COMMON CORE Locker LESSON 9.1 Adding and Subtracting Rational Epressions Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? Resource Locker Common Core Math Standards The student is epected to: COMMON CORE A-APR.D.7() Understand that rational epressions form a system analogous to the rational numbers,...; add, subtract, rational epressions. Also A-SSE.A. Mathematical Practices COMMON CORE MP.7 Using Structure Language Objective Eplain to a partner how to simplify a rational epression and how to add and subtract rational epressions. ENGAGE Essential Question: How can you add and subtract rational epressions? Possible answer: Convert them to like denominators, then add the numerators, all the while keeping track of the combined ecluded values. Eplore Identifying Ecluded Values Given a rational epression, identify the ecluded values by finding the zeroes of the denominator. If possible, simplify the epression. _(1 - A ) - 1 The denominator of the epression is - 1. B C D E F Since division by 0 is not defined, the ecluded values for this epression are all the values that would make the denominator equal to 0. - 1 0 Begin simplifying the epression by factoring the numerator. ( 1 - )( 1 ) - 1 (1 - ) _ - 1 Divide out terms common to both the numerator and the denominator. ( 1 - )( 1 ) - 1 (1 - ) _ 1 - (1 - ) The simplified epression is _(1 - ) - 1-1 -, whenever 1 -(1 ) -1 - What is the domain for this function? What is its range? Its domain is all real numbers ecept 1. Its range is all real numbers ecept y -. PREVIEW: LESSON PERFORMANCE TASK View the Engage section online. Discuss the photo and how the river s current can either help the kayaker go faster or slow the kayaker down. Then preview the Lesson Performance Task. Reflect 1. What factors can be divided out of the numerator and denominator? Common factors Module 9 45 Lesson 1 Name Class Date 9.1 Adding and Subtracting Rational Epressions Essential Question: How can you add and subtract rational epressions? Eplore Identifying Ecluded Values Given a rational epression, identify the ecluded values by finding the zeroes of the denominator. If possible, simplify the epression. _ - 1 (1 - ) A-APR.D.7() Understand that rational epressions form a system analogous to the rational numbers,...; add, subtract, rational epressions. Also A-SSE.A. The denominator of the epression is. Since division by 0 is not defined, the ecluded values for this epression are all the values that would make the denominator equal to 0. - 1 0 Begin simplifying the epression by factoring the numerator. (1 - ) _ - 1 ( ) ( ) - 1 Divide out terms common to both the numerator and the denominator. 1 (1 - ) _ - 1 ( ) ( ) - (1 - ) The simplified epression is (1 - ) _ - 1, whenever What is the domain for this function? What is its range? Reflect 1-1 - 1-1 1-1 - 1 - (1 ) -1-1. What factors can be divided out of the numerator and denominator? Resource Its domain is all real numbers ecept 1. Its range is all real numbers ecept y -. Common factors Module 9 45 Lesson 1 HARDCOVER PAGES 309 316 Turn to these pages to find this lesson in the hardcover student edition. 45 Lesson 9.1
Eplain 1 Writing Equivalent Rational Epressions Given a rational epression, there are different ways to write an equivalent rational epression. When common terms are divided out, the result is an equivalent but simplified epression. Eample 1 Rewrite the epression as indicated. Write 3 as an equivalent rational epression that has a denominator of ( 3) ( 5). ( 3) 3 The epression has a denominator of ( 3). ( 3) The factor missing from the denominator is ( 5). Introduce a common factor, ( 5). _ 3 ( 3) 3 ( 5) ( 3) ( 5) 3 ( 3) is equivalent to 3 ( 5) ( 3) ( 5). Simplify the epression ( 5 6) ( 3 ) ( 3). Write the epression. Factor the numerator and denominator. Divide out like terms. ( 5 6) ( 3 ) ( 3) _ ( ) ( 3) ( 1) ( ) ( 3) 1 1 EXPLORE Identifying Ecluded Values INTEGRATE TECHNOLOGY Students have the option of completing the Eplore activity either in the book or online. QUESTIONING STRATEGIES When identifying ecluded values, why is it not necessary to consider values of the variable that make the numerator equal to 0? If the numerator is 0 and the denominator is nonzero, the fraction is defined and is equal to 0, so values that make the numerator 0 do not need to be ecluded (as long as they don t also make the denominator 0). Your Turn 5. Write as an equivalent epression with a denominator of ( - 5) ( 1). 5-5 _ 5 5 5 5 5 ( 5) 1 5 1 ( 1) ( 5) ( 1) 1 ( 5) ( 1) 3. Simplify the epression ( 3 )(1 ). ( 6 ) _ ( 3 )(1 - ) ( - 6 ) (1 )(1 - ) (1-4 ) (1 )(1 - ) (1 )(1 - ) (1 )(1 - ) 1_ (1 )(1 - ) Module 9 46 Lesson 1 PROFESSIONAL DEVELOPMENT Integrate Mathematical Practices Operations on rational epressions are similar to operations on fractions. For eample, a () b () c () d () a () d () b () c () b () d () ecept where b () 0 and/or d () 0. As with fractions, it is generally best to simplify rational epressions before adding, subtracting, multiplying, or dividing. Note that in the above equation, the denominator b ()d() may not be the least common denominator of the two rational epressions. Why are there no ecluded values for the rational epression _? There are no real 4 numbers for which 4 equals 0. EXPLAIN 1 Writing Equivalent Rational Epressions AVOID COMMON ERRORS Students sometimes make errors writing the simplified form of a rational epression in which all factors of the numerator divide out with common factors from the denominator. These students may write the simplified denominator as the final answer, forgetting that there is a factor of 1 that remains in the numerator. To help students avoid this error, encourage them to write a 1 above or below any factor that divides out, and to multiply the 1 s with any remaining factors when writing the final rational epression. Adding and Subtracting Rational Epressions 46
QUESTIONING STRATEGIES When simplifying a rational epression, when might it be helpful to factor 1 from either the numerator or the denominator? when the epressions in the numerator and the denominator are not both written in descending form, or when the leading coefficient of one or both epressions is negative Eplain Identifying the LCD of Two Rational Epressions Given two or more rational epressions, the least common denominator (LCD) is found by factoring each denominator and finding the least common multiple (LCM) of the factors. This technique is useful for the addition and subtraction of epressions with unlike denominators. Least Common Denominator (LCD) of Rational Epressions To find the LCD of rational epressions: 1. Factor each denominator completely. Write any repeated factors as powers.. List the different factors. If the denominators have common factors, use the highest power of each common factor. Eample Find the LCD for each set of rational epressions. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 To help students understand why rational epressions such as cannot be simplified, have them substitute a number, such as 6, for and compare the value of 6 1 6 to. Point out that the 1 two epressions are not equivalent. Tell students they can use this substitute-and-check strategy in other situations when they are not sure whether their action produces an equivalent epression. _- 3-15 and _ 6 4 8 Factor each denominator completely. 3-15 3 ( - 5) Reflect 4 8 4 ( 7) List the different factors. 3, 4, - 5, 7 The LCD is 3 4 ( - 5) ( 7), or 1 ( - 5) ( 7). -14-11 4 and 9_ - 6 9 Factor each denominator completely. - 11 4-6 9 List the different factors. - 3-8 and Taking the highest power of ( - 3), ( - 3) ( - 8) the LCD is. 4. Discussion When is the LCD of two rational epressions not equal to the product of their denominators? When a factor appears one or more times in each denominator Your Turn Find the LCD for each set of rational epressions. 5. _ 6 8-4 and _ 14 10-30 8-4 8 ( - 3) 3 ( - 3) 10-30 10 ( - 3) 5 ( - 3) factors:, 5, - 3 Taking the highest power of, the LCD is 3 5 ( - 3) 40 ( - 3). 6. 1 _ 15 60 and ( - 3) ( - 8) ( - 3) ( - 3) 5 9 0 15 60 15 ( 4) 3 5 ( 4) 9 0 ( 4) ( 5) factors: 3, 5, 4, 5 The LCD is 3 5 ( 4) (5) 15 ( 4) (5). Module 9 47 Lesson 1 COLLABORATIVE LEARNING Peer-to-Peer Activity Pair advanced students with classmates who are struggling with adding and subtracting rational epressions with unlike denominators. Encourage the advanced students to start by trying to diagnose which part of the process is causing the most trouble and to focus on helping with the appropriate skills. By eplaining the process, advanced students will develop a deeper understanding of the concepts and struggling students will benefit from peer instruction. 47 Lesson 9.1
Eplain 3 Adding and Subtracting Rational Epressions Adding and subtracting rational epressions is similar to adding and subtracting fractions. Eample 3 Add or subtract. Identify any ecluded values and simplify your answer. _ 4 _ Factor the denominators. _ 4 _ ( 1) Identify where the epression is not defined. The first epression is undefined when 0. The second epression is undefined when 0 and when -1. Find a common denominator. The LCM for and ( 1) is ( 1). Write the epressions with a common denominator by multiplying both by the appropriate form of 1. ( 1) ( 1) _ 4 ( 1) _ Simplify each numerator. 3 5 6 _ 3 ( 1) ( 1) Add. 3 5 6 ( 1) Since none of the factors of the denominator are factors of the numerator, the epression cannot be further simplified. _ - 5 - _ 3-4 Factor the denominators. _ -_ 3-4 Identify where the epression is not defined. The first epression is undefined when 0 and when 5. The second epression is undefined when 0. Find a common denominator. The LCM for ( - 5) and is ( - 5). Write the epressions with a common denominator by multiplying both by the appropriate form of 1. ( - 5) _ ( - 5) - _ 3-4 Simplify each numerator. 3 _ ( - 5) - 3 - - 19 0 ( - 5) Subtract. 19-0 ( - 5) - 5 _ - 5 Since none of the factors of the denominator are factors of the numerator, the epression cannot be further simplified. Module 9 48 Lesson 1 DIFFERENTIATE INSTRUCTION Communicating Math It may be beneficial to have students verbally describe the steps involved in adding and subtracting rational epressions. Ensure that students descriptions are accurate, but allow them to use their own words to describe the steps. Pay careful attention, in students eplanations, to how they find the common denominator, and how they convert the numerators of the fractions to obtain equivalent epressions. Students may also benefit from hearing how other students describe these steps. _ 3 EXPLAIN 3 Adding and Subtracting Rational Epressions AVOID COMMON ERRORS Students may make sign errors when combining the numerators in a problem involving the subtraction of two rational epressions. To avoid this, reinforce the importance of writing the numerator being subtracted in parentheses and carefully applying the distributive property before combining like terms. QUESTIONING STRATEGIES How do you find the LCD of rational epressions? First, factor each denominator. Then write the product of the factors of the denominators. If the denominators have common factors, use the highest power of that factor found in any of the denominators. What steps do you use to rewrite the epressions with like denominators? Multiply each numerator by any factors of the LCD that were not factors of its original denominator. Use the LCD for the denominator. PEER-TO-PEER DISCUSSION Ask students to consider whether the product of the denominators can always be used as the common denominator when adding or subtracting fractions with unlike denominators. Have them eperiment to see what happens in situations in which the product is not the least common denominator. The product can be used, but the result will need to be simplified. Adding and Subtracting Rational Epressions 48
Your Turn Add each pair of epressions, simplifying the result and noting the combined ecluded values. Then subtract the second epression from the first, again simplifying the result and noting the combined ecluded values. 7. - 1 and (1 - ) 8. Addition: 1 (1 ) ( )(1 ) 1 (1 ) (1 ) 4 1 (1 ) _ 4 1 (1 ) (1 ), ±1 Subtraction: 1 - (1 ) ( )(1 ) 1 - (1 ) (1 ) 4-1 (1 ) (1 ), ±1 (4 - ) and 1 ( - ) Addition: (4 ) 1 ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ), ± Subtraction: (4 ) - 1 ( ) ( ) ( ) - ( ) ( ) ( ) - ( ) ( ) ( ) - - ( ) ( ) ( )( - 1 ) ( ) ( ) _ ( )] ( 1) [(-1 ) (( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) 1, ± Module 9 49 Lesson 1 49 Lesson 9.1
Eplain 4 Adding and Subtracting with Rational Models Rational epressions can model real-world phenomena, and can be used to calculate measurements of those phenomena. Eample 4 Find the sum or difference of the models to solve the problem. Two groups have agreed that each will contribute $000 for an upcoming trip. Group A has 6 more people than group B. Let represent the number of people in group A. Write and simplify an epression in terms of that represents the difference between the number of dollars each person in group A must contribute and the number each person in group B must contribute. _ 000 _ 000 6 000 ( 6) _ 000 ( 6) ( 6) 000-1, 000 000 ( 6) 1, 000 _ ( - 6) A freight train averages 30 miles per hour traveling to its destination with full cars and 40 miles per hour on the return trip with empty cars. Find the total time in terms of d. Use the formula t d r. Let d represent the one-way distance. Your Turn Total time: _ d 30 d_ 40 _ d. 40 d. 30 _ 30. 40 40. 30 d. 40 d. 30 100 7 _ d 10 9. A hiker averages 1.4 miles per hour when walking downhill on a mountain trail and 0.8 miles per hour on the return trip when walking uphill. Find the total time in terms of d. Use the formula t d r. Let d represent the one-way distance. Total time: d 1.4 d d 0.8 0.8 1.4 0.8 d 1.4 0.8 1.4 d 0.8 d 1.4 _ 1.1. 1.1 d 55 8 d Image Credits: (t) Larry Lee Photography/Corbis; (b) Henrik Trygg/Corbis EXPLAIN 4 Adding and Subtracting with Rational Models QUESTIONING STRATEGIES What is the significance of the ecluded values of a rational epression that models a real-world situation? The ecluded values are numbers that are not possible values of the independent variable in the given situation. INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can use a graphing calculator to compare the graph of the function defined by the original sum or difference with the graph of the function defined by the final simplified epression. The graphs should be identical. Module 9 430 Lesson 1 LANGUAGE SUPPORT Communicate Math Have students work in pairs. Provide each pair of students with some rational epressions written on sticky notes or inde cards and with some addition and subtraction problems. Have the first student eplain the steps to simplify a rational epression while the second student writes notes. Students switch roles and repeat the process with an addition problem, then again with a subtraction problem. Adding and Subtracting Rational Epressions 430
ELABORATE INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Ask students to discuss why values that are ecluded from an addend in a sum of rational epressions must be ecluded from the final simplified form of the sum, even if that value does not make the simplified epression undefined. 10. Yvette ran at an average speed of 6.0 feet per second during the first two laps of a race and an average speed of 7.75 feet per second during the second two laps of a race. Find her total time in terms of d, the distance around the racecourse. Total time: d 6. d 7.75 d 6. d _ (7.75 6.) d 7.75 7.75 6. ()(13.95) d 48.05 0.58d Elaborate QUESTIONING STRATEGIES Why does the rational epression _ 1-1 have two ecluded values, but the epression _ - 1 1 have none? The denominator of the first rational epression is equal to 0 when 1 and when -1. There is no real number that makes the denominator of the second epression equal to 0. SUMMARIZE THE LESSON How do you subtract two rational epressions? If the denominators are like, subtract the numerators and write the result as the numerator over the common denominator. If the denominators are not alike, find the LCD, convert each rational epression to an equivalent epression having the LCD, and then follow the steps described above. If the second numerator contains more than one term, be careful to apply the distributive property when subtracting. 11. Why do rational epressions have ecluded values? Ecluded values would make the epression undefined. 1. How can you tell if your answer is written in simplest form? When none of the factors of the denominator are factors of the numerator, the answer is written in simplest form. 13. Essential Question Check-In Why must the ecluded values of each epression in a sum or difference of rational epressions also be ecluded values for the simplified epression? You cannot add or subtract undefined epressions so any value that makes one epression in a sum or difference undefined has to be an ecluded value for the simplified epression. Module 9 431 Lesson 1 431 Lesson 9.1
Evaluate: Homework and Practice Given a rational epression, identify the ecluded values by finding the zeroes of the denominator. 1. _ - 1. 4_ 3-4 ( 17) 1 3 4 1 4 ( 4) ( 1) ( 17) 1, 4 0, 17 Write the given epression as an equivalent rational epression that has the given denominator. - 7 3. Epression: _ 8 Denominator: 3 8 Simplify the given epression. 5. 7. (-4-4) ( - - ) (-4-4) ( ) _ 4 (1 ) ( ) ( 1) 4 (1 ) _ ( ) (1 ) 6 5 1 3 4 1 Find the LCD for each set of rational epressions. 9. 3 8 ( 8) 7 8 ( 7) ( 8) ( 7) 3 8 4 ( ) _ 16 and _-4 3-7 4 ( ) or 3 7 3 8 6 5 1 3 4 1 (3 1) ( 1) (3 1) ( 1) ( 1) ( 1) 16 ( 8) 3-7 3 ( - 9) List the different factors., 3, 8, - 9 LCD: 3 ( 8) ( - 9) 6 ( 8) ( - 9) 6. 8. 10. - - 8 _ 9 8 4-1 _ 1 _ - 4 5-30 5 13 and _ 7-4 Online Homework Hints and Help Etra Practice 3 4. Epression: _ 3 3-6 Denominator: ( - ) ( 9) _ 3 3 3-6 3 - - 3 3 - _ - ( 9) ( - ) ( 9) - - 8 9 8-1( 8) ( 8) ( 1) -1 ( 1) 4-1 ( - 1) 1 ( 1) ( - 1) ( 1) Denominator factors: 5, 7, - 6 5 7 ( - 6) 35 ( - 6) Module 9 43 Lesson 1 Eercise Depth of Knowledge (D.O.K.) 1 1 Recall of Information MP.5 Using Tools 3 4 Skills/Concepts MP. Reasoning 5 7 Skills/Concepts MP.4 Modeling 8 3 Strategic Thinking MP.4 Modeling 9 30 Skills/Concepts MP.4 Modeling 31 3 3 Strategic Thinking MP.3 Logic COMMON CORE Mathematical Practices EVALUATE ASSIGNMENT GUIDE Concepts and Skills Eplore Identifying Ecluded Values Eample 1 Writing Equivalent Rational Epressions Eample Identifying the LCD of Two Rational Epressions Eample 3 Adding and Subtracting Rational Epressions Eample 4 Adding and Subtracting with Rational Models QUESTIONING STRATEGIES Practice Eercises 1 Eercises 3 8 Eercises 9 14 Eercises 15 4 Eercises 5 8 Why do some rational epressions have ecluded values while others do not? The denominators of some rational epressions contain factors that are equal to 0 for one or more values of. Those rational epressions will have ecluded values, since the rational epression is undefined for those values of. In other rational epressions, no value of makes the denominator equal to 0, so there are no ecluded values. AVOID COMMON ERRORS Students often list only the values that make the simplest form of a rational epression undefined. Stress the importance of recording all of the values for which the original epression is undefined. INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Remind students that they can check their answers to a subtraction problem by adding the answer to the subtrahend to see if the result is the minuend. (If a - b c, then c b a.) This is especially useful for checking that no sign errors were made when subtracting. Adding and Subtracting Rational Epressions 43
11. 13. 4 1 _ 5 6 5 15 and _ 10 0 5 6 ( ) ( 3) 10 0 10 ( ) 5 ( ) List the different factors., 5,, 3 LCD: 5 ( ) ( 3) 10 ( ) ( 3) 1 3-1 - 54 and _-1 1-84 3-1 - 54 3 ( - 9) ( ) 1-84 1 ( ) ( - ) 3 7 ( ) ( - ) List the different factors. 3, 7, - 9,, - LCD: 3 7 ( - 9) ( ) ( - ) 1 ( - 9) ( ) ( - ) 1. 14. - 11-3 - 8 and - - 4-3 - 8 ( - 7) ( 4) - - 4 ( 4) ( - 6) List the different factors. - 7, - 6, 4 LCD: ( - 7) ( - 6) ( 4) 3 5 40 60 and 17 - - 7 56-84 5-40 60 5 ( - 6) ( - ) -7 56-84 -7( - 6) ( - ) List the different factors. 5, -7, - 6, - LCD: 5 (- 7) ( - 6) ( - ) - 35 ( - 6) ( - ) Add or subtract the given epressions, simplifying each result and noting the combined ecluded values. 15. 1_ 1 _ 1-4 16. _ - 4 _ - - - 4 1 1 1-1 (1 ) 1 - (1 ) - 1, 0, -1 (1 ) (1 - ) (1 ) _ (1 ) 4-4 - - - 4 4 - - - 4 (- ) _ ( ) ( - ) (-) _ ( ) ( - ) _ -1 ( ), ± Module 9 433 Lesson 1 433 Lesson 9.1
17. 1_ - _ - 1 - - 1 ( ) - (- 4) ( ) ( - 4) ( ) - 4, 0, - ( ) 19. 0. 1. _ 4 - - _ - 1_ 3-4 - 1_ - 3 3_ - 4 - _ 5 3 _ ( - ) ( ) - _ ( ) 4-18. _ 4 4 4 1-8_ 1 _ 4 4 4 1-8 1 4 4 4-8 1 _ 4 4-4 1 4 ( 4-1) ( 1) 4 ( 1) ( - 1) ( 1) 4 ( - 1) ( 1) 0 has no real solutions so no values are ecluded. - - 4 - - - - 4-4 - ( - ) ( ) ( - ), ± 1-4 - 5 3 ( ) ( - ) - 5 3 _ ( - 5) ( - ) ( ) ( - ) ( ) ( - ) 3 - ( 5) ( - ) ( ) ( - ) 3 - ( 3-10) ( ) ( - ) 3 - - 3 10 ( ) ( - ) _ - - 3 13 ( ) ( - ), ± 3-4 - 1-3 1 ( - 1) ( 4) - 1 ( - 1) ( - ) ( - ) _ ( - 1) ( 4) ( - ) - _ 4 ( - 1) ( - ) ( 4) ( - ) - ( 4) _ ( - 1) ( 4) ( - ) _ -6, -4, 1, ( - 1) ( 4) ( - ). _-3 9-4 1_ 3 _ -3 9-4 1 3-3 (3 ) (3 - ) 1 (3 ) -3 (3 ) (3 - ) (3 - ) (3 ) (3 - ) - 3 (3 ) (3 - ), 0, ± _ MODELING When working with rational epressions that represent real-world situations, students should recognize that not only must they consider ecluded values that are based on the algebraic nature of the denominators of the epressions, but they also need to consider values that must be ecluded due to the limitations on the domain in the given situation. Module 9 434 Lesson 1 Adding and Subtracting Rational Epressions 434
VISUAL CUES When students simplify rational epressions, suggest that they highlight each different factor in the numerator and denominator with a different color by using highlighters or colored pencils. This process can help them keep track of common factors, as shown below. _ ( 1) ( ) ( 3) ( 1) ( ) 3 3. 4. _ - 1_ - 4 - _ - - _ 1-4 - - - _ 1 ( ) ( - ) - _ - 3 3-1_ - 3 _ 3 - ( - ) (-) ( ) ( - ) 1 ( ) ( - ) _ - 4 4 1 4 4 ( ) ( - ) 9 ( ) ( - ), ± _ - 3 3-1 - 3 3 - - 3 3-1 - 3 - - - 3 _ - 3 3 - - 3-3 ( - 3) ( - 3) (- - 3) ( 3) ( 3) ( - 3) ( - 3) ( 3) - 6 9 - - 6-9 ( - 3) ( 3) -1 ( - 3) ( 3), ±3 ( ) ( ) ( - ) ( ) Image Credits: Chris Crisman/Corbis 5. A company has two factories, factory A and factory B. The cost per item to 00 13q produce q items in factory A is q. The cost per item to produce q items in 300 5q factory B is. Find an epression for the sum of these costs per item. Then q divide this epression by to find an epression for the average cost per item to produce q items in each factory. 00 13q 300 5q q (00 13q) 300 5q q q q 400 6q 300 5q q 6. An auto race consists of 8 laps. A driver completes the first 3 laps at an average speed of 185 miles per hour and the remaining laps at an average speed of 00 miles per hour. Let d represent the length of one lap. Find the time in terms of d that it takes the driver to complete the race. 3d 185 5d 3d (00) 00 185 (00) 5d (185) 00 (185) 700 51q q 700 51q So the average cost per item to produce q items in each factory is. 4q 600d 95d 37,000 61d 1480 hours Module 9 435 Lesson 1 435 Lesson 9.1
7. The junior and senior classes of a high school are cleaning up a beach. Each class has pledged to clean 1600 meters of shoreline. The junior class has 1 more students than the senior class. Let s represent the number of students in the senior class. Write and simplify an epression in terms of s that represents the difference between the number of meters of shoreline each senior must clean and the number of meters each junior must clean. 1600 s - _ 1600 1600 (s 1) - 1600s s 1 s (s 1) (s 1) s 1600s 19,00-1600s s (s 1) 19,00 s (s 1) meters CONNECT VOCABULARY Have students eplain why a rational epression has ecluded values for a denominator of zero, using what they know about fractions or ratios to eplain. Help them by re voicing or clarifying their eplanations, as needed. 8. Architecture The Renaissance architect Andrea Palladio believed that the height of a room with vaulted ceilings should be the harmonic mean of the length and width. The harmonic mean of two positive numbers a and b is equal to 1 a 1. Simplify this b epression. What are the ecluded values? What do they mean in this problem? 1_ a 1 1b b ab 1a _ ba ( b a ab ) ab, ecluding a 0, b 0, a -b. a b These values do not occur because geometric lengths are positive. 9. Match each epression with the correct ecluded value(s). 3 5 a. _ C no ecluded values 1 b. _ D 0, - - 1 c. _ 3 4-1 B 1, -1 4 3 6 d. _ A - ( ) _ 3 5 A., - B. 1, 1, -1-1 C. 3 4-1, no ecluded values 4 D. 3 6, 0, - ( ) Image Credits: Klaus Vedfelt/Getty Images Module 9 436 Lesson 1 Adding and Subtracting Rational Epressions 436
JOURNAL Have students compare the method used to add two rational numbers to the method used to add two rational epressions. Have them use specific eamples to illustrate their eplanations. H.O.T. Focus on Higher Order Thinking 30. Eplain the Error George was asked to write the epression - 3 three times, once each with ecluded values at 1,, and -3. He wrote the following epressions: a. _ - 3-1 b. _ - 3 - c. _ - 3 3 What error did George make? Write the correct epressions, then write an epression that has all three ecluded values. George wrote epressions that had the correct ecluded values, but his epressions are not equivalent to - 3. The correct epressions should be the following: ( - 3) ( - 1) a. _ - 1 ( - 3) ( - ) b. _ - ( - 3) ( 3) c. _ 3 ( - 3) ( -1)( - ) ( 3) - 3 _, 1,, -3 ( - 1) ( - ) ( 3) 31. Communicate Mathematical Ideas Write a rational epression with ecluded values at 0 and 17. Answers may vary. Sample answer: a ( - 17) 3. Critical Thinking Sketch the graph of the rational equation y 3. Think 1 about how to show graphically that a graph eists over a domain ecept at one point. 8 y 4-8 -4 0 4 8-4 -8 This epression has an ecluded value at -1. This can be shown on the graph with a hole at that one point. Module 9 437 Lesson 1 437 Lesson 9.1
Lesson Performance Task A kayaker spends an afternoon paddling on a river. She travels 3 miles upstream and 3 miles downstream in a total of 4 hours. In still water, the kayaker can travel at an average speed of miles per hour. Based on this information, can you estimate the average speed of the river s current? Is your answer reasonable? Net, assume the average speed of the kayaker is an unknown, k, and not necessarily miles per hour. What is the range of possible average kayaker speeds under the rest of the constraints? CONNECT VOCABULARY Some students may not be familiar with the term kayak. Have a student volunteer describe a kayak. Eplain that kayak is used both as a noun and as a verb, and that the person paddling a kayak is called a kayaker. Total time is equal to time upstream plus time downstream. T r 1 r D ( 1 r 1 r 1 r 1 r 1 r ) T D ( r 1 r ) r 1 r Substitute for r 1 and r : T D T D (4 - c ) 4 (4 - c ) 4 T c - 4-4 T r ) D ( ( - c) ( c) ( - c) ( c) Plug in specific constants: c -4 T 4-4 ( 3_ 4) 4-3 4 1 c ± 1 ±1 The answer is 1 mile per hour only because the speed in this contet cannot be negative. To figure out the range of speeds of the kayaker in the river, we substitute k for above. T (k - c) (k c) (k - c) (k c) k D ( k - c ) k T c - k -k T c -k T k c ± -k T k -k T k 0 k k T k T 3_ 1.5 miles per hour 4 The kayaker has to go at least 1.5 miles per hour regardless of the speed of the current. AVOID COMMON ERRORS Students may divide the total distance (6 miles) by the total time (4 hours) to get an average speed of 1.5 miles per hour, but this is incorrect. Ask students why the upstream and downstream travel times need to be calculated separately. The kayaker is traveling at a different speed in each direction due to the river current. Ask students what quantity is the same in both directions. distance Ask students what quantities are different in each direction. the travel time and speed of the kayaker Module 9 438 Lesson 1 EXTENSION ACTIVITY Have students eplain how the presence of a strong wind might affect the time it takes the kayaker to travel up and down the river. Ask the students how they would account for the wind in their calculations. Add or subtract a term w for the effect of the wind on the kayaker s speed. For eample, if the wind s effect is acting in the same direction as the current, the kayaker s speed upstream would be r 1 - c - w and downstream would be r c w. It the wind s effect is acting in the opposite direction of the current, the kayaker s speed upstream would be r 1 - c w and downstream would be r c - w. Scoring Rubric points: Student correctly solves the problem and eplains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or eplain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Adding and Subtracting Rational Epressions 438