Arithmetic and Algebra

Transcription

1 Arithmetic and Algebra 16 + = false 10 5 = true 33 n = 1 open Write true if the statement is true or false if it is false. Write open if the statement is neither true nor false = = = = = = n = = = = n = = = = = n = = n = = n = n = = = = = 6. 8n = = n = = = by CompassLearning, Inc. Algebra AL001

2 Representing Numbers Using Letters Numerical expressions: Algebraic expressions: m 5 8d + Variables: n in 7n + 7 k in k 3 Operations: Multiplication and addition in y + 3 Division in Name the variable in each algebraic expression. 1. 4y + 1. k 6 3. x n r k x p 5. m y 6. 3(d) 1. m 5 Fill in the table. For each expression, write the expression type numerical or algebraic and list the operation or operations. Expression Expression Type Operation(s) d p k Solve the problem. 5. Only 17 members of Mr. Ricardo s class are going on the class trip. The class has a total of k students. Write an algebraic expression for the number of students who are not going on the trip. 004 by CompassLearning, Inc. Algebra AL00

3 Integers on the Number Line All the numbers on this number line are examples of integers. An example of a negative integer is 5 (see arrow). An example of a positive integer is 5 (see arrow). The number 0 is neither negative nor positive. 5 = 5. In other words, 5 is 5 units from 0 (count the units). 5 = 5. In other words, 5 is 5 units from 0 (count the units). Identify each integer as either negative, positive, or zero Write each absolute value Solve this problem. 5. On the number line, how could you represent $5 that you earned? How could you represent $5 that you had to pay? 004 by CompassLearning, Inc. Algebra AL003

4 Adding Integers Add + ( 5). Start at, move 5 units to the left. The answer is Add Start at 3, move 7 units to the right. The answer is Answer the questions. 1. To add a negative number, in which direction do you count on the number line?. To add a positive number, in which direction do you count on the number line? Write each sum on the blank ( 7) ( 5) ( 11) ( 1) 1. + ( 4) ( 6) ( 8) 004 by CompassLearning, Inc. Algebra AL004

5 Subtracting Integers Find the difference: 14 ( 15) Rule To subtract in algebra, add the opposite. 15 is the opposite of = 9 Rewrite each expression as addition. Solve the new expression ( 11) ( 5). 9 (+3) (+10) 5. 7 ( 10) 6. 4 (+4) 7. (+8) ( 1) 9. 6 (+) 11. (+9) 1. 1 (+4) ( 3) (+7) ( 7) (+5) Solve these problems. Write an expression and the answer. 19. Dara s kite is flying 67 feet high. Jill s is flying 40 feet high. What is the difference between the heights of these two kites? 0. A helicopter hovers 60 m above the ocean s surface. A submarine is resting 30 m underwater, directly below the helicopter. What is the difference between the positions of these two objects? 004 by CompassLearning, Inc. Algebra AL005

6 Multiplying Integers Notice the possible combinations for multiplying positive and negative integers. positive (positive) = positive 4(4) = 16 positive (negative) = negative 4( 4) = 16 negative (positive) = negative 4(4) = 16 negative (negative) = positive 4( 4) = 16 Multiplying any integer, positive or negative, by 0 gives 0 as the product. Tell whether the product is positive, negative,or zero. 1. (4)( 7). ( 6)(3) 3. ( 7)(0) 4. ( 9)( 9) 5. ()( 11) 6. (5)( 3) 7. ( 15)( 1) 8. (0)(14) 9. ( 5)(7) 10. ( 8)( ) 11. ( 3)( 9) 1. (4)(6) 13. (11)( ) 14. ()(9) 15. ( 6)(4) Find each product. Write the answer. 16. ()(9) 17. ( 5)(9) 18. (3)( 9) 19. ( 10)() 0. ( 9)( 7) 1. ( 5)(10) 5. ( 11)( ) 6. ( 4)(8) 7. ( 10)( 8) 8. (5)( 5) 9. ( 1)( 1) 30. (7)(10). (1)() 3. ( 3)(0) 4. (6)( 4) 004 by CompassLearning, Inc. Algebra AL006

7 Integers Data from a climbing expedition is shown in this table. Elevation in Feet (Compared to Sea Level) Base Camp 384 Camp 1 +5,07 Camp +7,511 Camp 3 +8,860 Camp 4 +10,103 Camp 5 +10,856 Camp 6 +11,349 Summit +1,015 During the climb, some climbers began at base camp and climbed to the summit. Other climbers also began at base camp but did not reach the summit these climbers moved back and forth between camps carrying supplies and other necessities. The movements of various climbers in the expedition are shown below. Find the number of feet climbed by each climber. 1. Climber A: Base Camp to Camp 1 to Base Camp. Climber C: Base Camp to Camp 5 to Base Camp 3. Climber F: Base Camp to Camp 3 to Base Camp 4. Climber B: Base Camp to Camp to Base Camp 5. Climber H: Base Camp to Camp 6 to Base Camp 6. Climber E: Base Camp to Summit to Base Camp 7. Climber G: Base Camp to Camp 4 to Base Camp 8. Climber D: Base Camp to Camp 5 to Camp 3 to Camp 4 to Base Camp 9. Climber I: Base Camp to Camp to Camp 1 to Camp 6 to Base Camp 10. How many feet above base camp is the summit? 004 by CompassLearning, Inc. Algebra AL006

8 Dividing Positive and Negative Integers Notice the possible combinations for dividing positive and negative integers. positive positive = positive 6 = 3 positive negative = negative 6 = 3 negative positive = negative 6 = 3 negative negative = positive 6 = 3 Dividing 0 by any integer, positive or negative, produces 0 as the quotient. Tell whether the quotient is positive, negative,or zero Find and write each quotient by CompassLearning, Inc. Algebra AL007

9 Simplifying Expressions One Variable Simplify n + + 4n. 1. Look for like terms. n and 4n are like terms, because they have the same variable, n.. Combine the terms: n + 4n = 6n 3. Rewrite the whole expression: 6n + Now you are finished, because 6n cannot combine with. In each expression, underline the like terms. 1. 3k 8 + k. p p w + 4w 4. 5m 3 + m 5. 7x + 5x c + c m + 3m 7 8. y ( 3y) x x 10. 8r + ( 3r) Simplify each expression b + b 1. 11y + y + y 13. 7j + 3j j 14. k 17 + k x + x d + 8d 17. 9g + ( g) h 3 h 19. m + ( 8m) k 3k 1. y + ( y) n 4n 3. x + 11x h + 7h ( 3k) + k 6. 7d d m ( m) 8. 3w + ( 5w) x x 30. 8g + ( 5g) by CompassLearning, Inc. Algebra AL008

10 Simplifying Expressions Several Variables Simplify j j 1 + 3k. 1. Scan for variables. The expression has two: j and k.. Combine j terms: j + j = 3j 3. Combine k terms: 3k (no combining required) 4. Combine integers: 4 + ( 1) = Rewrite the whole expression: 3j + 3k + 3 Now you are finished, because you cannot combine unlike terms. Check the column or columns to show which kinds of terms each expression includes. Expression x terms y terms Integers 1. 3x + x y. 3y x + y y x 8 Combine like terms. Simplify each expression. 6. 3k k r 7. c + 3b + 8b + c 8. 5k j k p 6 + m p 10. 4n + 4 ( d) + n ( 4x) + 1 3x + 8y 1. w + ( 11) + 18y + w h 1 9h + 8k + ( 3k) m 3m 8p 3p x 7 + 3y 7x 7x 004 by CompassLearning, Inc. Algebra AL009

11 Positive Exponents To multiply like variables having exponents, add the exponents. k 3 k 3 = k (3 + 3) = k 6 To divide like variables having exponents, subtract the exponents. m 4 m = m (4 ) = m You cannot use these rules to multiply or divide unlike variables. y 3 a b 5 j Is the rewritten expression on the right true or false? Write the answer. 1. m 4 m = m (4 + ). a a a = a ( ) 3. k n 3 = k ( + 3) 4. y 7 y = y (7 1) 5. b 8 b 3 = b (8 3) 6. a a = a 7. d 6 d = d (6 + ) 8. w w 4 w 5 = w ( ) Simplify each expression. 9. k 3 k w 5 w 11. j 14 j n 6 n 13. d d 5 d y 7 y 15. x x 3 x a a a 4 a 4 Use a calculator to find the value of each expression. 17. x when x = c 5 when c = 19. r 5 when r = 3 0. k 3 when k = by CompassLearning, Inc. Algebra AL010

12 Formulas with Variables Find the perimeter of a triangle with unequal sides. The perimeter is the sum of the length of all the sides. a c Perimeter = a + b + c b If a = 3, b = 5, and c = 4, then the perimeter of the triangle is = 1. Use the formula for the perimeter of a triangle with unequal sides to find each answer. 1. a = 8 cm b = 10 cm Perimeter = c = 6 cm. a = m b = 7 m Perimeter = c = 3 m Use the formula for the perimeter of a square. Write each answer. s 3. s = 5 km s s Perimeter = 4. s = 18 cm Perimeter = s Perimeter of a square = 4s 5. s = 9 m Perimeter = 004 by CompassLearning, Inc. Algebra AL011

13 Using a Formula The speed that an airplane travels is called air speed. Air speed is calculated by the formula g = a h. Ground speed (g) equals air speed (a) minus head wind speed (h). If you know any two values, you can solve for the missing variable. Ground speed (g) = Air speed (a) Head wind speed (h) g = 300 mph 50 mph g = = 50 The ground speed (g) equals 50 mph. 50 mph = a 50 mph 50 = a 50 a = = 300 The air speed (a) equals 300 mph. 50 mph = 300 mph h 50 = 300 h h = = 50 The head wind speed (h) equals 50 mph. Find the missing values in this table. Ground speed (g) Air speed (a) Head wind speed (h) mph 4 mph. 309 mph 31 mph mph 1 mph mph 1 mph mph 38 mph mph 45 mph mph 400 mph mph 310 mph 9. 1 mph 8 mph mph 14 mph mph 78 mph mph 47 mph mph 13 mph mph 45 mph mph 7 mph 004 by CompassLearning, Inc. Algebra AL01 AGS Publishing. Permission is granted to reproduce for classroom use only.

14 Modeling Expressions: Algebra Tiles Lesson at a Glance In this lesson, you will investigate representing and simplifying algebraic expressions using Algebra Tiles. Review the computer-based activity AL15 to access the Algebra Tiles and the interactive version of the lesson. Warm-up Cracking the Code Cryptology is the study of codes and ciphers. Messages are concealed by substituting letters, words, and phrases with other symbols. Even Julius Caesar was known to send secret messages using his own cipher. Above each symbol, write its corresponding letter. Use the given clue and see how quickly you can crack this coded message! 006 by CompassLearning, Inc. Algebra AL15

15 Case Studies Charlie is starting on a mathematical adventure. He is beginning algebra. Explore algebraic expressions with Charlie and make your own conclusions. Case Study 1 Charlie s Assumptions Charlie believed that the following statements were true: The sum of two negative integers is always a negative integer. The sum of a positive integer and a negative integer is always a negative integer. Charlie decided to use the unit blocks of the Algebra Tiles to prove his claims. He knew that a positive unit and a negative unit make a zero pair. Charlie used the main workspace to show his calculations. He represented four different equations. Did Charlie prove his assumptions? Why or why not? 006 by CompassLearning, Inc. Algebra AL15

16 Case Study More Assumptions Charlie wanted to use Algebra Tiles to investigate his assumptions about subtraction with negative integers. Charlie used this method: Step 1: Place tiles that represent the first number or minuend on the workspace. Step : Place zero pairs to match the second number or subtrahend. Step 3. Remove the subtracted value to find the solution. Charlie thought the following assumptions were true: ( integer) ( integer) ( integer) ( integer) ( integer) ( integer) Is Charlie correct? Connect his assumptions to known addition rules. List examples to support or disprove Charlie s assumptions. Case Study 3 Simply Tiles Charlie was simplifying an expression and using Algebra Tiles to find his answer. y 1 x y 1 xy He placed the matching tiles on the workspace and then grouped the tiles according to size. What is the simplified expression? How could Charlie check his answer? 006 by CompassLearning, Inc. Algebra AL15

17 Case Study 4 Commuting Problem Charlie had simplified three expressions, but he wasn t sure whether he was done. Review Charlie s expressions. Are they simplified expressions? If not, where did Charlie go wrong? Case Study 5 Self-Expression Charlie was investigating the additional features of the Algebra Tiles. He selected the options Enter an Expression and Factors. He entered - as the coefficient of x in the first factor, entered as the coefficient of x in the second factor, and clicked OK. The resulting expression was (-x)(x). He placed the x tiles in the axes to represent the factors x and -x. He then represented the product by placing four negative x tiles. Charlie clicked Check and verified that the tiles matched the entered expression. He reasoned that (-x)(x) was equal to -x -x -x -x and that its simplified expression was -4x. If Charlie wanted to simplify the expression (x)(y), what would he need to do? What is the simplified expression? 006 by CompassLearning, Inc. Algebra AL15

18 Commutative Property of Addition n + 3n = Expanded notation: n + n + n + n + n = n + n + n + n + n n + 3n = 5n Commutative property of addition: 7a + 6a = 6a + 7a Find each sum using expanded notation. 1. k + 3k. r + r 3. 4y + y 4. n + 5n Rewrite each sum showing the commutative property of addition. 5. b k x + 5x m + 3p 10. 7q y + 4y Solve these problems. 13. Torri weighs 98 pounds. Carrie weighs 116 pounds. Torri adds her weight to Carrie s weight. What sum will she get? 14. Suppose Carrie adds her weight to Torri s weight. What sum will she get? 15. What mathematical property do the sums in problems 13 and 14 illustrate? 004 by CompassLearning, Inc. Algebra AL013

19 Commutative Property of Multiplication Turning a rectangle two different ways shows that the order of factors in a product can change without affecting the result. In each case, the product is the area of the rectangle in square units. Factors Product Factors Product 5 3 = = For each pair of rectangles, write two products representing area in square units. 1. = sq. units 1 7 = sq. units = sq. units 6 = sq. units 3 Answer the questions. Mrs. Rossi buys 6 bags of apples. Each bag holds 8 apples. Mr. Lundgren buys 8 bags of apples. Each bag holds 6 apples. 3. How many apples did Mrs. Rossi buy? 4. How many apples did Mr. Lundgren buy? 5. What mathematical property do problems 3 and 4 illustrate? 004 by CompassLearning, Inc. Algebra AL014

20 Associative Property of Addition ( ) + 4 = 13 + (10 + 4) (b + c) + d = b + (c + d) Rewrite each expression to show the associative property of addition. 1. 3k + (k + 5). ( ) (11x + 10y) (5 + q) 5. (3 + 1n) + 6. g + (1 + h) Answer the questions. In a club, 3 members bring all of the sandwiches for a picnic. Mike and Lynn arrive together. They have already put together Mike s 3 sandwiches and Lynn s 5 sandwiches. Hosea comes a little later with 4 sandwiches. In all, the club has a total of 1 sandwiches. 7. Write an addition expression that shows the grouping described above. 8. Suppose that Mike had come first, alone, and that Lynn and Hosea had come with their combined sandwiches later. Write an addition expression to represent this grouping. 9. Would the club s total number of sandwiches be the same with either grouping? 10. What mathematical property does this story illustrate? 004 by CompassLearning, Inc. Algebra AL015

21 Associative Property of Multiplication (5 )3 = 5( 3) (pq)r = p(qr) Use the associative property of multiplication to place parentheses in two different ways. 1. mnq. abc 3. 4st 4. kvz bc 6. 9mn 7. aks Answer the questions to solve the problem. A college will build a theater with at least 30 balcony seats. Two architects have drawn up plans. Here are their designs for the balconies: Plan A: 3 rows }6 Seats } Plan B: 6 Seats rows 8. Which plan does (3 6) describe? Hint: Think of the parentheses as inside dimensions. 9. Which plan does 3(6 ) describe? 10. Do the two plans have the same number of balcony seats? 004 by CompassLearning, Inc. Algebra AL016

22 The Distributive Property Multiplication 4( + 8) = = = 40 5(x + y) = 5 x + 5 y = 5x + 5y Fill in the blanks in each rewritten expression. 1. (x + y) = x + y. 4(m + 4) = (8 + 1) = (4 + c) = (p + q) = q 6. 8(a + 1) = 8a + 7. ( j + k) = k 8. 10( ) = [ q + ( 3)] = ( 3 + n) = 4n 11. 3( 7 + b) = (d + k) = 7d (5 + 1) = (n + p) = 11p 15. k(13 + m) = 13k b(y + z) = + bz Rewrite each expression, using the distributive property. Simplify where possible ( + 1) 18. 6(r + z) 19. 1(d + k) 0. 3(11 + w) 1. ( 4 + m). 8[ a + ( 3)] 3. 9(x + y) 4. 7(g + 10) 5. 8( v + 8) 004 by CompassLearning, Inc. Algebra AL017

23 The Distributive Property Factoring 4j + 4k = 4( j + k) 3x 3y = 3(x y ) Identify the common factor in each expression k + 11w. jn + kn 3. q + ( r) 4. 1b 13b 5. 3p + 46p 6. dx 3 dy a + 1.9b 8. 3r 1.5r 9. 7m + 7v Draw a line to match the expression on the left with its factored form on the right x 7y a. a(x + y) 11. ax + ay b. (x y) 1. 4x + 4y c. 3(ax + y) 13. x + y d. 7(x y) 14. 3ax + 3y e. 4(x + y) Solve the problem. 15. Two children are paid their allowances in dimes and nickels. Each child receives exactly the same number of each type of coin. Let d stand for the number of dimes each child received. Let n stand for the number of nickels each received. One way to represent total amount of allowance to the children is d + n.what is another way? (Hint: Use the distributive property to factor.) 004 by CompassLearning, Inc. Algebra AL018

24 Properties of Zero Additive Property of Zero: = = Additive Inverse Property: = ( 5) = 0 Multiplication Property of Zero: 0(6) = 0 4(0) = 0 If the two numbers are additive inverses, write true. Otherwise, write false x x Write each sum k m k y Write each product (11) 17. ( 17)(0) q 19. (xy)(0) 0. (0)( 9) 1. (cde)(0). ( jk)(0) 3. n (0)(ab 3 ) Solve the problem. 5. Jenna said to Brett, I ll give you double the number of marbles you have in your pocket. Brett replied, But I don t have any marbles in my pocket. Jenna responded, So I ll give you double nothing, which is nothing. How could Jenna say the same thing in a mathematical expression? Underline one. a. 1 + = 3 b. 0 + = c. (0) = by CompassLearning, Inc. Algebra AL019

25 Properties of 1 Multiplication Property of 1: (1)(7) = 7 (1)(y) = y Multiplicative Reciprocals: 1 8, 8: = 1 1 x, x: 1 x x 1 = 1 Complete the table by writing the reciprocal of the term and then checking your answer. Term Reciprocal Check k 6. m 1 7. c 8. 3 Solve the problems. 9. Each wedge of apple pie is 1 of the pie. How many wedges make one 5 whole pie? Complete the equation to show your answer. 1 = In a geometry study group, 6 students were each given an identical puzzle piece of a hexagon (6-sided figure). The students assembled their pieces to make a whole hexagon. What fraction of the hexagon was each puzzle piece? Complete the equation to show your answer. 6 = by CompassLearning, Inc. Algebra AL00

26 Powers and Roots 3 = (3)(3) = 9 9 = = (3)(3)(3) = = 3 Fill in the blank in each sentence. 1. (19)(19)(19) = 6,859, so 3 = 6,859.. If = 89, then 17 = 89 and 89 =. 3. If 3 16 = 6, then 6 = If = 16, then the fourth of 16 is. 5. (4)(4)(4)(4)(4) = 1,04, so 4 = 1, = 11, which means that 11 = 11. Find each square root. You may use a calculator , Solve the problems. 19. Talia is sewing a quilt with a regular checkerboard pattern that is, all the squares are identical. In each square of the checkerboard, she plans to stitch a simple flower. Talia will have to stitch 36 flowers in all. How many squares lie along one side of the quilt? 0. The volume of a cube of sugar is.197 cm 3.Circle the letter of the expression that gives the length of one edge of the cube. a b. (1.3) 3 c by CompassLearning, Inc. Algebra AL01

27 More on Powers and Roots Simplify the expression: ( x) 3 Step 1 ( x) 3 = ( x)( x)( x) Step Multiply ( ) three times: ( )( )( ) = 8 Step 3 Multiply x three times: (x)(x)(x) = x 3 Step 4 Multiply the expanded number and variable: 8x 3 Note: x = x or x 3 x = 3 x 3 x 3 = x Simplify each term. You can use a calculator. 1. (8d). ( 10n) 3. ( y) 3 4. (3m) 4 Find each value. Write all the possible roots k Answer the questions to solve the problem. If a scientist built a machine that could transport people backward in time, then normal time might be represented as a positive number and backward time as a negative number. Suppose you could square or cube backward time. 14. What would be the square of 10 units of backward time? 15. What would be the cube of 10 units of backward time? 004 by CompassLearning, Inc. Algebra AL0

28 Order of Operations = Step 1 Calculate the cube, or third power, of : 3 = 8 Step Multiply: 1 8 = 96 Step 3 Add: = 104 Find the value using the order of operations ( + 8) (7 7 4) (6 + 6) (59 55) Answer the questions to solve the problem. Mr. and Mrs. Wang plan to knock out a wall between two rooms of their house to make one larger room. One room is a rectangle 10 feet by 15 feet, so its area in square feet is 10(15). The other room is a square, 9 feet on a side, so its area is 9 square feet. What will be the total area of the new room? Circle the letter of the expression that calculates the answer. a b. ( ) c State the order to perform the operations when calculating the answer. (1) () (3) 15. Work out the answer. (You can use a calculator.) square feet 004 by CompassLearning, Inc. Algebra AL03

29 Order of Operations The answer shown below for the computation is not correct The computation is not correct because the addition was performed first. The order of operations states that the division 6 should have been performed first. The answer is 13 when the computation is performed correctly In each problem below, the computation has been performed incorrectly. For each problem, tell why the computation is incorrect. Then give the correct answer = = = = = = = = = = by CompassLearning, Inc. Algebra AL03

30 Using Square Root Square roots can be used to explore the area of a square. Suppose a forest fire causes damage to an area enclosed by an imaginary square that has an area of 5,65 square miles. What is the measure of one side measured in miles? Step 1 Remember the formula for finding the area of a square is A = s. Step Substitute 5,65 for the area. 5,65 = s Step 3 Find the square root of the area. 5,65 = s 75 = s One side of the square measures 75 miles. Use the information about the damage caused by different forest fires to answer the following questions. Forest Fire A Forest Fire B Forest Fire C Forest Fire D Forest Fire E 11 square miles 34 square miles 79 square miles 3,969 square miles 1,936 square miles 1. If the damage caused by Forest Fire A is measured in the shape of a square, what is the measure in miles of one side of the damaged area?. If the damage caused by Forest Fire B is measured in the shape of a square, what is the measure in miles of one side of the damaged area? 3. If the damage caused by Forest Fire C is measured in the shape of a square, what is the measure in miles of one side of the damaged area? 4. If the damage caused by Forest Fire D is measured in the shape of a square, what is the measure in miles of one side of the damaged area? 5. If the damage caused by Forest Fire E is measured in the shape of a square, what is the measure in miles of one side of the damaged area? 004 by CompassLearning, Inc. Algebra AL04 AGS Publishing. Permission is granted to reproduce for classroom use only.

31 Properties of Real Numbers: Calculator Lesson at a Glance In this lesson, you will explore real number relationships and operations using a calculator. Review the computer-based activity AL154 to access the Ticker Tape Calculator and the interactive version of the lesson. Warm-up The Lost Boys It is time for the math club to leave the amusement park, but five of the members are still missing. Locate the boys by matching them to their rides and t-shirt colors. Use the given clues to find the boys before the bus leaves. Place an X in a box if it is not a match. Place an O in the box if it is a match. Clues: Jason is on the Tidal Wave, but he is not wearing yellow. Brad is wearing an orange t-shirt, but he is not on the Splash Down. The boy who is on the Crash Zone is wearing green. Brian is wearing a yellow t-shirt. The boy wearing the red t-shirt is on the Super Screamer. Matt is not wearing green. 006 by CompassLearning, Inc. Algebra AL154

32 Case Studies Jeff and Jane are preparing for a math exam. Participate in their study session and record your solutions. Case Study 1 Pizza Puzzle While studying for their math exam, Jane and Jeff had a pizza delivered to them. The pizza cost $1.50, and the tip cost $.00. To figure out how much each of them should pay, they used these keystrokes on their own calculators. Jeff got 13.5, while Jane got 7.5. Why did they get two different answers? Whose answer is correct? Try using Jeff and Jane s keystrokes on the Ticker Tape Calculator. Which answer do you get? What does this tell you about the Ticker Tape Calculator? Case Study Parenthesis Parable Jeff and Jane tried solving the expression 6 4 by using the parentheses key to group the numbers. They were able to create four expressions that gave different answers. ( 6) 4 15 (6 ) ( 6 ) 4 96 (( 6) ) Create your own expression using the numbers 3,, 1, and in the same order. How many different answers can you find by placing parentheses in the expression? Can you think of an expression that would not be affected by parentheses? 006 by CompassLearning, Inc. Algebra AL154

33 Case Study 3 Laws Defied Jeff knows the commutative and associative properties of addition and multiplication. a b b a ab ba (a b) c a (b c) (ab)c a(bc) He tried applying the same rules to subtraction and division. a b b a a b b a (a b) c a (b c) (a b) c a (b c) Are these rules valid? Can any of these expressions ever be true? Case Study 4 Distributive Dishes Jane ordered food for Jeff, her three friends, and herself. Here is the list of prices: The cashier entered the prices for five hamburgers, then five French fries, and then five large drinks on the cash register: 5(.50)? 5(0.95)? 5(1.5)?? Jane used the distributive property to quickly check the bill. What did Jane do? Use the calculator to add the bill using both the cashier s and Jane s methods. Were the answers the same? Record the keystrokes used. 006 by CompassLearning, Inc. Algebra AL154

34 Case Study 5 Backstroke Techniques Jeff and Jane are exploring different ways to write and enter an expression on their calculators. They figured out the keystrokes for the first two rules, but they need help on the last one. Here are their thoughts and entries. Dividing by a value is equal to multiplying by its reciprocal. Example: Keystrokes: Subtracting a value is equal to adding its opposite. Example: 5 5 Keystrokes: Finding the root of a value is equal to raising it to the reciprocal of the index. Example: 4 65 = 651/4= Help Jeff and Jane. How is the last set of expressions entered into the calculator? Record your keystrokes and answer. 006 by CompassLearning, Inc. Algebra AL154

35 Writing Equations 10 times some number equals x = 30 Write an equation for each statement. Let x be the variable in the equation times some number equals 30.. times some number plus 5 equals times some number minus 8 equals subtracted from some number equals times some number plus 7 equals subtracted from some number equals. 3x = 18 x = 4, 5, 6 (3)(4) = 18 F (3)(5) = 18 F (3)(6) = 18 T 7. 5p = 15 Find the root of each equation by performing the operation on each value for the variable. Write T if the equation is true or F if the equation is false k = 80 p = 1 k = 7 p = k = 8 p = 3 k = w = a = w = 5 a = 11 w = 6 a = 1 w = 7 a = by CompassLearning, Inc. Algebra AL05

36 Solving Equations: x b = c Solve m = 8 for m. Step 1 Write the equation. m = 8 Step Add to both sides of the equation. m + = 8 + Step 3 Simplify. m = 10 Step 4 Check. 10 = 8 Solve each equation. Check your answer. 1. x 4 = 0. b 7 = 1 3. n = 7 4. k 13 = 3 5. d 100 = c 11 = 0 7. y 4 = r 80 = 0 9. w 8 = 0 Read the problem and follow the directions. 10. A sports store buys a shipment of catcher s mitts at the beginning of the year. By year s end, the store has sold 100 mitts and has 150 left. How many mitts did the store have at the first of the year? Let x stand for the number of catcher s mitts the store had at the beginning of the year: x 100 = 150 How would you solve this equation? Circle the answer. a. Subtract 100 from both sides. b. Add 100 to both sides. c. Subtract 150 from both sides. 004 by CompassLearning, Inc. Algebra AL06

37 Solving Equations: x + b = c Solve k + 5 = 9 for k. Step 1 Write the equation. k + 5 = 9 Step Subtract 5 from both sides of the equation. k = 9 5 Step 3 Simplify. k = 4 Step 4 Check = 9 Solve each equation. Check your answer. 1. w + 3 = 4. r + 8 = 1 3. y + = 7 4. c = k + = d + 9 = n + 5 = b + 7 = x + 3 = 38 Read the problem and follow the directions. 10. Amy read 17 books over the summer, 11 more than Tim. How many books did Tim read? Let r stand for the number of books Tim read over the summer: r + 11 = 17 How would you solve this equation? Circle the answer. a. Add 11 to both sides. b. Subtract 11 from both sides. c. Subtract 17 from both sides. 004 by CompassLearning, Inc. Algebra AL07

38 Solving Multiplication Equations Solve 4y = 16 for y. Step 1 Write the equation. 4y = 16 Step Multiply both sides of the equation by 1, the reciprocal of 4. 4 ( 1 4 )4y = 16( 1 4 ) Step 3 Simplify. y = 4 Step 4 Check. 4 4 = 16 Note: Another way to do Step is to divide both sides of the equation by 4. 4 y 4 = Solve each equation. Check your answer. 1. q = k = x = d = w = j = y = n = p = 1.8 Read the problem and follow the directions. 10. Juanita is 15, five times the age of her brother, Frank. How old is Frank? Let a stand for Frank s age: 5a = 15 How would you solve this equation? Circle the answer. a. Subtract 5 from both sides of the equation. b. Multiply both sides of the equation by 5. c. Multiply both sides of the equation by by CompassLearning, Inc. Algebra AL08

39 Solving Equations with Fractions Solve 1 y = 5 for y. 3 Step 1 Write the equation. 1 3 y = 5 Step Multiply both sides of the equation by the reciprocal of the fraction. ( 3 1 ) 1 3 y = ( 3 1 )5 Step 3 Simplify. y = 15 Step 4 Check. 1 3 (15) = 5 Solve each equation. Check your answer x = w = q = 7 4. r = m = k = d = y = a = 4 5 Read the problem and follow the directions. 10. Spruceville received 35 inches of snow last winter, or 5 of its average 8 annual snowfall. What is Spruceville s average annual snowfall? Let n stand for Spruceville s average annual snowfall: 5 n = 35 8 How would you solve this equation? Circle the answer. a. Subtract 35 and then multiply by 5 8. b. Multiply both sides by 8 5. c. Subtract 5 from both sides by CompassLearning, Inc. Algebra AL09

40 Solving Equations More Than One Step Solve m 7 = 1 for m. Step 1 Add 7 to both sides. m 7 + (7) = 1 + (7) Simplify: m = 8 Step Divide both sides by. m = 8 Simplify: m = 4 Step 3 Check. (4) 7 = 1 Simplify: 1 = 1 Solve each equation. Check your answer. 1. 4b + 1 = 17. c + 18 = x = p + 8 = v 1 = w 16 = g + 5 = 35 One step is missing in the solution to each equation. In a complete sentence, write the missing step. 8. 7y 4 = 10 Step 1 Step Add 4 to both sides of the equation. 9. d + 1 = 19 Step 1 Step Divide both sides of the equation by k + 3 = 1 Step 1 Step Subtract 3 from both sides of the equation. 004 by CompassLearning, Inc. Algebra AL030

41 Equations Without Numbers ax b = c Solve for x. Step 1 Write the equation. x 5 = 7 ax b = c Step Add 5 or b to both sides. x = ax b + b = c + b x = 1 ax = b + c Step 3 Divide each side by or a. x = 1 a x = b + c a a Step 4 Check. x = 6 x = b + c a (6) 5 = 7 a( b + c ) b = c a 7 = 7 (b + c) b = c c = c Solve each equation for x.check your answer. 1. ax c = b. bc = ax 3. x b + a = c 4. abx = c Follow the directions to solve the problem. 5. Center School has won two more soccer games than the combined wins of River School and Bluff School. This statement can be turned into a mathematical equation. Let x stand for the number of games Center School has won. Let y stand for the number of games River School has won. Let z stand for the number of games Bluff School has won. x = y + z + Solve the equation for z to show the number of soccer games Bluff School has won. 004 by CompassLearning, Inc. Algebra AL031

42 Formulas Solve the formula for the perimeter of a triangle for c. P = a + b + c a + b + c = P (a + b) + c (a + b) = P (a + b) c = P (a + b) Solve each equation. 1. P = 6s for s. A = 1 h(m + n) for m 3. P = b + h for b 4. A = lw for l 5. P = a + b + c for b 6. A = 1 h(m + n) for n 7. V = hm for m 8. P = b + h for h Answer the questions to solve the problem. Mr. Jiang is building a deck on his house. He can calculate the perimeter of the deck using this formula: P = 4s 9. Solve the equation to show the length of a side that is, solve for s. 10. Is Mr. Jiang s deck in the shape of a square or a rectangle? Explain your answer. 004 by CompassLearning, Inc. Algebra AL03

43 The Pythagorean Theorem Use the Pythagorean theorem: c = a + b Find c when a = 4 and b = 5. Use a calculator, and round the answer to the nearest tenth. c = (4) + (5) c = c = 41 c = 41 = Round off: 6.4 Use the Pythagorean theorem and a calculator to find the missing side of each triangle. Round to the nearest tenth. 1. a = b = 7 c =. a = b = 6 c = a = b = 8 c = a = 9 b = c = 36 Solve the problem. 5. A sailboat has a sail in the shape of a right triangle. You know that side a is m long and side b is 4 m long. How long is side c of the sail? Substitute known values in the Pythagorean theorem and solve. Use your calculator and round to the nearest tenth. c b a 004 by CompassLearning, Inc. Algebra AL033

44 Using the Pythagorean Theorem The lengths of the sides of a triangle are 3 ft, 4 ft, and 5 ft. Is the triangle a right triangle? When the lengths of the sides of a right triangle are given, the longest length is the hypotenuse. Substitute 3, 4, and 5 into the formula a + b = c. a + b = c = = 5 5 = 5 True When the lengths of the sides of a triangle are 3 ft, 4 ft, and 5 ft, the triangle is a right triangle. The lengths of the sides of a triangle are 10 cm, 13 cm, and 15 cm. Is the triangle a right triangle? When the lengths of the sides of a right triangle are given, the longest length is the hypotenuse. Substitute 10, 13, and 15 into the formula a + b = c. a + b = c = = 5 69 = 5 False When the lengths of the sides of a triangle are 10 cm, 13 cm, and 15 cm, the triangle is not a right triangle. The lengths of the sides of various triangles are given below. Is the triangle a right triangle? 1. 4 in., 5 in., 7 in.. 5 cm, 1 cm, 13 cm 3. 1 mm, 4 mm, 3 mm 4. yd, 3 yd, 13 yd m, 68 m, 85 m 004 by CompassLearning, Inc. Algebra AL033

45 Inequalities on the Number Line Write a statement of inequality for the number line. Use x as the variable x 1. Write a statement of inequality for each number line. Use x as the variable. o o o Write T if the disjunction is True or F if it is False by CompassLearning, Inc. Algebra AL034

46 Solving Inequalities with One Variable Solve x 5 > 3 for x. Step 1 Write the inequality. x 5 > 3 Step Add 5 to both sides of the inequality. x > Step 3 Simplify. x > 8 Note: For inequalities with addition, multiplication, or fractions, solve in the same way as for equations with the same operations. Solve each inequality. 1. x 3 > 0. 5d > k + 11 < q > c g 1 < p < 1 8. w + 9 Solve the problems. 9. A school has arranged teaching loads so that no teacher ever has more than 5 students. Describe the school s teaching load using an inequality and the variable t. 10. The sponsor of a concert promises the concert singer a fee based on $5 per person in the audience. If attendance is below 00, however, the singer will be paid a minimum fee based on 00 seats filled. Using the variable f, write an inequality to represent the singer s minimum fee. 004 by CompassLearning, Inc. Algebra AL035

47 Using Equations You can solve problems by setting up equations. First identify what you need to find out that s your unknown. Assign a letter to represent the unknown and then create an equation that relates the unknown to the known. Finally, solve the equation. The Canada Trust Tower in Toronto has one-half as many stories as the Empire State Building in New York City. The Canada Trust Tower has 51 stories. How many stories does the Empire State Building have? Step 1 Read the problem carefully to set up the equation. j = number of stories in the Empire State Building 1 j = 51 Step Solve the equation. 1 j = 51 (multiply each side by the reciprocal of 1 ) 1 j = 51 j = 10 Step 3 Write your answer in sentence form. The Empire State Building has 10 stories. Use equations to solve the following problems about buildings around the world. 1. The Central Plaza is the tallest building in Hong Kong, measuring 1,7 feet, which is 18 feet taller than Hong Kong s Bank of China Tower. How tall is the Bank of China Tower?. The AT&T Corporate Center in Chicago is 60 stories tall. The Sears Tower in Chicago has 50 more stories than the AT&T Corporate Center. How many stories tall is the Sears Tower? 3. The Central Plaza in Hong Kong is 78 stories. The Moscow State University building in Moscow has 1 3 the number of stories of the Central Plaza. How many stories does the Moscow State University building have? 4. The Kompleks Tun Abdul Razak Building in Penang, Malaysia, is 760 feet tall. The world s tallest buildings, the Petronas Twin Towers in Kuala Lumpur, Malaysia, are 73 feet taller than Kompleks Tun Abdul Razak Building. How tall are the Petronas Twin Towers? 5. NationsBank Plaza in Atlanta stands 1,03 feet tall. One Ninety-One Peachtree Tower, also in Atlanta, is 53 feet shorter. How tall is One Ninety-One Peachtree Tower? 004 by CompassLearning, Inc. Algebra AL036 AGS Publishing. Permission is granted to reproduce for classroom use only.

48 Modeling Equations: Algebra Balance Lesson at a Glance In this lesson, you will represent and solve linear equations with one variable using the Algebra Balance. Review the computer-based activity AL156 to access the Algebra Balance and the interactive version of the lesson. Warm Up Five Times the Trouble The Quinn quintuplets are crying. Calm the babies! Match the babies with their colored blankets and favorite stuffed toys. No two babies like the same color or toy. Use the given clues to solve the puzzle. Place an X in the box if it is not a match. Place an O in the box if it is a match. Clues: Abby loves ponies but hates the color red. Billy only likes 4-legged animals and the color blue. Christy doesn t like dogs or ducks. The boy with the green blanket likes the bear. The baby that likes the cat has a yellow blanket. Eddy and Billy do not like bears. 006 by CompassLearning, Inc. Algebra AL156

49 Case Studies Jack and Wendy are exploring linear equations with one variable using the Algebra Balance. Participate in their study session and record your solutions. Case Study 1 Can We Both Be Right? Jack and Wendy are solving for x in the equation x 3 1. Wendy added a value to both sides to solve for x.jack subtracted a value from both sides to solve for x.can both Jack and Wendy be correct? Explain why or why not. Case Study Is Negative Lighter? While Jack was solving the equation x -3 10, he noticed something interesting. He knew that placing a unit on the Algebra Balance lowered the pan, but he found that placing a -1 unit on a pan raised it. He tried it with the x unit and found the same pattern. Placing an x unit on the pan lowered it and placing a -x unit on the pan raised it. He shared this observation with Wendy. Wendy thought that this would not work for all equations. Do you agree with Wendy? Why or why not? 006 by CompassLearning, Inc. Algebra AL156

50 Case Study 3 Mixing Positives and Negatives Now that Jack and Wendy have worked with negative and positive variables, they are creating general rules about how to solve for x using the least number of steps. They call this process isolating the variable. Review their steps: Did they solve for x using the least amount of steps? If not, what could they have done instead? Solve the equation x 3-5 using the least amount of steps. Record your procedure. Case Study 4 It s All the Same Wendy was solving the equation x -. After setting up the Algebra Balance, she continued to solve for x.by accident, she multiplied by rather than subtracting. The equation became x 4-4. Wendy worried that she made a mistake. Jack told her, No problem. It s an equivalent equation. You can still solve for x. What do you think Jack means? Create some more equivalent equations for x by CompassLearning, Inc. Algebra AL156

51 Case Study 5 No Balance? Jack and Wendy were entering their own equations into the Algebra Balance. They tried to enter the equation 5x 3, but they received a message that said, The equation is either not valid or cannot be displayed on the scale. Jack stated, That doesn t make sense. You should be able to place five x units on the left pan and three 1 units on the right pan. Wendy countered, Maybe it s not about whether you can put the units on the scale. Maybe it s whether you can solve the equation on the scale. Jack wasn t convinced. He wanted to see more examples. Solve the equation 5x 3. Then, help Jack by creating additional equations that cannot be solved using the Algebra Balance. 006 by CompassLearning, Inc. Algebra AL156

52 Writing Equations Odd and Even Integers Two times a number added to 4 is 14. What is the number? Step 1 Let n = the number. Then n is two times the number. Step Write and solve the equation. 4 + n = n = 14 4 n = 10 n = 5 Step 3 Check: 4 + (5) = = 14 Write an equation for each statement. Use n as the variable. 1. Three times a number added to is 5.. Four times a number decreased by 5 is Five added to 8 times a number is Seven times a number minus is Eleven times a number added to 10 is Eight times a number decreased by 5 is Four added to 9 times a number is Ten times a number decreased by 1 is Nine times a number minus 14 is. Solve the problem. 10. A cook in a cafeteria has only 7 slices of rye bread left at closing time. An assistant immediately goes to a store and buys 4 identical loaves of sliced rye bread. With this additional supply, the cafeteria now has 79 slices of rye bread. How many slices are in one packaged loaf of rye bread? 004 by CompassLearning, Inc. Algebra AL037

53 Using the 1% Solution to Solve Problems A train has traveled 150 miles toward its destination. This distance represents 30% of the total trip. What will the total mileage be? Step 1 30% of mileage = 150 Given. Step 30% 30 = Divide both sides by 30 to solve for 1%. 1% of mileage = 5 Step 3 100% of mileage = 500 Multiply both sides by 100 to find the total mileage. Use the 1% method to find a number when a given percentage of the number is known. 1. 5% of a number is % of a number is % of a number is % of a number is % of a number is % of a number is % of a number is % of a number is 90. Solve the problems. 9. At Mayville Community College, 490 students are enrolled in the computer program. If 70% of the students in the college are in the computer program, how many students does Mayville Community College have in all? 10. An airline reports that 9% of its flying customers last year were under 1 years of age. If 70,000 children under 1 years of age flew on the airline s planes, how many customers did the airline have last year? 004 by CompassLearning, Inc. Algebra AL038

54 Using the Percent Equation 30% of the pencils in a box of 0 pencils are red. How many red pencils are there? Step 1 Write the percent equation. ( 1 p 00 )(n) = r 30 Step Change the percent into a fraction. p = 30 ( 1 )(n) = r 30 Step 3 Write the total number. n = 0 ( 1 )(0) = r 3 Step 4 Simplify the fraction. ( 1 )(0) = r 0 3 Step 5 Solve the equation. ( 1 )(0) = % of 0 is 6. There are 6 red pencils in the box. Use the percent equation to find the percent of each given number % of 40. 0% of % of % of % of % of % of % of % of 1,000 Solve the problem. 10. Last summer, Jan gave away 15% of the tomatoes she raised in her garden. Jan picked a total of 10 tomatoes. How many did she give away? 004 by CompassLearning, Inc. Algebra AL039

55 Using Percents This chart displays population data of the world s ten largest cities. Population of the World s Ten Largest Cities Rank City/Country Population 1994 Projected Population Tokyo, Japan 6,518,000 8,700,000 New York City, U.S. 16,71,000 17,600,000 3 São Paulo, Brazil 16,110,000 0,800,000 4 Mexico City, Mexico 15,55,000 18,800,000 5 Shanghai, China 14,709,000 3,400,000 6 Bombay (Mumbai), India 14,496,000 7,400,000 7 Los Angeles, U.S. 1,3,000 14,300,000 8 Beijing, China 1,030,000 19,400,000 9 Calcutta, India 11,485,000 17,600, Seoul, South Korea 11,451,000 13,100,000 The chart shows the 1994 population in each city and a projection of the population in the year 015. Use the chart to answer these questions. 1. In which city is the population expected to increase the most? By how many people is the population expected to increase?. In which city is the population expected to increase the least? By how many people is the population expected to increase? 3. Which city is expected to experience the greatest percent of increase? By what percent, to the nearest whole number, is the population expected to increase? 4. Which city is expected to experience the least percent of increase? By what percent, to the nearest whole number, is the population expected to increase? 5. In 1996, the population of the world was 5,771,938,000 people. By 00, the world s population is projected to increase to 7,601,786,000 people. By what percent, to the nearest whole number, is the population of the world expected to increase between 1996 and 00? 004 by CompassLearning, Inc. Algebra AL039

56 Solving Distance, Rate, and Time Problems The distance formula is d = rt. d stands for distance, r for rate of speed, and t for time. Find d when r = 0 kilometers per hour (km/h) and t = hours. Solve: d = (0)() = 40 kilometers Use r = d t to solve for rate of speed. Find r when d = 33 miles and t = 3 hours. Solve: r = 3 3 = 11 miles per hour (mph) 3 Use t = d r to solve for total time. Find t when d = 450 kilometers and r = 90 km/h. 50 Solve: t = 4 = 5 hours 90 Use the appropriate version of the distance formula to find the unknown value. 1. d =? r = 5 mph t = 1 hour Answer in miles.. d =? r = 38 km/h t = 3 hours Answer in kilometers. 3. d = 90 miles r = 60 mph t =? Answer in hours. 4. d = 1,968 kilometers r =? t = 4 hours Answer in km/h. 5. d = 54 miles r = 18 mph t =? Answer in hours. 6. d =? r = 7 km/h t = 1 hour Answer in kilometers d = 33 kilometers r =? t = 4 hours Answer in km/h. 8. d = 14 miles r = 70 mph t =? Answer in hours. 9. d = 1,33 miles r =? t = 3 hours Answer in mph. 10. d =? r = 96 km/h t = 3 hour Answer in kilometers by CompassLearning, Inc. Algebra AL040

57 Using a Common Unit Cents In algebra equations, represent money as cent values. Change dollar amounts into cents by multiplying by 100. penny = 1 cent nickel = 5 cents dime = 10 cents quarter = 5 cents dollar = 100 cents Suppose you have $3.00. How many cents do you have? = 300 cents To represent money amounts, multiply the number of coins by value. Suppose you have n nickels + twice as many (n) dimes. What is the money value? n(5) + n(10) or 5n + 0n Write the value of each amount of money in cents. 1. $1.9. $ $ $.5 5. $ $0.89 For each group of coins, choose the expression that gives the total money value of the coins in cents. Circle the letter of your choice. 7. n quarters plus twice as many dimes a. n(5) + n(10) b. n + n 8. n nickels plus half as many quarters a. n + 1 n b. n(5) + 1 n(5) 9. n dimes plus 4 times as many pennies a. n(1) + 4n(10) b. n(10) + 4n(1) Write an expression for the money value of the given coins, in cents. 10. n dimes + 1 as many quarters by CompassLearning, Inc. Algebra AL041

58 Calculating Simple Interest Use the formula I = prt to calculate simple interest. How much interest will $300 at 6% interest earn in 1 year? I = $300(0.06)(1) = $18 I Use the formula p = r to solve for p and calculate principal. t Find the principal in an account that has a rate of 5% and earns $75 interest in years. $ 75 p = (0. = $750 05)() I Use the formula r = p to solve for r and calculate rate of interest. t What is the rate of interest if $1,000 earns $40 interest in 3 years? $ 40 r = ($1 = 0.08, or 8%,000)(3) Calculate interest on the given principal, rate, and time. 1. p = $00 r = 3% t = 1 year. p = $70 r = 4% t = 1 year 3. p = $1,400 r = 7% t = years Calculate principal from the given interest, rate, and time. 4. I = $90 r = 6% t = 3 years 5. I = $7 r = 3% t = 1 year 6. I = $56 r = 8% t = 1 year Calculate rate from the given interest, principal, and time. 7. I = $63 p = $700 t = 1 year 8. I = $4 p = $150 t = 4 years 9. I = $48 p = $960 t = 1 year Solve the problem. 10. For years, Will kept $655 in a savings account that earned 4% annual interest. How much interest did Will earn? 004 by CompassLearning, Inc. Algebra AL04

59 Deriving a Formula for Mixture Problems Use this formula to determine the cost of a mixture: cost of total mixture Price per pound = number of pounds Peanuts cost $3.00 per pound. Cashews cost $6.00 per pound. Suppose you mix 4 pounds of peanuts with pounds of cashews. What will the mixture cost, per pound? Price per pound = 4($3 )+ ($6) 4 + = $ 4 6 = $4 Use the information in each table to answer the questions that follow it. The formula you will need is in the example on this page. 1. Fill in the formula with this data. Item Cost per pound Number of pounds beans $.00 7 dried tomatoes $ Find the cost of this mixture. 3. Fill in the formula with this data. Item Cost per pound Number of pounds popcorn $9.00 peanuts $ Find the cost of this mixture. Solve the problem. 5. Suppose a grocery store mixes 4 pounds of oat cereal with 1 pound of almonds. The oat cereal costs $1.0 per pound, and the almonds cost $4.80 per pound. What should the mixture cost per pound? 004 by CompassLearning, Inc. Algebra AL043

60 Ratio and Proportion 3 = 4 because the cross products are equal = 4 3, so 1 = 1 6 Find the missing term in the proportion 5 = x by making an 10 equation from the cross products. Then solve the equation. 5 x 10 0 = 5x or 5x = 0 x = 4 Therefore, 5 = Tell whether each equation is a proportion. Prove your answer = = = = = = = = 3 18 Find the missing term in each proportion = x 15 x = = 6 x x = = x 7 x = 1. 1 x = 10 x = = 4 x x = = x 14 x = x = x = 004 by CompassLearning, Inc. Algebra AL044

61 Using Proportions If the large gear on a bicycle turns 3 times, how many times will the small gear turn? Step 1 Find how many times the small gear turns for one turn of the large gear. L arge g Smallg ear ear = = 9 4 = 1 4 For one turn of the large gear, the small one turns 9 4 times. Large gear or 1 4 Small gear 1 turn of large gear 3 turns of large gear Step Write the proportion. = 9 4 turns of small gear x Step 3 Rewrite the proportion algebraically. Then solve for x. 1 3 = 9 4 x x = (3)( 9 4 ) x = 7 = For three turns of the large gear, the small one turns times. Solve each problem. 1. Lara can run 8 laps in 0 minutes. At that rate, how many laps can she run in 80 minutes?. Colin rides his bike 39 miles in 3 hours. If he rides at the same speed, how many miles can he ride in 5 hours? 3. If 4 gallons of gas cost $5.44, how much do 11 gallons cost? 4. A recipe that makes dozen cookies calls for 4 1 cups of flour. How much flour would be needed to make 7 dozen cookies? 5. The indoor track at Sylvia s gym is short, and she has to run 14 laps to go one mile. How many laps must Sylvia run to go 1 miles? 6. Juan can swim 16 laps at the pool in 4 minutes. If he keeps swimming at the same rate, about how many laps can he swim in one hour? 7. The scale on a map reads 3 inches = 100 miles. If two cities measure 4 1 inches apart, how far apart are they in actual miles? 8. The scale on a map reads 3 inches = 100 miles. If the actual distance between two cities is 30 miles, what is the distance on the map? 9. The scale on a map reads inches = 5 miles. If two towns measure 7 3 inches apart, how far apart are 8 they in actual miles? 10. The scale on a map reads 4 cm = 30 miles. If the actual distance between two towns is 10 miles, how far apart are they on the map? 004 by CompassLearning, Inc. Algebra AL045 AGS Publishing. Permission is granted to reproduce for classroom use only.

62 Using Formulas: Calculator Lesson at a Glance In this lesson, you will investigate manipulating formulas and conserving the order of operations when using a calculator. Review the computer-based activity AL158 to access the Ticker Tape Calculator and the interactive version of the lesson. Warm-up Please Excuse My Dear Aunt Sally My dear aunt enjoys creating puzzles. She made this one especially for math lovers. To solve her puzzle, you must correctly apply the order of operations. It s time to begin the challenge! Write arithmetic operators in the empty squares between the numbers. Each row and column must form a correct equation. 006 by CompassLearning, Inc. Algebra AL158

63 Case Studies Logan and his grandfather are planning a hiking trip. They are using formulas and calculators to compute costs and travel time. Join Logan and Grandpa Al as they use the Ticker Tape Calculator to evaluate formulas and expressions. Case Study 1 Hiking Sense Logan and Grandpa Al are planning to backpack 8 miles over 4 days. They want to complete 30% of the distance on the first day. Logan is trying to figure out how far they should hike before they set up camp on their first day. He is using the percent equation. Percent Equation Percent Number Result p n r p percent n number r result Logan then notices that there is no percent key on the Ticker Tape Calculator. He knows that percent means per hundred. How can Logan change the equation so the percent key is unnecessary? What keystrokes should Logan enter into the Ticker Tape Calculator to find how far he should hike the first day? 006 by CompassLearning, Inc. Algebra AL158

64 Case Study Hiking up the Trail Grandpa Al thinks he and Logan could easily hike 1 miles the first day and wants to lengthen the total number of miles of their trip. How should Logan rearrange the percent equation to solve for the total number? Percent Equation Percent Number Result p n r p percent n number r result If 1 miles is 30% of the hike, what keystrokes should Logan enter to find the total number of miles? What is the answer? Case Study 3 Simply Interested Logan wants to purchase trekking poles for his grandfather. A pair costs $97, but Logan doesn t have the money right now. The sporting goods store offered a credit plan where Logan would be charged % interest per month. Logan doesn t want to pay more than $30 in interest and wants to calculate how soon he needs to pay back the money. He rearranges the simple interest formula to solve for time. He also converts % into the decimal 0.0. I Simple Interest Formula t p r Logan s entry and the resulting display look like this: Logan knows that this is not right. Identify the error made and the keystroke sequence Logan should have used. 006 by CompassLearning, Inc. Algebra AL158

65 What is the correct answer? Case Study 4 Leisurely Rate Logan and Grandpa Al are enjoying their backpacking trip. They walked miles in 45 minutes. But, when they hit a steep and rocky climb, they traveled 3 miles in 4 hours. Logan plans to calculate their average speed using the rate formula. average rate to tal distance total time miles + 3 miles 45 minutes + 4 hours Logan s keystrokes into his portable calculator give a decimal answer that he struggles to understand. Logan figures that they are traveling about one-tenth of a mile per hour. Do you agree with his answer? Explain why or why not. Case Study 5 Mixing Ratios Logan and Grandpa Al are debating how much GORP, also known as Good Ole Raisins and Peanuts, to prepare. Logan wants to create a mixture that has a ratio of :1 or twice as many peanuts as raisins. Logan found how many calories are provided by a 000-gram bag of GORP by using the information in the chart. Logan created a formula for this mixture problem and found the total to be 9800 calories. peanuts raisins (( 3 )(000)( 5 85 )) + (( )(000)( 4 98 )) = by CompassLearning, Inc. Algebra AL158

66 Exponents To multiply terms with exponents, add the exponents. n n = n + = n 4 To raise a power to a power, multiply the exponents. (n ) 3 = n 3 = n 6 To divide terms with exponents, subtract the exponents. n 5 n = n 5 = n 3 (Note: n 0.) Use the rule for dividing terms with exponents to find each answer y 5 y 3, y 0 3. m 8 m 4, m 0 4. r 9 r, r (j + k) 4 (j + k),(j + k) w 7 w 7, w 0 Answer the questions to solve the problem. A square has a side s that is 3 m long. The formula for area of a square is A = s.fill in the blank to show how to calculate the area of this square. s = 3 m s 9. Area = ( ) square m Next calculate the area of the square by using the rule for raising a power to a power. (See the previous answer.) 10. Area = square m 004 by CompassLearning, Inc. Algebra AL046

67 Negative Exponents Rewrite with a negative exponent: 10. Rewrite x 1 3 with a negative exponent, x 0: x 3. Rewrite 5 with a positive exponent: 15. Rewrite n with a positive exponent, n 0: n 1. Rewrite using a negative exponent c, c y 1 7, y (3j k) Rewrite using a positive exponent x y (d + 3k) n (m 3n) Simplify each power of. 1. =. 1 = 4. 1 = 5. = 3. 0 = 004 by CompassLearning, Inc. Algebra AL047

68 Exponents and Scientific Notation Write and in scientific notation. Step 1 Step Step Count decimal places: to the left (Rule: Use a positive exponent if the decimal point moved left.) Count decimal places: 3 to the right (Rule: Use a negative exponent if the decimal point moved right.) Write each number in scientific notation. 1. 9, ,000 6.,30, , , ,839, ,000,000, ,000, Solve the problems. 19. A certain bacteria cell is mm thick. Write this measurement in scientific notation. 0. Dinosaurs became extinct (that is, died out) about 65 million years ago. This number is written out as 65,000,000. Rewrite it in scientific notation. 004 by CompassLearning, Inc. Algebra AL048

69 Computing in Scientific Notation Multiply: ,150,000 Step = ,150,000 = Step ( )( ) = ( )( ) Step 3 (35.75)( ) = (35.75)(10 3 ) Step 4 (3.575)(10)(10 3 ) = (3.575)( ) = (3.575)(10 4 ) Find each product. Write the answer in scientific notation ,400, ,500, , Divide: 110,000, Step 1 110,000,000 = = Step = (1.1.5)( ) Step 3 (0.44)(10 8 ) = 0.44(10 6 ) Step 4 4.4(10 1 )(10 6 ) = (4.4)( ) = (4.4)(10 5 ) Find each quotient. Write the answer in scientific notation ,000,000 3, , , , by CompassLearning, Inc. Algebra AL049

70 Defining and Naming Polynomials The chart summarizes the kinds of polynomials. The greatest power of a variable is called the degree of a polynomial. Expression Name of the Polynomial Degree y monomial y + 5 binomial y 3y + 5 trinomial y 3 + y 3y + 5 polynomial 3 Fill in the missing data in the chart. Write on each numbered blank. Expression Name of the Polynomial Degree 3n + n 1. k 3 k + k 4 polynomial. 5x monomial 3. 3x 4. 7y + 4y 5 trinomial 5. n 3 + n 8n n k k b Each expression is described incorrectly. Write what is wrong with the description. 13. k + 5 binomial in k,degree 14. y 4 trinomial in y,degree 15. r 3 r 3r + 7 polynomial in x,degree by CompassLearning, Inc. Algebra AL050

71 Adding and Subtracting Polynomials Add (x 3 + 4x + 8) and (x 3 x x + 3). x 3 + 4x x 3 x x + 3 3x 3 + x x + 11 Subtract (x 3 4x 1) from (4x 3 + 7x 3x + 4). Find the opposite of the expression to be subtracted: ( 1)(x 3 4x 1) = x 3 + 4x + 1 Add: 4x 3 + 7x 3x x 3 + 4x + 1 x 3 + 7x + x + 5 Find each sum. 1. (5k 3 9k + 1k 3) and (k k + k + 14). (3y 4 + y 7y) and (5y 3 y + y + 9) Find each difference. Remember to add the opposite. 3. (4n 8n + 3) (3n + 5n 4) 4. (6x 4 x 3 + 1x 16) (5x 3 + ) Solve the problem. 5. A nut company will close one of its two stores and combine all of the inventory (the nuts in stock) from the two stores. The following polynomials give the number of bags of dry-roasted peanuts in each store: Lakewood store: x + 4x Downtown store: 3x x + 4 Find the combined inventory of the dry-roasted peanuts. 004 by CompassLearning, Inc. Algebra AL051

72 Multiplying Polynomials Multiply (n + )(n 3). Use the distributive property. (n + )(n 3) = n(n 3) + (n 3) = n 3n + n 6 = n n 6 Find each product. 1. (k + 4)(3k 6). ( x + 5)( x 10) 3. (c + c) 4. (d )(d + 5) 5. (y + 9)(y y + 6) 6. (n 3)(n 3 + 6n 3) 7. (w 3 + 1)(w 4) 8. ( x 5)(3x 5 + x 4 4x 3 + 6x ) 9. (c 3 )(c 3c + 9) 10. (k + 7)(k 5) 11. (m 3)(m 4 3m 3 7m 1) 1. (8b + 3)(b 9) 13. (k 3 )(k 6 + k 5 3k 4 8k 3 + 4k 9k 9) 14. (7n 4)(3n 3 8n 5n + 8) Solve the problem. 15. A machine in a factory turns out a large metal grid (a crisscross or checkerboard pattern), which later gets cut into small pieces for computer parts. The measurements of this large grid are as follows: length: (3x + 1) width: (x 3) Find the area of this grid, using the formula: Area = length width. 004 by CompassLearning, Inc. Algebra AL05

73 Special Polynomial Products Each of these polynomial products forms a pattern. (a + b) = (a + b)(a + b) = a(a + b) + b(a + b) = a + ab + b (a b) = (a b)(a b) = a(a b) b(a b) = a ab + b (a + b)(a b) = a(a b) + b(a b) = a b (a + b) 3 = (a + b)(a + b)(a + b) = (a + b)[(a + b)(a + b)] = a 3 + 3a b + 3ab + b 3 Study each product. Decide what polynomials were multiplied to give the product. Use the example above as a guide. Write your answer in the blank. 1. m + mn +n. j k 3. x + xy +y 4. c 3 + 3c d + 3cd + d 3 5. w x 6. g gh + h Find each product. Compare your solutions with the patterns in the example, above. 7. (p + r)(p r) 1. (t + u)(t + u)(t + u) 8. (f + g) 9. (y + z) (a + d)(a d) 13. (n + p)(n + p) 14. (c d) 15. (k + m)(k m) 11. (p q)(p q) 004 by CompassLearning, Inc. Algebra AL053

74 Exponents and Complex Fractions It is possible to simplify complex fractions that contain exponents. Simplify. 1 a b a Step 1 Step Rewrite the complex fraction horizontally. Recall that the fraction bar separating the numerator from the denominator means "divide." 1 a b a 1 a = b a Multiply each term in the expression by the reciprocal of the divisor. 1 a b a b a 1 = ( a )( b a ) ( )( ) b a Step 3 Simplify. ( a 1 )( b a ) ( b a 1 a 1 )( a b ) = ( a )( b ) = ( a b ) Simplify these complex fractions. 1. b 1 b a 4. g g 7. 4c de 4 d 8 e 3 d 10. 8c 9 b 5 d cd 3 b. c d 5. x zy x y z 8. 3r st u r u 3. a b 6. 10m 3 n p p n 9. 1g 4hi ah g h a i 004 by CompassLearning, Inc. Algebra AL053

75 Dividing a Polynomial by a Monomial Find the quotient of (3x 3 6x + 9x) 3x. Step 1 Rewrite the problem. Step Divide each term of the numerator by the term in the denominator. 3 x 3 3 x 6 x 3 x 9x 3 x = x x + 3 (quotient) 3x 3 6x + 9x 3x Step 3 Check by multiplying quotient by divisor. The answer should be the dividend. (x x + 3)(3x) = 3x 3 6x +9x (dividend) Find each quotient. Check your work using multiplication. 1. (15n 5n + 45) 5. (16y 3 4y) (4y) 3. (k 7 4k k 5 k 3 6k ) (k ) Solve the problems. 4. An engineer in a paper-clip factory represents the number of paper clips that come out of a machine in one hour by the following polynomial expression: x 5 + 4x x 18x.The paper clips are packed in boxes, each of which holds x paper clips. How many boxes will be filled by an hour s run of the paper-clip machine? 5. A textile factory produces a bolt (roll) of cloth 40 yards long. The expression 16k 4 + 8k 3 + 4k 3k gives the number of threads in this bolt of cloth. If the bolt is cut into 8 equal pieces of cloth, how many threads will each piece have? 004 by CompassLearning, Inc. Algebra AL054

76 Dividing a Polynomial by a Binomial Find the quotient of (x 3x 5) (x + 1). x 4 x + 1 x 3 x 5 (x + x) 4x 5 ( 4x 4) 1 remainder Check by multiplication: (x + 1)(x 4) 1 = x 3x 5 Find each quotient. Identify any remainder. 1. (x x 15) (x + 5). (14a 6a 4) (7a + 1) 3. (5y + 4y 1) (y + ) 4. (3d 3d 5) (d + ) Tell what is wrong with the following division work. Show how to correct the error. 5. x 4 x + 3 x 5 x 5 + 3x 4? + 3 x by CompassLearning, Inc. Algebra AL055

77 Polynomials in Two or More Variables Evaluate P(x, y) = x + xy + y for x = and y = Step 1 Substitute the variables with their values. P(, ) = () + ()( ) + ( ) Step Step Follow the order of operations. Add = 4 P(x, y) = 4 Evaluate P(x, y) = x + xy + y for each set of values. 1. x = 1, y =. x = 1, y = 6 3. x = 1, y = x = 6, y = 5 5. x = 1, y =8 Evaluate P(x, y) = x 3 y + x y + xy 3 at 6. P(1, ) 7. P( 3, ) 8. P(7, 0) 9. P(8, ) 10. P( 1, 5) Evaluate P(x, y, z) = x 3 yz + x y z + xy 3 + yz for 11. x = 1, y =, z = 1 1. x = 1, y = 0, z = x =, y = 1, z = x = 0, y = 5, z = x =, y = 4, z = by CompassLearning, Inc. Algebra AL056

78 Polynomial Interest Savings accounts pay interest on the money you deposit in the account. The amount of money in the account is called the principal. When the interest that is earned is added to the principal, it is called compound interest. Some interest is compounded yearly, some quarterly, some monthly, and some daily. If a principal, P, is invested at an interest rate, r,and interest is compounded annually for t years, the total savings will be S = P(1 + r) t. You place $00 in a savings account. The annual interest rate is 5 percent and is compounded three times per year. What is your total savings after one year? In this example, P = $00; r = 5 percent; n = 3, the number of times the interest is compounded in a year; and t = 1, the number of years. r S = P(1 + n ) nt S = 00( ) S = 00( ) 3 S = 00( ) = $10.17 Find the answer to each question. 1. At the end of one year, what would the total savings be on a principal of $500 at 4 percent annual interest, compounded quarterly?. At the end of one year, what would the total savings be on a principal of $500 at 4 percent annual interest, compounded biannually (twice a year)? 3. At the end of two years, what would the total savings be on a principal of $500 at 4 percent annual interest, compounded biannually (twice a year)? 4. If you deposited $500 for three years at 6 percent annual interest, compounded annually, how much money will you have saved? 5. If you deposited $500 for four years at 6 percent annual interest, compounded biannually, how much money will you have saved? 004 by CompassLearning, Inc. Algebra AL057 AGS Publishing. Permission is granted to reproduce for classroom use only.

79 Modeling Polynomials Lesson at a Glance In this lesson, you will represent polynomials using Algebra Tiles. Review the computer-based activity AL160 to access the Algebra Tiles and the interactive version of the lesson. Warm-up Puzzle The Power of Two You have the answers. Can you find the questions? You have 7 tiles. Arrange one or more to form an expression that equals a target value of any number from 1 to 9. To earn the maximum number of points, build the longest expression possible for each of the target numbers! Find all the targets and earn a rank! 006 by CompassLearning, Inc. Algebra AL160

80 Case Studies Joe and Ann are exploring adding and multiplying polynomials using Algebra Tiles. Explore the cases with Joe and Ann. The cases are based on the exploratory activity. Possible solutions have been provided for each case. Case Study 1 Third-Degree Investigations Joe is adding the polynomials x x 4 and x 5. He predicts that the sum will be a 3rd degree polynomial. Ann thinks that the degree of the sum cannot be higher than the degree of either addend. In this case, Ann predicts that the sum will be a 1st degree polynomial. Joe and Ann decide to use Algebra Tiles to solve the problem. They represent each expression by placing the matching tiles in the workspace. Joe and Ann discover that this sum is a nd degree polynomial, but they still wonder if their predictions could ever be true. What do you think? Can a polynomial sum be a higher degree than the addends? Can the sum be a lesser degree than the addends? Case Study Filling the Square Ann is representing (x)(x) with Algebra Tiles. She knows that the x tile can be placed in the workspace to represent the product. But, she notices that the length of an x tile is the same as the length of three unit tiles. Instead of using an x tile, can Ann fill the area with unit tiles? Why or why not? 006 by CompassLearning, Inc. Algebra AL160

81 Case Study 3 Be Counted Ann is investigating multiplying polynomials. She distributes the terms from one factor across the terms of the second factor and simplifies the resulting polynomial. ()(x) ()(x 1) ()(x y 1) (x 1)(x 1) She comes to the conclusion that the number of terms in the simplified product is the same as the number of terms in the largest factor. Is Ann s conclusion correct? Explain why or why not. Case Study 4 Both Ways Ann and Joe are finding the product of (x 1) and (x ) using the Algebra Tiles. To represent factor (x 1), Ann places an x tile and a unit tile along the horizontal axis. To represent the factor (x ), she places an x tile and two unit tiles along the vertical axis. She fills the rectangle with an x tile, three x tiles, and two unit tiles. The final expression is (x 3x ). Joe does the opposite. He puts the factor (x ) along the horizontal axis and the factor (x 1) along the vertical axis. Will Joe s final expression be different from Ann s? Is this the general rule or an exception to the rule? Explain your thinking. 006 by CompassLearning, Inc. Algebra AL160

82 Case Study 5 Dividing Time Joe is dividing a polynomial. (x x 6) (x ) He wonders if Algebra Tiles can be used to represent division. Ann thinks this is a great challenge! She tells Joe her ideas. Represent the divisor on the horizontal axis and the dividend in the workspace. Arrange the dividend tiles into a rectangle so its (x x 6) (x ) length matches the tiles on the horizontal axis. If needed, add zero pairs of tiles to fill the rectangle. The height of the rectangle should be the quotient. Translate Ann s ideas into a list of the steps that Joe should perform with the Algebra Tiles. Try it yourself. What is the final result? What is the quotient? 006 by CompassLearning, Inc. Algebra AL160

83 Greatest Common Factor Find the GCF. 140 and 56 49k 4 and 1k Step 1 Write the factorizations. 140 = = 7 49k 4 = 7 7 k k k k 1k = 3 7 k k Step Identify common prime factors. 7 7 k k Step 3 Write the GCF as a product. 7 = 8 7k Find the GCF for these groups of integers , , , , , 90 Find the GCF for these groups of expressions x 5 y 4,7xy j 3 k 4,54j k a 3 b,18a b 9. 5m 6 n,30m 5 n Solve the problem. 10. Dad just had a birthday. Before this birthday, dividing Dad s age by left a remainder of 1. How do you know that Dad s new age is not a prime number? 004 by CompassLearning, Inc. Algebra AL058

84 Factoring Polynomials Factor 35x 3 y 14x y 3. Step 1 Find the GCF: 35x 3 y = 5 7 x x x y y 14x y 3 = 7 x x y y y The GCF is 7x y. Step Rewrite the expression using the GCF. 35x 3 y 14x y 3 = 7x y (5x)(1) 7x y ()(1)(y) = 7x y (5x y) by the distributive property Step 3 Check. 7x y (5x y) = 35x 3 y 14x y 3 Find the GCF and factor these expressions. 1. 6a + 9a. b 4 4b 3. 4d + 8d cd 4. 6x 3 9x y 5. 1a 3ab + 9a b 6. j k jk 7. 1xyz 3 18xy z 8. 1m 3 n p + 6m np 3mp Solve the problems. 9. Suppose you have the following in your fruit bowl: x apples 6x peaches x pears In all, you have x + 7x pieces of fruit. Factor this expression. 10. With the same contents in your fruit bowl, suppose you eat all of the x apples? Write an expression to represent the fruit you will now have left. Can this expression be factored? If it can, factor it. 004 by CompassLearning, Inc. Algebra AL059

85 Factoring Trinomials: x + bx + c Factor a + a 15. Step 1 a + a 15 = ( + )( ) Step a + a 15 = (a + )(a ) to give a Step 3 Find factors of 15 whose sum is. (5)( 3) = 15, and (5) + ( 3) = a + a 15 = (a + 5)(a 3) Step 4 Check by multiplying. (a + 5)(a 3)= a + a 15 Factor the expressions. Check by multiplying. 1. y + 7y + 1. w 4w 1 3. b 9b x 11x n 7n z z d + 5d a 5a m m 15 Solve the problem. 10. A grid (checkerboard pattern) is printed on each sheet of graph paper produced in a paper factory. The total number of squares on the printed grid is x 6x 7. What is the length, in squares, of each side of the grid? (Hint: factor the trinomial.) 004 by CompassLearning, Inc. Algebra AL060

86 Factoring Trinomials: ax + bx + c Factor 3x + 13x + 4. Step 1 3x + 13x + 4 = ( x + )( x + ) to give x Step Find factors of 3 and 4 whose sum is 13. Factors of 3 = (1)(3) Factors of 4 = (1)()() After trying out the possible combinations, the following factors of the trinomial are found: (3x + 1)(x + 4) Step 3 Check by multiplying. (3x + 1)(x + 4) = 3x + 13x + 4 Factor these expressions. 1. 3a + 4a x 4x d d x 18x n + 13n y 7y x + 11x 3 8. n 5n b + 7b 0 Solve the problem. 10. A pretzel-maker fills identical bags with an equal number of pretzels. In one hour, the pretzel-maker bags (4k + 17k + 18) pretzels in all. Factor this trinomial to find the number of bags (larger factor) and the number of pretzels per bag (smaller factor). 004 by CompassLearning, Inc. Algebra AL061

87 Factoring Expressions: a b Find the factors of x 4. Step 1 Find the square roots of x and 4. x = x and 4 = Step Place the values in the model. a b = (a + b)( a b). x 4 = (x + )(x ) Step 3 Check by multiplying. (x + )(x ) = x 4 Factor these expressions. Check your answers. 1. y 144. x w b x 4y 6. 4m 9n 7. j 4 k 8. a b c n 4 5p x 900y 1. 36a 8 49b k 16 5k n 49p Solve the problem. 15. A town has an exactly rectangular shape. If the town s area is (p 11) square kilometers, what is the length of the town border on each side of the rectangle it forms? 004 by CompassLearning, Inc. Algebra AL06

88 Factoring Expressions: a + ab + b Find the factors of n + 8n Model for factoring: a + ab + b = (a + b) Step 1 Assign values to n. a = n or a = n b = 16 or b = 16 ab = (n 4) = 8n Step Place the values in the model. n + 8n + 16 = (n + 4) Step 3 Check by multiplication. (n + 4)(n + 4) = n + 8n + 16 Find the factors of each polynomial. Check your answers. 1. r +10r + 5. b + 0b k + k y + 14y x + 4xy + 4y 6. 9v + 4vw + 49w 7. c + cd + d 8. 4m + 8mn + 49n Solve the problems. 9. A square parking lot has a surface area of (w 4 w x + x ) square feet. Factor this trinomial to find the length of one side in feet. 10. The surface of a square window pane is (k + 8k + 16) cm in area. Factor this trinomial to find the length of one side of the pane in cm. 004 by CompassLearning, Inc. Algebra AL063

89 Zero as a Factor Find the value of the variable in each example. 3n = 0 Since 3 0, n must be 0. 4(y + 1)= 0 Since 4 0, (y + 1) must be 0. (x 1)(x )= 0 implies that one of the factors is 0. If (x 1) = 0, solve for x: x = x = 1 If (x ) = 0, solve for x: x + = 0 + x = Find the value of the variable in each expression. Check your work. 1. 8y = 0. 4k = p = x = v = 0 6. b 4 = x = r 3 = n = n 3 = 0 Solve these equations for the variable. Check your solutions. 11. (13)(b + 5) = 0 1. (n 30)(6) = (a + 4)(a 5) = (x + 4)(x + 7) = (x 30)(x 10) = (d 6)(5d + 5) = (x 1)(x + ) = (x + 6)(x 1) = (3y + )(y 1) = 0 0. (k 5)(3k + 6) = by CompassLearning, Inc. Algebra AL064

90 Solving Quadratic Equations Factoring Solve x 7x + 10 = 0. Step 1 Factor the equation. x 7x + 10 = (x )(x 5) = 0 Step Set each factor equal to 0, and solve each factor for x. x = 0 x 5 = 0 x = x = 5 Step 3 Check. Let x = : x 7x + 10 = () 7() + 10 = = 0 True Let x = 5: x 7x + 10 = (5) 7(5) + 10 = = 0 True Find the solutions. Check your work. 1. x x 3 = 0. b + b 1 = 0 3. w 6w 16 = 0 4. n n 10 = 0 5. x 4x 1 = a + 5a 6 = x 7x 3 = 0 8. n + 5n 3 = 0 9. y + 5y 14 = 0 Solve the problem. 10. The square of a number d plus 5 times d plus 6 equals zero. Write an equation for this puzzle. Then factor the equation and solve for the factors to find the possible values of d. 004 by CompassLearning, Inc. Algebra AL065

91 Frame Factor Factoring equations is useful in a wide variety of workplace applications. Many of these applications involve computing the area of a shape without knowing the exact dimensions. The overall dimensions of this picture frame are 10 in. by 7 in. The picture inside the frame has an area of 40 in. How wide should the mat surrounding the picture be? Step 1 Area = length width 40 = (10 x)(7 x) = (10)(7) (10)(x) (7)(x) + 4x 4x 34x + 70 = 40 Step Set equation equal to zero. 4x 34x = x 34x + 30 = 0 10 in. 10 x x Factor from each term. (x 17x + 15) = 0() x 17x + 15 = 0 (x 15)(x 1) = 0 So, either x = 1 or x = 7.5. Check 40 = (10 x)(7 x) for x = 7.5. Substituting 7.5 in the equation leads to a negative length, which is impossible, so 7.5 is not a solution. Check 40 = (10 x)(7 x) for x = 1 40 = (10 1)(7 1) 40 = (8)(5) True. The mat should be 1 inch wide. 7 x x 7 in. Factor to solve each problem. 1. The outside edges of a 6 ft rectangular frame are 6 ft by 8 ft. The picture has an area of 35 ft.how wide is the mat?. The area of a walkway around a rectangular garden is equal to twice the area of the garden alone. The garden measures 4m by 6 m. What is the width of the walkway? 4 m 6 m x x 8 ft x x 3. If the sides of a square are lengthened by ft, the area of the square becomes 100 ft.what is the length of the side of the original square? x + 4. The area of a rectangle is 64 cm.its length is 4 times the width. What is its length and width? 5. The area of a square is 3 less than the perimeter of the same square. What is the length of a side of the square? x x x by CompassLearning, Inc. Algebra AL066 AGS Publishing. Permission is granted to reproduce for classroom use only.

92 Factoring Polynomials: Algebra Tiles Lesson at a Glance In this lesson, you will explore factoring polynomials using Algebra Tiles. Review the computer-based activity AL16 to access the Algebra Tiles and the interactive version of the lesson. Warm-up Puzzle Spy Girls Agency The Spy Girls Agency consists of 5 agents. They are on a secret mission and must be brought back safely. Locate them by figuring out each agent s rank, first name, and last name. Each agent has a unique rank and name. The highest rank is 1, and the lowest is 5. Place an X in a box if it is not a match. Place an O in the box if it is a match. Clues: Susan is ranked higher than Agent Adams, but is not the highest. Hailey s rank is higher than Agent Davis, but not as high as Agent Bingham. Allison is ranked higher than either Kate or Hailey, but is not the highest. Agent Adams is ranked higher than agent Miller, but is ranked lower than Agent Cooper. Agent Miller is not ranked the lowest. Jen and Agent Bingham are best friends. 006 by CompassLearning, Inc. Algebra AL16

93 Case Studies Charlie has asked his friend Ted to explain factoring polynomials. Ted has decided to demonstrate using Algebra Tiles. Work through the cases with Ted and Charlie. The cases are based on the exploratory activity. Possible solutions have been provided for each case. Case Study 1 Distributing a Factor Ted tells Charlie that in order to understand factoring, he must also understand the distributive property of multiplication. They are inverse procedures. To apply the distributive property, each term in the first factor is multiplied by each term in the second factor. Then, the like terms are combined. Ted and Charlie use Algebra Tiles to represent each pair of factors and its distributed form. Charlie notices that 3x is equivalent to x x x.he wonders if writing the factor in expanded form affects the product. What do you think? Does it matter if the factors are written in simplified or expanded form? 006 by CompassLearning, Inc. Algebra AL16

94 Case Study Fear Factor Charlie tells Ted that he fears that he s finding partial solutions. He knows that an integer can have more than one set of factors. For example, 6 has four pairs of factors: (1)(6), (-1)(-6), ()(3), and (-)(-3). Charlie wonders if the same is possible when factoring a trinomial like x 5x 6. Ted writes the general form to show Charlie the relationship of the trinomial product to its factors. He then tells Charlie that a trinomial in this form has only one set of factors. (x f 1 )(x f ) x bx c? x 5x 6 Charlie is unconvinced. What do you think? What are the factors for x 5x 6? Does this type of trinomial always have only one set of factors? Case Study 3 Negative Factor Charlie factors two expressions using the Algebra Tiles. Each time he checks his answers, he gets an error message. What error or errors is Charlie making? What is the correct way to factor the expressions? 006 by CompassLearning, Inc. Algebra AL16

95 Case Study 4 The Missing Tiles Charlie is having trouble factoring x 9 using the Algebra Tiles. He cannot arrange the tiles to form a rectangle. Ted tells Charlie there is a way to form a rectangle. He reminds Charlie of the general form of a trinomial. (x f 1 )(x f ) x f 1 x f x c x bx c In this situation, the terms (and tiles) that combine to form bx cancel each other. What do you think Ted means? How can the tiles be arranged to form a rectangle? What is the factored expression? Case Study 5 The Hidden Factor Charlie factors a trinomial into two binomial factors using Algebra Tiles. x 6x 4 (x )(x ) Ted congratulates Charlie in finding a valid solution, but then tells him that there is a greater challenge awaiting him. Ted tells Charlie that this particular expression has additional sets of factors. Identify another way that the given expression can be factored. x 6x 4 (x )(x ) How can you tell when an expression has more than one set of factors? 006 by CompassLearning, Inc. Algebra AL16

96 Organizing Data A bar graph compares data by heights of bars. Study this bar graph. Mrs. Chung s Class Pets Owned by Students 10 9 Number of Pets Birds Cats Dogs Snakes Hamsters Answer these questions about the bar graph in the example. 1. Which pet do the greatest number of Mrs. Chung s students own?. Which pet do the least number of Mrs. Chung s students own? 3. If the number scale on the left of the graph were covered up, would you still be able to answer questions 1 and? Explain. 4. How many birds do students in Mrs. Chung s class own? 5. How many hamsters do the students own? 6. If the number scale on the left of the graph were covered up, would you still be able to answer questions 4 and 5? Explain. Suppose 18 people are asked how much money they have in their pockets. Their answers are collected as data to fill the chart on the left. Use this data to complete the frequency table. One is done as an example for you. How much money do you have in your pocket? (in cents) Frequency Table Interval Tally Frequency by CompassLearning, Inc. Algebra AL067

97 Range, Mean, Median, and Mode Find range, mean, median, and mode of the following set of data: {$3.98, $1.44, $3.15, $5.39, $1.44}. Range Find the difference between the greatest and least values. $5.39 $1.44 = $3.95 range Mean Find the sum of the values. Then divide the sum by the number of data values (5). Sum = $ $ = $3.08 mean Median Mode Arrange the data from least to greatest. Cross off greatest and least pairs until one value remains in the middle. That value is the median. $1.44 $1.44 $3.15 $3.98 $5.39 $3.15 median Find any repeated values. They make up the mode. $1.44 mode Use a calculator to find the mean of each set of data. 1. {17, 4, 9, 14}. {17, 51, 100, 87} 3. {4.5, 11.6, 8.8} 4. {553, 700, 97, 644, 199} 5. {600, 400, 100, 300, 500, 00} 6. {18.1, 3.7,10.5, 8.8, 1.4} Answer the questions to solve the problem. Jenny collected data on the number of bulls-eyes she hit in archery practice over a 7-day period. Here is her data set: {7, 4, 9, 1, 6, 19, 6}. 7. Find the range of Jenny s data. 8. Find the arithmetic mean of the data. 9. Find the median of the data. 10. Find the mode of the data. 004 by CompassLearning, Inc. Algebra AL068

98 Box-and-Whiskers Plots The following data represents the number of strikes (knockouts of all bowling pins) the members of a bowling club bowled in their best game. Organize this data for a box-and-whiskers plot. {11, 3, 4, 8, 6,, 9} Step 1 Arrange the data from least to greatest. Label the lower extreme and upper extreme. Step Find and label the median of the data. Step 3 Find the median of all the values below the median. Label this item the lower quartile. Step 4 In a similar way, find and label the upper quartile lower lower median upper upper extreme quartile quartile extreme For each data set, arrange the data from least to greatest value on the blank. Then answer the questions. Data set: {$3.95, $8.15, $6.95, $14.69, $4.88, $4.0, $9.9, $13.1, $7.67, $10.99, $1.79} 1. What is the median?. What is the lower extreme? 3. What is the upper extreme? 4. What is the lower quartile? 5. What is the upper quartile? Data set: {66, 7, 15, 7, 1, 44, 39, 55, 48, 35, 19, 45, 40, 58, 30} 6. What is the median? 7. What is the lower extreme? 8. What is the upper extreme? 9. What is the lower quartile? 10. What is the upper quartile? 004 by CompassLearning, Inc. Algebra AL069

99 The Probability Fraction The probability fraction: number of favorable outcomes number of possible outcomes A board game has a spinner with an arrow and 6 numbered regions. When the player spins the arrow, it lands on one of the six numbers (assume that it never stops on a line). Use the probability fraction to find the probability that the arrow will land on number 6. Step 1 Find the denominator. The number of possible outcomes is 6 because there are 6 regions on the spinner. Step Find the numerator. The number of favorable outcomes is 1 because the problem asks for number Step 3 Simplify the fraction if possible: 1. No simplification is necessary. 6 The probability of spinning to number 6 is 1 6. Use the probability fraction to solve these problems. 1. Suppose you drop a photograph on wet pavement. What is the probability that it will land image-side down on the pavement?. Suppose a class of 4 has one student named Brad. Each day, the teacher lines up the students in random order. What is the probability on any day that Brad will be in front? 3. In the same class, what is the probability on any day that Brad will be at the end of the line? 4. In the same class, if Brad has a twin brother named Jackson, what is the probability that either twin will be at the front of the line? 5. Suppose you come to a fork in the road and have no idea which fork to take. One fork leads directly to your destination, but the other leads away from it. What is the probability you will choose the correct fork? 004 by CompassLearning, Inc. Algebra AL070

100 Probability and Complementary Events Suppose you toss a 1 6 number cube. It is certain that the outcome will be in the set {1,, 3, 4, 5, 6}. Suppose you toss a coin. It is impossible that the outcome will be both heads and tails. Suppose you close your eyes and point at random to a key on your computer keyboard. It is likely that you will point to a letter or number key. In the same situation, it is not likely that you will point to the letter Q. Write one of the following words on the blank to describe the probability of each event: certain, impossible, likely, not likely. 1. With eyes closed, you pick a crayon at random from your box of 48 crayons. The color you pick is green.. The sun will come up tomorrow morning. 3. The first card you draw from a deck of regular playing cards is an ace. 4. Opening a book randomly, you open it to page Your book has 86 pages. You open the book randomly to page A pollster sends a questionnaire to 40 households in your community of 800 total households. One of these questionnaires arrives in your mailbox. 7. If you roll a 1 6 number cube, the number that rolls up will be the square of another integer. 8. If you roll a 1 6 number cube, the number that rolls up will be the square root of an integer. 9. The next person you pass on the sidewalk has a birthday in January. 10. You take one egg out of a dozen eggs in the refrigerator. It is not the last egg in the carton. 004 by CompassLearning, Inc. Algebra AL071

101 Tree Diagrams and Sample Spaces A child is asked to select one crayon and one picture for coloring. Crayon choices are blue or red. The picture choices are a balloon or a star. What is the probability that the child will select red and a star? Color choices: blue red Picture choices: balloon star balloon star The 4 possible choices: blue and balloon red and balloon blue and star red and star The probability is 1 4 : P (red, star) = 1 4 Suppose the child is still asked to choose between a blue or red crayon but is now offered 3 picture choices: balloon, star, or box. Use a tree diagram to determine the probability of each outcome. 1. Find P (red, box). Find P (red, star) 3. Find P (blue, not star) 4. Find P (blue, balloon) 5. Find P (not blue, not balloon) 6. Find P (blue or red, box) 7. Find P (red, any picture) 8. Find P (any color, any picture) Solve the problems. 9. Suppose that runners may choose to run in the 5-km or 10-km race. What is the probability that the next runner to sign up will be female and will choose the 5-km race? 10. For the same event, what is the probability that the next runner to sign up will be of either sex and will choose the 10-km race? 004 by CompassLearning, Inc. Algebra AL07

102 Dependent and Independent Events Suppose children take one pencil each from the same box of 10 pencils. Half of the pencils have erasers, half do not. The first child chooses a pencil, then the second child chooses. What is the probability that both will choose a pencil with an eraser? These events are dependent. 5 The probability of an eraser for child A s choice is 1, or The probability of an eraser for child B s choice is 1, or Suppose instead that each child chooses from an identical separate box of pencils. These events are independent, so each probability is identical. Write whether the events are dependent or independent. 1. Each of 5 children chooses and keeps a marble from a bag of 5 marbles.. A player in a board game rolls a number cube. Then a different player rolls the cube. 3. A clothing store has one of a particular shirt left. One man buys the shirt. Then another man comes in, asking to buy the same shirt. 4. Three children always sit on the backseat of their family car. Today, the first child sits in the middle. Then the second child sits down. 5. One person draws a card from the deck, looks at it, and puts it back into the deck. The next person then draws from the deck. 6. At the start of a board game, one person selects her playing piece from a bag of 7 pieces. Then you select your piece. 7. A grab bag holds 3 wrapped gifts: one red, one blue, and one green. You take the gift wrapped in red. Then the person on your right takes one. 8. A friend shows you a card trick, having you select 1 card out of 5. Then your friend repeats the same trick with someone else. 9. Two trains are on the same track line. Train number one slows down. Train number two then slows down. 10. A vase in a flower shop holds 3 flowers. After you take one, the florist replaces it. Then another person takes one. 004 by CompassLearning, Inc. Algebra AL073

103 The Fundamental Principle of Counting Find 4 factorial, or 4! The factorial of 4 is the product of all positive integers from 4 down to 1. Here is the calculation: = 4 4! = 4 Find the following factorials. You may use a calculator. 1. 8!. 11! 3. 9! 4. 5! 5. 7! 6. 3! 7. 6! 8. 10! How many different ways can Ben rearrange the letters in his name? Possible letters in first position: 3 Possible letters in second position: 3 1 = Possible letters in third position: 1 = 1 Ben can arrange the letters in his name 3 1 = 6 ways. You can also say that Ben can arrange the letters in 3! ways. Solve the problems. 9. How many different ways can Kristy rearrange the letters in her name? 10. The last 4 digits of Juan s phone number are 138. In how many different ways can Juan rearrange these digits? 004 by CompassLearning, Inc. Algebra AL074

104 Choosing the Best Measure of Central Tendency Some measures of central tendency more accurately describe a data set than others. Suppose four children and one grandparent are in a room. The ages of the children are, 4, 3, and 4 years old. The grandparent is 67 years old. Which measure(s) of central tendency best describes the ages of the people in the room? Determine the mean, median, mode, and range of the ages. Then choose the best measure(s). mean = 16 median = 4 mode = 4 range = 65 If the mean were used to describe the ages of the people in the room, the impression would be given that the people in the room were teenagers, and this is not true. If the range were used to describe the ages of the people in the room, the impression would be given that the people in the room were much older than they actually are. Since most of the people in the room are very young, the median or the mode would provide the best description of the ages of the people in the room. Use the data in the table for Problems 1 5. Time Spent Studying Last Night in minutes Find the mean of the data.. Find the median of the data. 3. Find the mode of the data. 4. Find the range of the data. 5. Which measure(s) best describes the length of time the students shown in the table studied last night? Explain. 004 by CompassLearning, Inc. Algebra AL074

105 Multistage Experiments To find the probability that any outcome in an experiment will occur, use the fraction P = number of favorable outcomes number of possible outcomes Some probability experiments are multistage experiments. There are two steps to find the probability of a multistage experiment. First, determine the probability of each stage. Second, find the product of the probabilities. The following box contains the letters that spell the word mathematics. m a t h e m a t i c s Suppose you were to draw a letter from the box without looking and not replace it, then you repeat the procedure four more times. What is the probability that you will first draw t, then e, then a, then m, then s to spell the word teams? Step 1 Find the probability of each stage. Remember that you do not replace the letter after each draw. Stage 1: Drawing the letter t. P = Stage : Drawing the letter e. P = 1 0 Stage 3: Drawing the letter a. P = 9 Stage 4: Drawing the letter m. P = 8 Stage 5: Drawing the letter s. P = 1 7 Step Find the product of the probability for each stage = 55, 8 1 or 440 6,9 30 Use the information below to answer the following questions. Suppose a launched rocket must burn three separate stages of fuel to reach orbit. The probability of failure for the first stage is 1 5,for the second stage is 1 1 1, and for the third stage is For which stage is the probability of success the least? Why?. For which stage is the probability of success the greatest? Why? 3. Determine the probability of an overall success. (Stages 1,, and 3 all succeed.) 4. How many times less likely is the third stage to fail than the first or second stage? Explain. 5. In one million launches of this rocket, how many launches would you expect to fail? 004 by CompassLearning, Inc. Algebra AL075 AGS Publishing. Permission is granted to reproduce for classroom use only.

106 Factorials and the FCP: Calculator Lesson at a Glance In this lesson, you will explore the fundamental counting principle and factorial notation using the Ticker Tape Calculator. Review the computer-based activity AL164 to access the Ticker Tape Calculator and the interactive version of the lesson. Warm-up Tracking the Paths Test your trailblazing skills and find the maximum number of paths that form a series of the same letters. Find the answer by tracing each path or by discovering the mathematical pattern. 006 by CompassLearning, Inc. Algebra AL164

107 Case Studies Jeff and Jane are thankful for living in a free country. In fact, they like to figure out how many choices are available to them in every situation. They have heard that the fundamental counting principle can help with these calculations, and they want to learn more. Join Jeff and Jane as they apply this principle to their lives. Read each case, and experiment using the Ticker Tape Calculator. Case Study 1 Combo Counting Jeff is ordering the lunch special. He can choose to have a hamburger, a cheeseburger, or a hot dog. The lunch includes an order of fries or onion rings. It also comes with a soda, milk, or juice. Jeff wants to use the fundamental counting principle to compute how many different ways that the lunch can be ordered. The fundamental counting principle states that multiplying the number of choices for each task finds the total number of choices. How can Jeff apply the fundamental counting principle to his problem? What are the choices for each task? What is the total number of ways that Jeff could order the lunch special? 006 by CompassLearning, Inc. Algebra AL164

108 Case Study Arranging the Count Jane has 10 books she wants to place on the bookshelf. She wonders how many arrangements are possible. Jeff tells her to use the fundamental counting principle. Since Jane has 10 books to put away and there are 10 positions on the bookshelf, she should multiply 10 by Jane is not convinced. She says that this would be true if she were placing one of the ten books in one of the ten positions. But, she wants to arrange all the books. She thinks that she has 10 tasks with a decreasing number of choices as each book is placed Who do you think is correct? Explain why. What is the possible number of arrangements? Case Study 3 Factorial Facts Jeff tells Jane that most calculators have a factorial key, which saves time when entering permutations and combinations. Jane wants Jeff to give her some examples. He tells her that permutations are used to find the number of ordered arrangements, like organizing 5 books on a bookshelf. The question for a permutation is, How many possible ways can 5 books be arranged? Permutation: 5! 10 In a combination, order doesn t matter, like selecting 3 books from the bookshelf with 5 books. It doesn t matter in which order the books are selected, only that they are selected. The question for a combination is, How many possible ways can 3 books be selected from a bookshelf that contains 5 books? 5! Combination: 10 (! )( 3!) How would Jane write the problems if she did not use factorial notation? Experiment with the Ticker Tape Calculator. Do you prefer using the factorial notation? Why or why not? 006 by CompassLearning, Inc. Algebra AL164

109 Case Study 4 Permanent Permutations Jeff and Jane are creating posters for student council elections. They must decide to either buy stencils or stickers. Stencils are reusable, but they are more expensive. Jane calculates that they can either buy 5 stencils or 8 stickers. They know that they will be creating 3-letter words. Jeff and Jane created the equations for finding the possible number of arrangements of 3-letter words. 5 stencils stickers ! 5! Why did Jeff and Jane construct the equations this way? What is the possible number of arrangements using the stencils? What is the possible number of arrangements using the stickers? Which set would you buy? Why? Case Study 5 A Winning Combination Jeff and Jane are discussing the state lottery. The winner must match six numbers from 1 to 45. They think that this would be an example of a combination problem. They examine the general formula for the number of possible combinations of r objects taken from a group of n objects. n! (n r)!(r!) Using the numbers given, how many possible combinations are there? Is the state lottery an example of a combination problem? 006 by CompassLearning, Inc. Algebra AL164

110 Fractions as Rational Numbers 6 Write 1 in simplest form. 8 Step 1 Prime factorization of 6 and 18: = Step Identify common prime factors and calculate the GCF. Common prime factors = 3 GCF = 3 = 6 Step 3 Divide the fraction numerator and denominator by the GCF, = Step 4 Check. 1 = 1, so 6 3 = 18 1, and 18 = Write each fraction in simplest form. Check your work Solve the problems. 13. Jamille found that 3 members of her class of 7 are younger than she, 3 so she exclaimed, of the class is younger than I am. How could 7 Jamille have simplified her statement mathematically? 14. A baker bakes 48 dozen doughnuts each morning. She sells 18 dozen in her store and fills orders with the rest. Write a fraction to show the portion of doughnuts the baker sells in her store. Simplify your answer. 15. Nick observed that he had finished 15 out of 36 homework 5 problems, or 1 of the total. Simplify Nick s fraction by CompassLearning, Inc. Algebra AL076

111 Algebraic Fractions Rational Expressions Simplify Step 1 Find the GCF of numerator and denominator. 6 n 3 n n 9n 3 = 3 3 n n n The GCF is 3 n n or 3n Step Divide both the numerator and denominator by the GCF. 6 n 9n 3n 6n 9n 3 3 = 3n 3 6n n Step 3 Check. = 3 n 18n3 = 18n 3 True 9n 3 Simplify these expressions. Check your work. c c 4 1. y(x + 3) y (x + 8) 7. b 3 (k 33) b 3 (k + 14) x 7x 3. m 4 m c 3 d 1c 5 d ab 4a 3 b 5. 7w yz 3 11w 3 y 4 z 8 6. (x + ) (x + ) 8. y 9 y 11y (x + 1) x + 4x r + 10 r a + 4 a + 8a x 5 x y (z + 1) 4y 3 (z 1) 15. w 3 w 6w b 9 (b + 3)(b 3) 17. k + 49 (k 7)(5k 5 + 7) 18. Solve the problems. 19. The storage area of warehouse A is x y square m. The storage area x + y of warehouse B is x + y square m. The expression square m shows x y the relationship between these two areas in fraction form. Is the fraction in its simplest form? If not, simplify. 0. Factory A packages 4x pencils a day. Factory B packages 6x 3 pencils a 4x day. The expression shows the relationship between the output 6x 3 of the two factories. Simplify the expression. 004 by CompassLearning, Inc. Algebra AL077

112 Multiplying and Dividing Algebraic Fractions Find the quotient of Step 1 Multiply by the reciprocal = 1 1 Step Simplify Step 3 Check. = = = = = 1 1 = True Find and check each quotient. Simplify your answer whenever possible c d d 3 c 7. b 3 c w 3 3 ab x ab x y 3 4 n 10. n m 3 10 n m 11. x + 1 x y w 9 x 1 x y w + 3 w k ab 1 cd 7 Answer the questions to solve the problem. In math class, Raphael learned that dividing by a number is the same as multiplying by its reciprocal. Four is really 4, said Raphael. So dividing 1 by 4 is the same as multiplying by 1. Use Raphael s idea about reciprocals 4 to complete the blanks. 17. Dividing by 8 is the same as multiplying by Dividing by is the same as multiplying by Dividing by is the same as multiplying by. 0. Dividing by is the same as multiplying by by CompassLearning, Inc. Algebra AL078

113 Complex Fractions and the LCM Simplify 3 1 Step 1 Find the LCM of each denominator. Multiples of 3: 3, 6, 9, 1 Multiples of :, 4, 6, 8 The LCM = 6. Step Multiply by the LCM and simplify. = = 4 3 or Find the LCM for each pair. 1., 3. 6, , , , , , , , 10. 9, x,5x 1. 3a,7a 13. 5b,8b 14. k,1k 15. n,13n 16. 4k,7k Simplify each complex fraction n w w x. 1 7 x 1 8 a by CompassLearning, Inc. Algebra AL079

114 Least Common Multiples and Prime Factors Find the LCM of 1 and 10. Step 1 List prime factors of the denominators, 1 and = 3 10 = 5 Step Count prime factors: greatest number of times appears: twice ( ) greatest number of times 3 appears: once (3) greatest number of times 5 appears: once (5) Step 3 Find the product of the above: 3 5 = 60 = LCM of 1 and 10 Using prime factorization, find the least common multiple for each pair. 1. 3, 8. 15, , , , , 7 7. x 4 y, xy 8. cd 4, c d 3 Solve the problems. 9. A store display has blinking lights. One blinks every 15 seconds and the other blanks every 1 seconds. After how many seconds will the lights blink at the same instant? (Hint: find the LCM of the numbers.) 10. Geri has play blocks that are 4 inches tall. Bette has blocks that are 6 inches tall. Suppose the two tots each stack their own blocks into towers, side by side. What is the least height at which both towers can be the same height? 004 by CompassLearning, Inc. Algebra AL080

115 Sums and Differences Find the sum of Step 1 Find the LCM of the denominators, 9 and = = 3 5 The LCM of 9 and 15 is = 45. Step Multiply each fraction by 1 in a form that will make the denominator = = 45 Step 3 Add the fractions and simplify = cannot be further simplified. Find the LCM, then add or subtract. Write your answer in simplest form w w c 3 c n 5n n 1 + m n 4. 1y 5y 14 1 k 8. k k Solve the problems. 9. Estelle estimated that she mowed of the yard on Friday. Then she 5 estimated that her brother Juaquin mowed another 1 7 the next day. Together, what portion of the yard had they mowed? 10. Roger figured that he had done 1 of his homework. His friend Mike 0 said he had done 3 8 of his homework. What is the difference between the amount of homework Roger and Mike had completed? by CompassLearning, Inc. Algebra AL081

116 Proportions and Fractions in Equations Solve 4 x =. 16 Step 1 Set up the cross products. (4)(16) = (x)() Step Solve for x. x = (4)(16) Commutative Property ( 1 )(x) = (4)(16)( 1 ) x = 3 4 Step 3 Check. 3 = 1 6 ()(3) = (4)(16) 64 = 64 True Solve for the variable. Check your work. a 1. 1 = = 5 x. 9 x = n = 4 n k 3. 1 = r = = y (w +5) = 5(w ) 3 Solve the problems. 9. A slaw recipe for 1 servings uses 6 cups of chopped cabbage. How much cabbage will be needed for 8 servings? 10. A farmer uses 1 1 bushels of wheat seed to plant acres of wheat. How 3 much will he need to plant 14 acres? (Hint: change to an improper fraction first.) 004 by CompassLearning, Inc. Algebra AL08

117 More Solutions to Equations with Fractions Solve 1 x = 3 using multiplication and using division. 5 Multiplication Division ( 1 5 )x = 3 ( 1 5 )x = 3 (5)( 1 )x = (3)(5) = 5 Check solution. 1 5 (15) = 3 3 = 3 True 1 5 x x = 15 x =3( 5 )= 15 1 Solve using division or multiplication. Check your answers. 1. b = k = x + 4 = p 1 = a = a + 1 = m 1 = 8. 1 y + 1 = 17 5 Solve the problems. 9. Danny subtracted the fraction 1 8 from 1 of a certain number to get a 3 result of 1. What was the number? 10. Tim scored,00 on a video game, or 8 of the total points that Dru 9 scored. How many points did Dru score? 004 by CompassLearning, Inc. Algebra AL083

118 The Greatest Common Factor of Large Numbers Using prime factorization to find the GCF of two whole numbers is difficult when the whole numbers are large. To find the GCF of two large whole numbers, use the following theorem. If x and y are two whole numbers and x y, then GCF(x,y) = GCF(x y, y). Find GCF(403, 78) Apply the theorem repeatedly. GCF(403, 78)= GCF(403 78, 78) = GCF(35, 78) GCF(35 78, 78) = GCF(47, 78) GCF(47 78, 78) = GCF(169, 78) GCF(169 78, 78) = GCF(91, 78) GCF(91 78, 78) = GCF(13, 78) The GCF of 403 and 78 is 13. Use the theorem shown above to find the GCF of each pair of whole numbers. 1. (33, 153) 6. (630, 180). (135, 54) 7. (648, 144) 3. (34, 7) 8. (954, 44) 4. (189, 45) 9. (1,440, 88) 5. (31, 96) 10. (15,015, 1,365) 004 by CompassLearning, Inc. Algebra AL083

119 Denominators and Zero 3 For what value(s) of n is n undefined? 4 If (n 4) = 0, the fraction is undefined. Solving the equation: n 4 = 0 n = n = 4 Therefore, the fraction is undefined if n = 4. Determine the value(s) for which each expression is undefined d 3 15k 45k k. a x 9x c c w a 7a 10. 5b b 1 5x 4. 4x x x + x 3x y y x 5b 6. b y x x 5 1x 48x by CompassLearning, Inc. Algebra AL084

120 Working Fractions Work problems rely on the fact that work done by one person + work done by second person = total amount of work completed. You can solve work problems with fractions. It takes Andrew 4 hours to pick up trash along a mile of road. Rosa can pick up trash along the same mile of road in 5 hours. If Andrew and Rosa work together, how quickly can the trash be picked up? Let x = the time, in hours, for both workers to clear the riverbank. Because Andrew can pick up the trash in 4 hours, you know that in one hour he can pick up 1 4 of the trash, in two hours 4 x of the trash, and in x hours 4 of the trash. Because Rosa can clear pick up the trash in 5 hours, you know that in one hour she can pick up 1 5 of the trash, in two hours 5 x of the trash, and in x hours 5 of the trash. Set up an equation that describes the work problem: part of the trash cleared by Andrew in x hours + part of trash cleared by Rosa = 1 (entire mile of road cleared) 4 x + 5 x = 1 Use what you know about fractional equations to solve the equation. 4 x + 5 x = 1 0 [ 4 x + 5 x ] = 0 1 5x + 4x = 0 9x = 0 x = 0 or 9 9 hours Find the solutions to the following problems. 1. Lali can mow her neighbor s yard in hours. Mark can mow the same neighbor s yard in 3 hours. How long will it take Lali and Mark, working together, to mow their neighbor s yard?. Haley and her sister Kelsey are earning money this summer by painting houses. They are trying to figure out how long it would take the two of them to paint their grandmother s house. Haley thinks she can paint the house in 4 days. Her sister thinks she can paint the house in 6 days. If they work together, how long would it take them to paint their grandmother s house? 3. Orlando can survey the West River in 3 hours. John can survey the same area in 4 1 hours. If they work together, how long will it take them to survey the river? 4. Alex and Mathilda are working at a nature center. Alex can make 15 bird feeders in 9 hours. It takes Mathilda 7 hours to make the same number of bird feeders. If they work together, how long will it take Alex and Mathilda to make 15 bird feeders? 5. Taj and Greta are members of the garden club. It takes Taj 1 1 hours to plant 3 rows in a garden. It takes Greta hours to plant 3 rows in a garden. If the garden has 9 rows, how long will it take Taj and Greta to plant the whole garden? 004 by CompassLearning, Inc. Algebra AL085 AGS Publishing. Permission is granted to reproduce for classroom use only.

121 Simplifying Expressions: Calculator Lesson at a Glance In this lesson, you will explore simplifying expressions using the Ticker Tape Calculator. Review the computer-based activity AL166 to access the Ticker Tape Calculator and the interactive version of the lesson. Warm-up Dealt Cards Pedro invites Bernie, Chip, and Kate to play cards. He deals four cards numbered two through five. Each card is a different suit. Use the given clues to deduce who has been dealt which card. Place an X in the box if it is not a match. Place an O in the box if it is a match. Clues: Chip holds the highest red card, but his card is one lower than Kate s. Bernie and Kate hold cards of the same color. Four is a black card. The sum of the spade and diamond is higher than the sum of the club and heart. 006 by CompassLearning, Inc. Algebra AL166

122 Case Studies Tim and Ilene are familiar with fractions. Now, they are taking on the challenge of complex fractions. Join Tim and Ilene as they learn how to simplify and divide rational expressions. Read each case and experiment using the Ticker Tape Calculator. Case Study 1 The Great Divide Ilene and Tim have converted fractions into decimals by entering the numerator divided by the denominator Another method they used is multiplying the numerator by the reciprocal of the denominator Using what they already know, Ilene and Tim are trying to determine if they can create a general rule about fractions. 1 b a a b What do you think? Is this a valid formula? 006 by CompassLearning, Inc. Algebra AL166

123 Case Study Avoiding the Point Ilene wonders what she should do when she wants to simplify a complex fraction rather than convert it into a decimal Tim suggests using the calculator for the intermediate steps rather than entering the whole expression. He reminds Ilene that they can apply what they learned in elementary school about multiplying fractions and simplifying an improper fraction into a mixed fraction. Apply the process to Ilene s fraction and find the simplified fraction. Record the keystrokes used in the process. Case Study 3 Simply Complex Tim is helping Ilene make fudge. Ilene has a half-pound of chocolate and each fudge bar requires five-sixths of a pound. Tim recommends getting another three-quarters pound of chocolate. He also tells her that she should reduce the amount of chocolate in a fudge bar by a third of a pound. Ilene quickly jots the recipe as an expression so she can figure out how many bars can be made Create a sequence of keystrokes for Tim and Ilene to find the solution to this expression. Give the solution as a decimal and as a simplified fraction. 006 by CompassLearning, Inc. Algebra AL166

124 Case Study 4 Calculator Crossing Tim knows that one of the powerful applications of fractions and complex fractions is solving proportions. Traditionally, Tim would set up the cross products first. Then, solve for the variable. Proportion x Set up cross products. (5)x (15)() 5x 330 Solve for x. 5 x Instead of performing separate steps, help Tim create one equation to solve for x using the given proportion. Record the keystrokes to enter into the Ticker Tape Calculator. Case Study 5 Clean Break Ilene and Tim decided to take a break from math and clean their rooms. It took Ilene 4 hours to clean her room and Tim 5 hours to clean a same-sized room. They wonder how long it would have taken if they cleaned one room together. Create the equation and solve. How long would it take if they were to work together to clean 15 same-sized rooms? 006 by CompassLearning, Inc. Algebra AL166

125 The Coordinate System Graph (, 3). y 4 3 P x 3 4 Point P is located at (, 3). Write the ordered pair that represents the location of each point on the graph. 1. Point W. Point K 3. Point R 4. Point G N 4 3 y 5. Point T 6. Point A 7. Point N 8. Point D 9. Point Q 10. Point B G R K W 3 4 T Q D A B x 004 by CompassLearning, Inc. Algebra AL086

126 Graphing Equations Graph y = 3x. Step 1 Assign values to x. Let x = 1 and x = 1. Step Solve for y in y = 3x. y = 3( 1) y = 3(1) y y = 3x (1, 3) y = 3 y = 3 So ( 1, 3) So (1, 3) Step 3 Plot the points ( 1, 3) and (1, 3). Then graph and label the line ( 1, 3) 3 4 x Solve each equation for y when the value of x is given. 1. y = 4x 6; x = 1. y = 3x 3; x = 1 3. y = x 4; x = 4. y = 4x ; x = 1 Given the x-values, solve for y.then graph the equation and label the line. 5. y = x + 1 y 4 x = 1 x = x by CompassLearning, Inc. Algebra AL087

127 Intercepts of Lines Find the following x- and y-intercepts. Find the y-intercept of y = x. Substitute x = 0 into the equation. Solve for y. y = (0) y = This is the y-intercept. Find the x-intercept of y = 3x + 1. Substitute y = 0 into the equation. Solve for x. 0 = 3x x = This is the x-intercept. 3 Find the x-intercept and y-intercept of each graph. y = x x-intercept. y-intercept y = 3x 4 3. x-intercept 4. y-intercept y = x + 5. x-intercept 6. y-intercept y = x x-intercept 8. y-intercept y = x x-intercept 10. y-intercept 004 by CompassLearning, Inc. Algebra AL088

128 Slopes of Lines Find the slope of a line that passes through ( 3, ) and (, 4). Step 1 Label one ordered pair (x 1, y 1 ) and the other (x, y ). (x 1 = 3, y 1 = ) (x =, y = 4) Step Substitute in the slope formula and solve. y m = y 1 x x 1 = 4 ( ) = 6 ( 3) 5 or Find the slope of a line that passes through the following points. 1. (1, 3) (, 4). ( 3, 1) (3, 13) 3. (1, 3) (5, 6) 4. (, 5) (3, 4) 5. (6, ) ( 3, 4) 6. (6, 4) (, 1) 7. ( 3, ) ( 1, 1) 8. (0, 1) (5, ) y y y = x (0, 0) x (0, 0) y = -x x Positive Slope Negative Slope Solve the problems. Refer to the graphs of slopes shown. 9. Think of a clock as a graph with the pivot of its hands at (0, 0). When the time is 10:0, the hands form a straight line. Does this line have positive or negative slope? 10. When the time is 8:10, is the slope of the line the hands form negative or positive? 004 by CompassLearning, Inc. Algebra AL089

129 Writing Linear Equations Write the equation of a line that passes through (6, 1) and (3, ). Step 1 Find the slope, m. y m = y 1 x x 1 = = 3 3 = 1 Step Find the y-intercept, using known point (6, 1). y = mx + b 1 = 1(6) + b b = 5 Step 3 Substitute slope and y-intercept in y = mx + b: y = (1)x + ( 5) or y = x 5 Write the equation of the line that passes through each pair of points. 1. (6, 0) (0, ). (, 0) ( 1, 3) 3. (1, ) (5, 8) 4. (6, 6) (3, ) 5. (0, 4) (, 0) 6. (, 3)(1, 5) 7. (, 7) (, 1) 8. (6, 6) (8, 3) Graph the line that passes through the following points. Then find the equation of the line and label it on the graph. 9. (, 1) (, ) 10. ( 4, 4) (, 0) y y x x by CompassLearning, Inc. Algebra AL090

130 Lines as Functions A function is a rule that associates every x-value with one and only one y-value. If a vertical line crosses a graph more than once, the graph is not a function. A circle is not a function. A vertical line will cross it at two points. A straight line is a function. A vertical line crosses it at one point only. Is each graph an example of a function? Write yes or no. Explain your answer. 1. y x. y x 3. y x 4. y x 5. y x 004 by CompassLearning, Inc. Algebra AL091

131 Domain and Range of a Function Find the range of the function y = f(x) = 3x + 1 for the domain, 0, 3, 6 Substitute the domain values in f(x) x = y = f( ) = 3( ) + 1 = 5 so y = 5 x = 3 y = f(3) = 3(3) + 1 = 10 so y = 10 x = 0 y = f(0) = 3(0) + 1 = 1 so y = 1 x = 6 y = f(6) = 3(5) + 1 = 16 so y = 16 Determine the domain and the range of a function from a graphed line and its end points. y (5, 4) (, 1) and (5, 4) Domain Range (, 1) x Domain = x 5 Range = 1 y 4 Determine the range for each function with the given domain. 1.f(x) = x + 5 domain: 1,0,3,7,10 range: 4.f(x)=x +3x 4 domain: 3,0,,4,6.f(x) = x 3 domain: 1, 0,, 5, 8 range: 3.f(x) = 1 x domain: 1,0,3,5,9 range: 5.f(x) = 3x 9 6.f(x) = x x Determine the domain and the range from the graph and the given ordered pairs. 7. domain: range: 8. domain: range: y range: domain: 4, 3,0,1,8 range: domain: 3, 1,0,,3 range: y ( 3, 4) (4, 7) x x (5, 3) ( 5, ) 9. domain: range: 10. domain: range: y y (, 4) x x ( 4, 3 ) (6, 3) (, ) 004 by CompassLearning, Inc. Algebra AL09

132 Graphing Inequalities: y < mx + b, y > mx + b Graph the region represented by y > x + 4. Step 1 Use y = x + 4 and substitution to find two points on the line. Let x = 0, and then let x = 1. The two points are (0, 4) and ( 1, 3). y > x + 4 (0, 4) ( 1, 3) y Step Plot the two points and connect them with a broken line. Step 3 Shade the region above the line. Label it y > x x Graph the region represented by each line. 1. y > x 1. y < x + 3. y > x + 3 y y y x x x Write an inequality to label the shaded region in each graph y y x x by CompassLearning, Inc. Algebra AL093

133 Graphing Inequalities: y mx + b, y mx + b The two graphs show the inequalities y 3x 1 and y > 3x 1. Can you see a difference between these graphs? The only difference between the two graphs is that the line of the equation y 3x 1 is a solid, unbroken line. This solid line indicates that the points on the line of the equation are also included in the graph of the inequality. y y y > 3x 1 y > 3x 1 x x Write the inequality that describes the shaded region. 1. y. y y = 3x x y = x x 3. y 4. y y = x x y = x + 4 x Answer the question. 5. Suppose you were to graph the inequalities y x and y < x.what would be the difference between the two graphs? 004 by CompassLearning, Inc. Algebra AL094

134 Graphs Without Numbers Every graph is a picture of something that has happened sometime in the past or is happening now. You can often determine what a graph is about just by its general shape. The first item below is done for you. It shows how to read the shape of a graph. Match each written description with a graph. Write the letter of the graph on the blank. Speeding baseball being caught by an outfielder D A. 1. Helicopter rising, moving off in a direction for a while, then lowering. Helicopter rising, hovering briefly, then descending 3. Wave motion such as an ocean wave or sound wave B. C. D. Answer the questions. 4. On all the graphs that appear on this page, what is the understood point of origin? 5. Why do you think it may be convenient in many situations to use a graph with only positive points? 004 by CompassLearning, Inc. Algebra AL095

135 Graphing You can connect points in the coordinate system with line segments to form polygons and other figures. For example, the points (1, 5), (6, 5), (0, ), and (5, ) form a parallelogram when connected with line segments. To graph any point in the coordinate system: Step 1 Locate the x-value of the pair by starting at the origin and moving left or right the appropriate number of units on the x-axis. y (1,5) (6,5) Step Locate the y-value of the point by moving up or down the appropriate number of units from the x-value. (0,) (5,) Step 3 Mark the point. x Make each graph to find the answer. 1. Graph these points: (1, ), (1, 3), (4, 4). Then draw a line from (1, ) to (1, 3), from (1, 3) to (4, 4), and from (4, 4) to (1, ). What shape is the figure?. Graph these points: (, 3), (1, 5), (3, 6), (5, 5), and (4, 3). Connect the points using line segments. What figure is formed? 3. Name four points in the coordinate system that form a square when connected. 4. Name four points in the coordinate system that form a rectangle when connected. 5. Name six points in the coordinate system that form a hexagon, or six-sided figure, when connected. 004 by CompassLearning, Inc. Algebra AL096 AGS Publishing. Permission is granted to reproduce for classroom use only.

136 Slope and Intercept: Grapher Lesson at a Glance In this lesson, you will explore the slope and y-intercept of linear equations. Review the computer-based activity AL168 to access the Grapher and the interactive version of the lesson. Warm-up Catch Me if You Can Detective Joe is on a thrilling mission. A gang of five thieves is at large, and Joe has to catch them all. Help Detective Joe match the thieves to their real identity and to the rewards posted for their capture. Place an X in a box if it is not a match. Place an O in the box if it is a match. Clues: Eagle Eye got her name from being able to spot a rich tourist. Bugger s reward money is 10 times as much as Herb s and 50 times as much as Rick s. Herb s alias refers to his fast driving. Drama Mama threw a fit when he realized that he had the lowest reward money posted. Phil s reward is the median of all the posted rewards. 006 by CompassLearning, Inc. Algebra AL168

137 Case Studies Hannah and Pete are getting ready to learn about graphing linear equations. Join them by reading each case and experimenting with the Grapher. Case Study 1 Grappling with the Grapher Hannah and Pete notice that there are several formats for entering linear equations into the Grapher: the standard form, the point-slope form, the slope-intercept form, and the table form. They don t know which form to select when graphing an equation with a y-intercept of 3 and an x-intercept of. Which form would you recommend and why? Describe the steps needed to draw the line using the Grapher. Case Study Perception of Interception Pete plots several lines, and each line intersects the y-axis as well as the x-axis. He concludes that all lines have two intercepts, one x-intercept and one y-intercept. Hannah is sure that this is incorrect, but she cannot figure out how to show this on the Grapher. What do you think? Is Pete or Hannah correct? Provide examples of equations to support your reasoning. 006 by CompassLearning, Inc. Algebra AL168

138 Case Study 3 Slippery Slopes Hannah is graphing linear functions written in the slope-intercept form, y mx b. She knows that m represents the slope and b represents the y-intercept. She plots several functions to determine how changing m and b affects a line. y mx b y x 3 Experiment with the Grapher and describe what happens as m and b are changed. Were you surprised by the result? Why or why not? Case Study 4 Setting the Standard Pete and Hannah have learned that the standard form of a linear equation is Ax By C,where A, B, and C are integers and A 0. And, they wonder how this form is related to others. Help them discover the relationship between the standard form and the slope-intercept form. Find the relationship between the values of A, B, and C of the standard form with m and b of the slope-intercept form. 006 by CompassLearning, Inc. Algebra AL168

139 Case Study 5 Lining Up Hannah and Pete are challenged to write an equation of a line that passes through three points. They are given the ordered pairs (, 3), (4, 7), and (-1, -1). Help Hannah and Pete write an equation that contains these points. Or, if these points cannot lie on the same line, change a single coordinate in one of the ordered pairs so that they do. Describe your strategy. 006 by CompassLearning, Inc. Algebra AL168

140 Parallel Lines Lines having the same slope are parallel. In an equation of the form y = mx + b, coefficient m of the variable x gives the slope. For example, in y = 3x + 4, the slope is 3. y = y 4 3 A line for an equation in the form y = constant is a horizontal line. A line for an equation in the form x = constant is a vertical line. Study the example lines in the graph y = 3x + 4 y = 3x 3 4 x = 4 x Write the equation of the line parallel to the given line and passing through the given point, which is the y-intercept. 1. y = x + 7; (0, 4). y = x ; (0, ) 3. y = 4x + 1; (0, 4) 4. y = x + 3; (0, 5) 5. y = 3x + 3; (0, 3) 6. y = 4x ; (0, 1) 7. y = x 3; (0, 1) 8. y = 5x 1; (0, 1) Solve the problems. 9. If you plotted the following equations on a single graph, which line would stand out? Write the letter of the answer and explain. a. x = 7 d. y = 3 g. x = 8 b. y = 6 e. x = 4 h. y = x + 4 c. x = 3 f. y = 9 i. y = 10. Will graphs for these equations show parallel lines? Explain. y = 6x + 1 y = 6x by CompassLearning, Inc. Algebra AL097

141 Describing Parallel Lines Write the equation of a line that is parallel to the line y = x + 3 and passes through the point (, 1). Step 1 The slope is, so y = x + b. Step Substitute the values from the point (, 1) and solve for b. y = x + b 1 = ( ) + b b = 5 Step 3 Substitute this value for b in the equation. y = x + 5 Write the equation of the line parallel to the given line and passing through the given point. 1. y = x + 4; (, 5). y = 3x 1; (1, 4) 3. y = x ; (, 8) 4. y = x 7; (4, 6) 5. y = 3x 3; (, 5) Rewrite each equation in the form y = mx + b.then write an equation for a line parallel to the first and passing through the given point. 6. 6y = 3x 1; (, 4) 7. 1 y = 3x; (, 9) 8. 5y = 4x + 5; ( 5, 4) 9. 10y = 0x 0; (1, 3) 10. 3x = 3y 1; ( 5, 11) 004 by CompassLearning, Inc. Algebra AL098

142 Intersecting Lines Common Solutions Look at these examples of graphed equations. y A. B. 5 5 y y = x y = x x y = x y = x x Equations having the same slope are represented by parallel lines that do not intersect. In Example A, y = x + and y = x have the same slope,. Equations having unlike slope are represented by lines that do intersect. In Example B, y = x has slope of 1 and y = x has slope of. Do these systems of equations have a common solution? Tell why or why not. 1. y = x + 3 y = x 1. y = 3x 4 y = 3x y = 3 x + y = 3 x + Answer the questions to solve the problem. Explain your answer. A large state in a desert country consists of flat land with few towns. The state has a rectangular shape, and the major roads are straight lines. Engineers use a grid map with x-axis and y-axis to design roads in this desert state. They describe road positions by equations, as follows: Road A-1: y = 3x + 5 Road A-: y = x Road B-1: y = 3x 1 4. Will roads A-1 and B-1 ever intersect? 5. Will roads A-1 and A- ever intersect? 004 by CompassLearning, Inc. Algebra AL099

143 Solving Linear Equations Substitution Find the common solution for the system: y = x + 4 y = x + 1 Step 1 From the second equation, substitute the value of y into the first equation. x + 1 = x + 4 Step Solve for x. x = 3 Step 3 Substitute this value of x into the first equation to solve for y. y = x + 4 y = (3) + 4 y = The common solution is the ordered pair (3, ). Step 4 Check. Substitute the x and y values in each equation. y = x + 4 y = x + 1 = (3) + 4 = (3) + 1 = True = True Find the common solution for each system of equations. Check each solution. 1. x + y = 4 x + 3y = 0. x + y = 0 x y = x + 4y = 11 7x 5y = 3 4. x + y = 1 x + y = x 5y = 4 4x + 3y = x y = 5 4x + 3y = x 4y = 5 5x + 4y = x = 3y 10 y = 5x 9. x + 6y = 14 3x 4y = y = x 6x = y by CompassLearning, Inc. Algebra AL100

144 Graphing Systems of Linear Equations Use a graph to find the common solution for these equations: x + y = 3 3x y = 1 Step 1 Find the x- and y-intercepts for each equation. x + y = 3 3x y = y = 3, so y = 3 3(0) y = 1, so y = 1 x + 0 = 3, so x = 3 3x 0 = 1, so x = 1 3 x-int. = 3, y-int. = 3 x-int. = 1, y-int. = 1 3 Step Plot the intercepts for each equation ( points). Draw the line connecting them. Read the point of intersection from the graph: (1, ). Step 3 Check by substituting the solution in the equations. Find the x- and y-intercepts for each equation. 1. y = x x + y = 3. 5x 3y = 11 x-intercept: x-intercept: x-intercept: y-intercept: y-intercept: y-intercept: Graph each system of equations and identify the point of intersection. 4. 3x y = 7 5. x + y = 5x + 5y = 5 x y = 4 y y x x by CompassLearning, Inc. Algebra AL10

145 And Statements Conjunctions Study the patterns in the following conjunctions = 10 and 10 = 100 T T = True = 3 5 and 1 1 = 1 4 T F = False 100 = 50 and ( 5) = 5 F F = False Complete each conjunction in the chart by choosing an appropriate statement from the Statement Box and writing it on the blank. (Hint: In the box, left-side statements are true, right-side are false.) p 4 3 = 64 (3)(9) = 7 3. x has a value of 3 in 3x = 9 q 1.. x = 3 is a vertical line 4. p q T F F T Statement Box 4 = 16 1 (14) = 14 (x )(x 3 ) = x 5 16 = ±4 17 is not a prime number 16 = 4 a 4 a = a = 16 Write your own conjunction so that the value of p q will be true by CompassLearning, Inc. Algebra AL103

146 Problem Solving Using Linear Equations Dean s cat is one year less than twice the age of Drina s cat. The difference in the cats ages is 7 years. Find the ages of the two cats. Step 1 Let x = age of Dean s cat y = age of Drina s cat x = y 1 one less than twice the age x y = 7 difference in the cats ages x = y + 7 last equation rewritten to put x on left Step Solve by substituting y + 7 for x in the first equation. y = 8 age of Drina s cat x = 15 age of Dean s cat Step 3 Check by substituting x and y values in both equations. x = y 1 x y = 7 15 = (8) = 7 15 = 15 True 7 = 7 True Use any method to solve each system of equations. Check your answer. 1. x + y = 9 x y = 6. x + y = 14 x y = x + 3y = 8x 3y = 3 Use systems of equations to solve the problems. 4. A farmer raises wheat and oats on 180 acres. She plants wheat on 0 more acres than she plants oats on. How many acres of each crop does the farmer plant? 5. Enrico says, I m thinking of mystery numbers. One number is 3 times the other. The sum of the two numbers is 48. What are Enrico s mystery numbers? 004 by CompassLearning, Inc. Algebra AL104

147 Introduction to Matrices: Addition and Subtraction Add the matrices Subtract the matrices Step 1 Add corresponding entries or members of each matrix. Step 1 Subtract corresponding entries or members of each matrix ( 6) (8) Step Write the sums in matrix form. Step Write the differences in matrix form Add or subtract the matrices by CompassLearning, Inc. Algebra AL105

148 Multiplication of Matrices Let A =. Find 4A Multiply each entry by 4. 4A = Let X =. Let Y = Multiply matrix X by matrix Y Step 1 Multiply the rows in X by the columns in Y. Step Add the products X Y = Find the product of each matrix and the number shown x y x 9 0 3x 1 y 4 0 Multiply the two matrices x y n r by CompassLearning, Inc. Algebra AL106

149 Using Venn Diagrams Venn diagrams represent one of the most common applications of conjunction and disjunction. A Venn diagram is a way to illustrate ideas in logic. A Venn diagram consists of a universal set and various subsets. In this Venn diagram, the universal set U contains the integers 1 9. Subset A contains the integers 1,, 4, 5, and 7. Subset B contains the integers, 3, 5, 6, and 7.The union of sets A and B is written A B and represents the set of all elements in A or in B or in both A and B. In this Venn diagram, A B = {1,, 3, 4, 5, 6, 7}. U A 5 7 B 6 3 The intersection of sets A and B is written A B and represents the set of all elements common to both A and B. In this Venn diagram, A B = {, 5, 7}. A set that contains no elements is called an empty set, or null set. The symbol or { } is used to designate the empty set. Find the answer to each problem. 1. Use the Venn diagram to find U, A B and A B. U 6 A B 7 3 U = A B = A B =. Draw a Venn diagram that displays these elements: U = {c, y, n, r, x}, A B = {c, y, r}, and A B = {c}. 3. How many different Venn Diagrams can be drawn to show U = {1,, 3, 4}, A B = {1, 3, 4}, and A B = {3}? 4. A survey of 30 high school freshmen found that 1 of the students surveyed were taking mathematics and science courses, students were taking science and no mathematics, and 11 were taking mathematics and no science. How many students were not taking a science or mathematics course? 5. Given A = the set of all the letters in the name of your city or town, and B = all the letters in the name of your state, what is U? A B? A B? 004 by CompassLearning, Inc. Algebra AL107 AGS Publishing. Permission is granted to reproduce for classroom use only.

150 Common Solution: Grapher Lesson at a Glance In this lesson, you will explore the relationship between systems of equations and their common solutions. Review the computer-based activity AL170 to access the Grapher and the interactive version of the lesson. Warm-up Rangers to the Rescue Five wild animals have escaped from Riverview Forest Zoo. Rangers have been sent out to track them. The animals have been sighted at specific coordinates. Help the rangers by matching them with their correct quarry and coordinates. Place an X in a box if it is not a match. Place an O in the box if it is a match. Clues: The hippo is in the 1st quadrant, also known as the northeast territory. The ranger that tracked the bear says that she found it was still in the zoo, otherwise known as the origin of the coordinate grid. The lion traveled east from the zoo. Jeremy tracked the legless animal southwest from the zoo. Cedric found his animal at the coordinates (, 5). Dawn found her animal in quadrant IV. 006 by CompassLearning, Inc. Algebra AL170

151 Case Studies Nicole and Ralph wonder how many weeks it would be until the amount paid to rent a computer becomes the same as the cost of buying one. Help Nicole and Ralph investigate this problem as well as others. Use systems of linear equations and the Grapher to help solve each case. Record your solutions. Case Study 1 To Buy or Not to Buy Nicole reviews the amount that it would cost to buy or rent a computer. Buying a computer system costs $600. Renting a computer system has an initial cost of $50 and $0 per week. Ralph says that these can be set up as equations. The independent variable x is the number of weeks. The dependent variable y is the dollar cost. He writes the equations using slope-intercept form.he then uses the substitution method to find the common solution, x 7.5 weeks. 0x x Nicole verifies that this is a reasonable answer by entering the original equations into the Grapher. How is the common solution shown on the Grapher? What would the solution be if buying the computer were to cost $650? 006 by CompassLearning, Inc. Algebra AL170

152 Case Study How Many Solutions? Nicole and Ralph know that there can be one solution for a system of linear equations. But, they wonder if there can be no solution or more than one solution. What do you think? Provide supporting examples. Case Study 3 Where the Lines Meet Ralph and Nicole are exploring using the Grapher tool. They enter two equations and the point of intersection. By accident, Ralph erases one of the equations. The remaining equation is y x 3, and the point is (, 5). Ralph wonders if there is enough information here to find the original equation. Nicole says that there are many systems of equations with a common solution of (, 5). Using the given equation and common solution, write two possible equations that would satisfy this condition: intersect y x 3 at point (, 5). What additional information is needed to find Ralph s original equation? 006 by CompassLearning, Inc. Algebra AL170

153 Case Study 4 Tallying T-shirts Nicole and Ralph have been asked to help with fund-raising and to sell school shirts. The general price is $18, but students and teachers can buy them for $1. Yesterday, Nicole and Ralph collected $5 for selling 16 shirts, but they forgot to keep track of how many were sold at the regular price and how many were sold at the reduced price. Nicole wants to use a system of linear equations to solve this problem. Ralph suggests using x to represent the number sold at the regular price and y to represent the number sold at the reduced price. Using those variables, write the system of equations that Nicole and Ralph should use to find the common solution. What is the answer? Case Study 5 Lining Up Nicole wonders what else can be solved using a system of linear equations. Ralph challenges her with a problem that he s been unable to figure out. Each week, he earns $98 from his part-time job. His mother lets him use her credit card to buy food and supplies. Each day he spends $15. Ralph wants to know when he ll no longer need the credit card. Nicole thinks about Ralph s problem and tells Ralph that there is a common solution to the equations, but it is not a valid solution to his problem. What do you think Nicole means? Solve the system of equations. Is Nicole correct? 006 by CompassLearning, Inc. Algebra AL170

154 Rational Numbers as Decimals Is the decimal form of these fractions terminating or repeating? Step 1 Step Divide numerator by denominator = 0.0 Terminating 5 Divide numerator by denominator = 0.6 Repeating 3 Write the decimal expansion for these rational numbers. Tell whether each is terminating or repeating Using a calculator, perform the division to change each fraction into an expanded decimal. Tell whether each is terminating or repeating by CompassLearning, Inc. Algebra AL108

155 Rational Number Equivalents What rational number is equal to 0.583? Step 1 Let x = Step Multiply to place the first repeating digit(s) to the left of the decimal. (1000)x = (1000) Simplify: 1000x = Step 3 Multiply to place the repeating digit(s) to the right of the decimal. (100)x = (100) Simplify: 100x = Step 4 Subtract the smaller from the larger result. 1000x = x = x = x = 5 Simplify: = Find the rational number equivalents for these decimal expansions. Show your work by CompassLearning, Inc. Algebra AL109

156 Irrational Numbers as Decimals Find each root. Tell whether it is rational or irrational Using a calculator: 3 = Using a calculator: 3 7 = 3.0 The number is irrational because it neither ends in zeroes nor has a repeating pattern. The number is rational because it ends in zeroes. Complete the chart. Find each root and tell whether it is rational or irrational.you may use a calculator. Radical Root Rational or Irrational? Solve the problem. 5. Caitlin has cut out a square piece of graph paper that contains a total of 81 blocks. How many blocks are there along one side of the square? 004 by CompassLearning, Inc. Algebra AL110

157 Products and Quotients of Radicals Simplify = 16 3 = 16 3 = 4 3 Simplify 7x. 3 7x 3 = 7 x 3 = ( 9 3 )( x x ) = (3 3 )(x x ) = 3x 3x Check. (3x 3x ) = 9x 3x = 7x 3 True Simplify the following radicals. Check your answers a x y x 3 y x y a b xy 17. 1, x 19. 9x y 0. 3k a 7 b 7. 18x 3 y a 3 b Solve the problem. 5. To repair a wall, Van and Mai have cut out a square piece of wallboard whose area is 396 square inches. The length of one side of this piece is 396 inches. Use the first rule of radicals to simplify this expression. 004 by CompassLearning, Inc. Algebra AL111

158 Sums and Differences of Radicals Find the sum of Step 1 Simplify = (9 ) = 9 = 3 Step Add. + 3 = (1) + 3 = (1 + 3) = 4 Subtract 1 3. Step 1 Simplify. 1 = (4 ) = = 3 Step Subtract. 3 (1) 3 = (1) 3 = 3 Add or subtract. If you cannot add or subtract, write not possible x + 8x x y + 18x y , x x by CompassLearning, Inc. Algebra AL11

159 Radicals and Fractions 3 4 Rationalize the denominator of = = 3 8 Rationalize the denominator of each fraction. Be sure your answer is in simplest form x x 1 Rationalize the denominator of. The conjugate of is = = + You cannot further simplify, because the in the numerator is a separate term. Use a conjugate to rationalize the denominator of each fraction. Be sure your answer is in simplest form by CompassLearning, Inc. Algebra AL113

160 Radicals in Equations Solve for x: x + = 13 Step 1 Isolate the variable, x. x + = 13 x = 11 Step Square both sides. ( x ) = 11 x = 11 Step 3 Check = = = 13 True Solve each equation for the variable. Check your answers. 1. x = 5. n = 8 3. k + 3 = 4. a = r + 8 = 1 6. y 5 = 5 7. m = n 3 = 3 Solve the problems. Show the equation as well as your answer. 9. Kristen challenges you with this puzzle: Add the square root of a mystery number to the square root of 100. The result is 19. What is the mystery number? 10. Jaime buys a square tablecloth. The package label declares, The area of this tablecloth is 800 square inches. What is the length of a side of the cloth? (Express your answer as a simplified radical.) 004 by CompassLearning, Inc. Algebra AL114

161 Simplifying Equations with Radicals One way to simplify an equation containing a radical sign is to raise each side of the equation to the second power. Suppose an object is dropped from a tall building. At the moment the object reaches a velocity of 4 feet per second, how far has the object fallen? Use the formula V = 64d where V = velocity in feet per second and d = distance in feet. Solution: V = 64d 4 = 64d (4) = ( 64d ) 576 = 64d 9 = d The object has fallen 9 feet. Use a calculator to solve these problems. 1. Suppose the formula V = 3d is used to find the distance in feet (d) an object falls at a velocity (V) measured in feet per second. An object is dropped from the edge of a roof. At the moment the object reaches a velocity of 36 feet per second, it hits the ground. How far did the object fall?. Suppose the formula S = 5.5 d is used to determine the distance in feet (d) it takes an automobile to stop if it were traveling a certain speed in miles per hour (S). Find the distance it would take an automobile traveling 70 miles per hour to stop. Round your answer to the nearest whole number. 3. Suppose the formula d = 0.5 h is used to determine the height in inches (h) that a submarine periscope must be for an observer looking through that periscope to see an object that is a distance of (d) miles away. How far does a submarine periscope have to extend above the water to see a surface ship that is 1 mile away? 4. A rectangle measures 4 inches by 6 inches. What is the length in inches, to the nearest tenth, of a diagonal of that rectangle? Use the formula a + b = c, where a and b represent the legs of a right triangle and c represents the hypotenuse. 5. A 16-foot ladder is leaning against the side of a building. If the bottom of the ladder is 8 feet from the side of the building, how far above the ground does the ladder touch the building? Use the formula a + b = c,where a and b represent the legs of a right triangle and c represents the hypotenuse, and round your answer to the nearest tenth. 004 by CompassLearning, Inc. Algebra AL114

162 Radicals and Exponents Rewrite 3 3a using exponents. 3 3a = a 1 3 Rewrite each expression using exponents x y. 7b 5. 17xy d ab Write w 3 w with exponents and simplify. w 3 w = w 1 w 1 3 = w( ) = w 4 3 Simplify using exponents. Then find the products. 7. c c 8. n 3 n 9. d 3 d 10. x 3 x 11. y 6 y 1. b 3 7 b Simplify = (3 3 ) 1 = a k n m 3 Rewrite each expression using exponents. Then find the product c b x n by CompassLearning, Inc. Algebra AL115

163 Drawing and Using a Square Root Graph Use the square root graph to find the value of x when x = 0. Step 1 Find y = 0 on the y-axis. Follow the dashed horizontal line to the square root graph (curved solid line). The point at which the dashed line meets the graph is (x, 0), where 0 = x. Step Follow the dashed vertical line from (x, 0) to the x-axis. The dashed line intersects the x-axis at the value, x = 0. Step 3 Read the approximate value: x 4.5. y (x, 0) (0,0) x Use the square root graph to find the following square roots. Estimate to the nearest tenth by CompassLearning, Inc. Algebra AL116

164 Falling Objects and Soaring Rockets The famous Italian scientist, Galileo Galilei made several discoveries about velocity. He did several experiments with falling objects. He discovered that the farther an object falls, the faster its velocity when it reaches the ground. The velocity, v,ofa falling object can be computed using the following expression: v = gh.in this formula, h = the distance the object has fallen, and g = the acceleration due to gravity. On Earth, the acceleration due to gravity is about 3 feet per second squared, 3 ft/s. Find the velocity of an object after it has fallen 15 feet. Use the expression v = gh. In this example, h = 15 feet. (Use g = 3 ft/s.) v = gh = (3)( 15) = 960 = feet per second For a rocket to travel from Earth to the moon or another planet, the rocket has to escape Earth s gravity. The escape velocity (v e ), or velocity needed to escape a planet s gravitational forces, can be computed using the following expression: v e = gr.in this formula g = the acceleration due to gravity, and R = the radius of the moon or planet in meters. Solve the following problems. 1. Find the velocity of a falling object after it has fallen 100 feet.. What is a falling object s velocity after it has fallen 50 meters? (Use g = 9.8 m/s.) 3. Find the velocity of a falling object after it has fallen 900 meters. (Use g = 9.8 m/s.) 4. Find the velocity of a falling object after it has fallen 00 feet. 5. Find the escape velocity for Earth. Earth s radius, R, is 6,38 kilometers; g for Earth is km/sec. 004 by CompassLearning, Inc. Algebra AL117 AGS Publishing. Permission is granted to reproduce for classroom use only.

165 Powers and Roots: Calculator Lesson at a Glance In this lesson, you will explore the relationship between powers and roots using the Ticker Tape Calculator. Review the computer-based activity AL17 to access the Ticker Tape Calculator and the interactive version of the lesson. Warm-up Number Crossword Take a shot at completing the great number crossword. Solve this number crossword by filling in the arithmetic operators to form correct equations. Remember to apply the order of operations! 006 by CompassLearning, Inc. Algebra AL17

166 Case Studies Jeff and Ann know that mathematicians use powers to write numbers in scientific notation and sometimes use roots to express irrational numbers. They want to learn more about them. Join Jeff and Ann as they explore powers and roots using the Ticker Tape Calculator and record your solutions. Case Study 1 Interesting Inverses Jeff and Ann know that addition and subtraction are inverse operations. They also know that multiplication and division are inverse operations. But, they are unsure about powers and roots. They wonder if there is any way to get back to the original value if you take the root of a number. Develop a rule to identify the number of times that the root, n x must be multiplied by itself to equal x. Explain why the rule will work. 006 by CompassLearning, Inc. Algebra AL17

167 Case Study Take Your Pick Jeff is trying to find another way to represent an expression with a fractional exponent. x a b He makes a list of possible equivalent expressions. He s sure that one of them must be correct. Ann warns that there may be more than one equivalent expression. Identify which option or options are correct and explain why. Case Study 3 Deep Rooted Error Jeff was trying to evaluate the square root of negative numbers using the calculator. But, for every expression, the calculator display read ERROR. Ann was experimenting with squaring a negative number and then finding its square root to return to its original value. But, for every expression, the calculator returned a positive value. Jeff has concluded that it is not possible to find the square root of negative numbers. Ann has concluded that the square root of a number is always positive. Are they correct in their assumptions? Explain why or why not. 006 by CompassLearning, Inc. Algebra AL17

168 Case Study 4 Radical Reasoning Jeff and Ann are playing a game to see who can more accurately predict the square root of a number. They re allowed to use all of their calculators keys except the power and root buttons. And, they can keep notes of any previous calculations. The next number is 6. What would your guess be? What if the next number is 30? How can you use the information in the chart to guide your predictions? Find the actual answer on the calculator. How close were you? Case Study 5 Simple Sums? Jeff and Ann are simplifying roots with a variable. 18x 3 98x They factor the terms under the radicand. x x x x They then bring out a root for each pair of identical factors in the radicand. x 3 7 x 16x 1x Jeff and Ann know that the square root of is an irrational number, so the terms cannot be simplified further. Now, they wonder if the two terms can be combined. What do you think? What is the simplified expression? How can you check that the simplified expression equals the original equation? 006 by CompassLearning, Inc. Algebra AL17

169 Angles and Angle Measure Identify angle pairs. a b a and b are vertical angles. d c c and d are complementary angles. f e e and f are supplementary angles. Describe each pair of angles. Use one of the following words: vertical, complementary, supplementary. Diagram 1 Diagram Diagram 3 Diagram g h 155 p q r s 48 j k 1. g, h. p, q 3. r, s 4. j, k Study these angles. The letter m stands for measure of. Acute angle Right angle Obtuse angle 0 < m < 90 m = < m < 180 Refer to diagrams 1 4 on this page and answer the questions. 5. Is g acute? 6. Is r a right angle? 7. Is h obtuse? 8. Is j a right angle? 9. Is k acute? 10. Is s acute? 004 by CompassLearning, Inc. Algebra AL118

170 Pairs of Lines in a Plane and in Space Intersecting lines, m and n Parallel lines, m and n Skew lines, m and n m n n m n m Identify lines r and s in each figure. Write intersecting, parallel,or skew. r r s r s s r s r s r s A transversal, t, intersects two or more lines. t p q If t is a transversal, write yes.otherwise, write no. t a b a t b s r t t v w by CompassLearning, Inc. Algebra AL119

171 Angle Measures in a Triangle Theorem m A + m B + m C = 180º B 60 Theorem m A + m B = m y B? A 55? C A C y Find the measure of C. Let n = m C. n + 60º + 55º = 180º Solve: n = 65º Find the measure of B. Let k = m B. 85º + k = 140º Solve: k = 55º Find m x in each triangle. x x x x x by CompassLearning, Inc. Algebra AL10

172 Naming Triangles Triangles can be classified according to their sides. Scalene Isosceles Equilateral Fill in the chart by writing the classification word to describe the triangle with the given sides. Triangle Measurements of the Sides Classification.5 inches, 1.5 inches,.0 inches cm, 8 cm, 6.5 cm. feet, 3 feet, 3 feet mm, 35 mm, 35 mm 4. 5 units, 10 units, 5 units 5. Triangles can be classified according to their angles Acute Equiangular Obtuse Right Fill in the chart by writing the classification word to describe the triangle with the given angles. Triangle Measurements of the Angles Classification 60,60, , 110, ,15, ,70, ,30, by CompassLearning, Inc. Algebra AL11

173 Quadrilaterals Find m D in the parallelogram. D C A 50 B Step 1 Because A B C D, A and D are supplementary. Thus, 50º + m D = 180º. Step Solve for m D. 50º + m D = 180º 50º + m D 50º = 180º 50º m D = 130º Step 3 Check. 50º + m D = 180º 50º + 130º= 180º True Find the measures of the angles in the isosceles trapezoid. D C A 45 B 1. A. C 3. D Tell whether enough information is given to calculate the measures of the angles in each described figure. Explain your answer. 4. Given: The figure is a rectangle. B A C D 5. Given: The figure is a trapezoid in which no sides are equal. D C A 45 B 004 by CompassLearning, Inc. Algebra AL1

174 Congruent and Similar Triangles Are these triangles congruent? Give a reason for the answer. 3 in. 3 in. 4 in. 4 in. Yes. They are congruent by the Side-Angle-Side (SAS) theorem, which states that if two sides and the included angle of two triangles are equal, the triangles are congruent. Tell whether each pair of triangles is congruent. If the pair is congruent, name the theorem that proves congruence (SAS, SSS, ASA). 7 units 74 5 units 7 units 74 5 units 4. cm 4 cm 1.8 cm 4. cm 4 cm 1.8 cm in. 11 in. 10 units 10 units units 45 9 units Answer the question. 5. Are all right triangles similar? Tell why or why not. 004 by CompassLearning, Inc. Algebra AL13

175 Trigonometric Ratios Find the value of x, using trigonometric (trig) ratios. Step 1 Set up an equation using the appropriate trig ratio. Since the side opposite from the given angle and the hypotenuse are known, use the sine (or sin) ratio. x sin 50º = 9 9 Step Using a calculator, find and substitute the sine value. 50 x = 9 Step 3 Solve for x. x = or approximately 6.9 units. x Find the value of x to the nearest tenth. Use a calculator x 43 x 68 5 x 3. x Solve the problem On a summer afternoon, a smokestack casts an 8-meter shadow. At this time of day, rays from the sun are striking the ground at an angle of 75.To the nearest tenth of a meter, how high is the smokestack? sun ray 75¼ smokestack 8m shadow 004 by CompassLearning, Inc. Algebra AL14

176 Using Geometric Shapes There is an infinite variety of geometric figures in all shapes and sizes. In some geometric figures, diagonals can be drawn. Two different diagonals can be drawn in a rectangle. No diagonals can be drawn in a triangle. In a hexagon, nine diagonals can be drawn. Draw all of the diagonals that can be drawn in each figure. Then count them and write the number on the line. 1. an octagon. a pentagon Think again about the number of diagonals that can be drawn in a triangle (three sides), rectangle (four sides), pentagon (five sides), hexagon (six sides), and octagon (eight sides). Find a pattern then use the pattern to predict the number of diagonals that can be drawn in the following shapes. 3. a dodecagon (1 sides) 4. a heptagon (seven sides) 5. a decagon (ten sides) 004 by CompassLearning, Inc. Algebra AL15 AGS Publishing. Permission is granted to reproduce for classroom use only.

177 Trigonometric Ratios: Calculator Lesson at a Glance In this lesson, you will explore trigonometric ratios using the Ticker Tape Calculator. Review the computer-based activity AL174 to access the Ticker Tape Calculator and the interactive version of the lesson. Warm-up Cryptic Cryptogram Cryptograms are coded messages. Each symbol represents a letter. Using the given clue, test your skills and see how quickly you can crack this cryptogram. Above each symbol, write the corresponding letter. 006 by CompassLearning, Inc. Algebra AL174

178 Case Studies Logan and Jane are learning some basic trigonometry. They are exploring relationships between the sides and angles of right triangles. Join them as they investigate trigonometric ratios using the Ticker Tape Calculator and record your solutions. Case Study 1 Acute Question Logan and Jane wonder if it is possible for the sine or cosine of an acute angle to be a whole number. What are the possible values for the sine and cosine of acute angles? Explain how you arrived at your answer. 006 by CompassLearning, Inc. Algebra AL174

179 Case Study Rise and Fall Jane wonders what happens to the values of the sine and cosine as an angle increases from 0 to 90. Logan thinks that they would move in opposite directions. Jane thinks that they would rise and fall together. What do you think? Create a chart and record the results. What pattern emerges? Does it fit your prediction? A 30 A 45 A 60 Case Study 3 Ratio Relationships Logan and Jane had discovered that the sine of an angle and the cosine of a complementary angle are identical. sin A cos (90 A) Now, they re wondering about the relationship of an angle s tangent to its sine and cosine. tan A? Help Logan and Jane use the sine and cosine of an angle to calculate its tangent. Provide supporting examples. sin A the length of the side opposite A the length of the hypotenuse cos A the length of the side adjacent to A the length of the hypotenuse tan A the length of the side opposite A the length of the side adjacent to A 006 by CompassLearning, Inc. Algebra AL174

180 Case Study 4 The Lost Arcsine As Logan was exploring trigonometric functions on his calculator, he noticed three more buttons. He logically concluded that these functions must undo the,, and keys. Jane tells Logan that the keys are called arcsine, arccosine, and arctangent. She also says that this feature is just what she needs to solve her problem. Jane must find all the angle measurements of a right triangle. She knows that its hypotenuse is 17 feet and one of the sides is 10 feet. Describe how Logan and Jane can find the missing angles. Round the answers to the nearest hundredth. Case Study 5 More Than Right Jane noticed that she could only draw acute angles with right triangles. She wonders if this means that only angles between 0 and 90 have trigonometric ratios. Logan thought that this was an interesting challenge. He tried entering angle measures greater than 90 in the Ticker Tape Calculator. When he pressed the and keys, they returned real values. Jane and Logan decided to track the sine and cosine of angles. What pattern did they find? Predict the sine and cosine of 405. Check your answers using the Ticker Tape Calculator. Were you correct? 006 by CompassLearning, Inc. Algebra AL174

181 Solutions by Factoring Put 6x = x + 15 in standard quadratic form. Step 1 Match the terms with terms in the standard form. Standard Quadratic Form: ax + b x + c = 0 6x = (1)x + 15 Step Subtract x from both sides of the equation. 6x x = x + 15 x Result: 6x x = 15 Step 3 Subtract 15 from both sides of the equation. 6x x 15 = Result: 6x x 15 = 0 Rearrange the terms of each equation to put it in standard quadratic form: ax + bx + c = x = 6 5x. x + 15 = x 3. y 7y + 6 = x = x Solve x 3x 4 = 0 by factoring. Step 1 Factor: (x 4)(x + 1) = 0 Step Set each factor = 0, and solve for x. x 4 = 0 x + 1 = 0 x = 4 x = 1 Step 3 Check the results. (4) 3(4) 4 = 0 ( 1) 3( 1) 4 = 0 0 = 0 True 0 = 0 True Use factoring to solve the equation you rearranged in problem 1 above by CompassLearning, Inc. Algebra AL16

182 Writing the Equations from Their Roots The roots of a quadratic equation are 1 and 3. What is the general form of the equation? Step 1 Given the roots, x = 1 or x = 3. Step Set the factors equal to zero. (x + 1) = 0 (x + 3) = 0 Step 3 Multiply the factors. (x + 1)(x + 3) = 0 Step 4 Use the distributive property to place the equation in general form. x + 4x + 3 = 0 Find the quadratic equation that has these roots. 1. 3,. 1, , 3 4., , , 7., , 9. 3, , , 1. 5, , , , by CompassLearning, Inc. Algebra AL17

183 Solving by Completing the Square Find the roots of x + 4x 3 = 0 by completing the square. Step 1 Rewrite the equation so that the constant is isolated. x + 4x = 3 Step Find the constant that must be added to complete the square. Take 1 of the x coefficient and square it. 1 (4) = = 4 Step 3 Add the constant to both sides of the equation. x + 4x + 4 = Result: x + 4x + 4 = 7 Step 4 Factor the trinomial on the left side, and solve for x. (x + ) = 7 Therefore, x + = ± 7 x = + 7 or x = 7 Step 5 Check by substituting the roots in the equation. Find the roots of each equation by completing the square. 1. x + 4x + 3 = 0. x x = 0 3. m 6m 7 = 0 4. k + 1k = 0 5. y + 8y 9 = 0 6. x 10x = 0 7. x 6x + = 0 8. a 4a 1 = 0 9. x x 15 = d d 4 = by CompassLearning, Inc. Algebra AL18

184 Solving Using the Quadratic Formula Use the quadratic formula to find roots of x + 5x + 6 = 0. b ± b ac x = 4 a Values of a, b, and c from the equation: a = 1 b = 5 c = 6 5 ± 5 (1)(6) Substitute: x = 4 (1) x = or 3 To check, substitute the roots in the original equation. Use the quadratic formula to find the roots of these equations. Remember to write the equation in standard form first. 1. y 5y + 6 = 0. x + 7x + 1 = 0 3. n n 1 = 0 4. x = 3x + 4 Check that the roots of the equation are valid. x + 4x = 0 Roots: 1 + or 1 Substitute for x: ( 1 + ) + 4( 1 + ) = 0 0 = 0 Therefore, 1 + is valid. Substitute for x: ( 1 ) + 4( 1 ) = 0 0 = 0 Therefore, 1 is valid. Check the roots of the equation. Tell whether they are valid. 5. x x 4 = 0 Roots: or by CompassLearning, Inc. Algebra AL19

185 Using Quadratic Equations You can model the path of a rocket in flight using the quadratic equation. A model rocket is launched vertically with a starting velocity of 75 feet per second. After how many seconds will the rocket be 40 feet above the ground? Step 1 Use the vertical motion formula h = 16t + vt + s. h = height above ground in feet t = time in seconds v = starting velocity in feet per second s = starting height in feet h = 16t + vt + s 40 = 16t + 75t = 16t + 75t 40 Step Use the quadratic formula to find t. x = t = b ± b ac 4 a 75 ± (75) 6)( 40 (4)( 1 ) ( 16) 75 ± 3065 t = 3 t = 0.61 seconds and 4.07 seconds, rounded to the nearest hundredth. The rocket will be 40 feet above the ground after 0.61 seconds and 4.07 seconds. Answer the questions using the vertical motion formula, the quadratic formula, and a calculator when needed. 1. After how many seconds will the rocket in the example be 60 feet above the ground?. After how many seconds will the rocket be 75 feet above ground? 3. Look again at the example. Why is the model rocket 40 feet above the ground at two different times? 4. Explain how you could estimate the total flight time of the rocket. 5. Estimate the total flight time of the rocket. 004 by CompassLearning, Inc. Algebra AL131 AGS Publishing. Permission is granted to reproduce for classroom use only.

186 Modeling Quadratic Equations: Grapher Lesson at a Glance In this lesson, you will explore the attributes of quadratic equations through graphing. Review the computer-based activity AL176 to access the Grapher and the interactive version of the lesson. Warm-up The Great Treasure Hunt Sheikh Al Jabra has hidden four fabulous jewels in the Great Desert. But, he cannot remember where they are. Help the Sheikh find the jewels by matching them with their location and their coordinates. Place an X in a box if it is not a match. Place an O in the box if it is a match. Clues: The Dangerous Dunes and the Odd Oasis are a distance of [6, 4] from each other. The Star Sapphire and the Emerald Eye are a distance of [3, 0] from each other. The Dual Diamond is a distance of [1, 5] from the Shifting Sands. The red jewel is located in the Dangerous Dunes. The blue jewel is not found at (4, ). 006 by CompassLearning, Inc. Algebra AL176

187 Case Studies Sarah is watching her two brothers play ball. She recognizes that the path of each throw resembles a parabola. Join Sarah as she explores graphing quadratic equations and record your solutions. Case Study 1 Root of Everything Sarah has been graphing several parabolas. She is trying to identify a pattern in the quadratic equations that predicts what a graph of a parabola will look like. Sarah found the roots of a quadratic equation by setting y 0 and factoring x 4. y x 4 Roots If y 0, 0 x 4 0 (x )(x ) x or x - What is the connection between the roots of a quadratic equation and the graph of a parabola? What is the least and the greatest number of roots a quadratic equation can have? Case Study Variable Differences Sarah knows that the standard form of a quadratic function is y Ax Bx C. She knows that the x and y values represent the coordinates on the parabola. She is now wondering what A represents. Experiment with various values of A to see how it affects the graph of the parabola. Record your trials and your conclusion. 006 by CompassLearning, Inc. Algebra AL176

188 Case Study 3 Varying Values Now that Sarah knows how the value of A affects the graph of the quadratic function y Ax Bx C,she wonders what the value of C represents. Create a variety of equations with different values for C.Graph the equations and record your observations. What does the value of C tell you about the graph of a parabola? Explain this relationship in mathematical language. Case Study 4 Line of Symmetry Sarah knows that parabolas are symmetrical. She wonders if the equation for the line of symmetry of the parabola is related to the coefficients A, B,or C. y Ax Bx C Can the values of A, B,or C be used to describe a parabola s line of symmetry? Explain your thinking. 006 by CompassLearning, Inc. Algebra AL176

189 Case Study 5 Alternative Graphing Sarah noticed two other forms for entering a quadratic function on the Grapher tool. y a(x h) k (y y 0 ) a(x x 0 ) She knows that the x and y values always represent the coordinates on the parabola, but she wonders what the other variables represent. Explore graphing quadratic equations using these forms. What do the other variables represent on a graph? 006 by CompassLearning, Inc. Algebra AL176