How many 0.7-pound packages can you make with 6 pounds of sunflower seeds? 30 6 Chapter Rational Numbers

6 Evaluate (.9) ( 3) ( + 3 ) OPEN-ENDED Write two fractions whose product is 3. 0 yd 30 yd 9 0. ( 3) 3 ( 3). FENCING A farmer needs to enclose two adjacent rectangular pastures. How much fencing does the farmer need? 3. GASOLINE A.-gallon gasoline tank is 3 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.3) (.7) 9. 9 ( 7 0) 0. 6 ( + 3. MULTIPLE CHOICE What are the coordinates of the point in Quadrant IV? (Skills Review Handbook) A (, ) B ( 3, 3) C C (0, ) D (3, 3) y A 3 D 3 O 3 x B F 3 E 9) Section. Multiplying and Dividing Rational Numbers 69

7 .3. Quiz Progress Check Subtract. Write fractions in simplest form. (Section.3). 7 ( 6 7). 7 ( ( 9.) 9) Find the distance between the two numbers on the number line. (Section.3) Divide. Write fractions in simplest form. (Section.) 7. 3 ( 6). 9 ( (.) Multiply. Write fractions in simplest form. (Section.). ( ) (.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.3) 6. TEMPERATURE Use the thermometer shown. How much did the temperature drop from :00 p.m. to 0:00 p.m.? (Section.3) 7. LATE FEES You were overcharged $. on your cell phone bill 3 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 3.. 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 70 Chapter Rational Numbers

8 Chapter Review Review Key Vocabulary rational number, p. 6 terminating decimal, p. 6 repeating decimal, p. 6 Vocabulary Help Review Examples and Exercises. Rational Numbers (pp. 9) a. Write 3 as a decimal. Notice that 3 = 3. Divide 3 by. The remainder is 0. So, it is a terminating decimal So, 3 =.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 7

9 . Adding Rational Numbers (pp. 0 ) 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 ( + ) (.).3 Subtracting Rational Numbers (pp. 63) Find ( 3 ( 3 ). ) = + 3 = + 3 = + 3 = 9, or 3 Add the opposite of 3. Write the mixed number as an improper fraction. Write the sum of the numerators over the common denominator. Simplify. The difference is 3. 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 3 inches. What is the change in the turtle s position? 7 Chapter Rational Numbers

10 . Multiplying and Dividing Rational Numbers (pp. 6 69) a. Find = 6 3 = 6 3 = 3 6 =, or 3 Write mixed numbers as improper fractions. Multiply by the reciprocal of 3. Multiply the numerators and the denominators. Simplify. The quotient is 3. b. Find The product is 3.. The decimals have different signs. The product is negative. Divide. Write fractions in simplest form ( 6 ) ( 3.) 9.. (.) Multiply. Write fractions in simplest form ( 7 9). ( 3)..9( 9.7) 3..(.6). 3 ( ) ( 3 )..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 73

11 Chapter Test Test Practice Write the rational number as a decimal 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 ( 3) ( 6.0) 6.. ) 7. 3 ( 7) ( 3 3 ). 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 3 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 $6.3, and Account B changes by $6.9. Which account has the greater balance? Explain. 7 Chapter Rational Numbers

12 Cumulative Assessment. When José and Sean were each years old, José was inches taller than Sean. José grew at an average rate of 3 inches per year from the time that he was years old until the time he was 3 years old. José was 63 inches tall when he was 3 years old. How tall was Sean when he was years old? Test-Taking Strategy Estimate the Answer A. 39 in. C. 3 in. B. in. D. 7 3 in.. Which expression represents a positive integer? Using estimation you can see that there are about 0 tabbies. So about 30 are not tabbies. F. 6 H. ( ) G. ( 3) 3 I What is the missing number in the sequence below? 9 6, 9, 9, 9, 9,. What is the value of the expression below? (.) A.. C. 0. B. 0. D... What is the distance between the two numbers on the number line? F. 3 G. H. 3 I. Cumulative Assessment 7

13 6. Sandra was evaluating an expression in the box below. 3 = 9 = 9 = 9 = 0 09 What should Sandra do to correct the error that she made? A. Rewrite 9 as 9 and multiply by. B. Rewrite as and multiply by 9. C. Rewrite as D. Rewrite 3 and multiply by 9. as 3 and multiply by. 7. What is the value of the expression below when q =, r =, and s =? 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? in. in. in. 76 Chapter Rational Numbers

14 9. What is the area of a triangle with a base length of inches and a height of inches? A. in. C. in. B. in. D. in. 0. What is the value of the expression below? ( ) 3 F. 6 H. G. I. 6. Four points are graphed on the number line below. R S T U 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? =. A. C. 0. B. 0. D. Cumulative Assessment 77