Algebra SUMMER PACKET Ms. Bank

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1 SUMMER PACKET Ms. Bank Just so you know what to expect next year We will use the same text that was used this past year: published by McDougall Littell ISBN-13: Summer Packet To help students retain the math skills listed on the following page, the math department requires rising students to complete this Algebra Summer Math Packet. The skills required to answer the questions in this packet are ones that should have been mastered by students in previous math courses. They are the skills covered in the first two chapters of the Algebra textbook. The packet contains a brief review and example problems for each skill. Students are to complete all of the questions in the Practice and Quiz sections for each skill. Please note: 1. Working through these problem sets is mandatory. 2. Students are to bring completed problem sets with them to turn in on the first day their math class meets next school year. 3. All work is to be done in pencil on notebook paper. ALL WORK MUST BE SHOWN. 4. Students should NOT use calculators on any portion of the packet. 5. Students should check their work upon completion of each section. Answers to the questions are located in the back of the packet. If an answer is incorrect, students should return to the work shown, attempt to find the source of the error(s), and correct the problem. 6. If students need more explanation and practice, they should read the corresponding section in the first or second chapter of the text and work odd numbered problems to check for understanding. (Answers to odd numbered problems are in the back of the text.) 7. Students will be given a homework grade for completion of this packet. 8. Students will take a test covering this material in the first week of school. Remember it is truly beneficial to keep your mathematical mind oiled over the summer. Should you lose this packet, it can be found on my school web page. Should you have questions, I will be checking throughout the summer. WORK IS TO BE DONE ON NOTEBOOK PAPER IN PENCIL. You will not turn in your copy of the packet, just your work. pbank@altamontschool.org Summer Packet

2 Summer Packet

3 A. Expressions, Equations, and Inequalities (pp. 1 3) A variable is a letter used to represent one or more numbers. An algebraic expression is made from numbers, variables, and algebraic operations. The following examples describe how expressions can be evaluated, combined, written, and used to write algebraic equations and inequalities. 1. Evaluate Expressions Evaluate an expression Substitute a number for the variable, perform the operation(s), and simplify the result if necessary. A. Expressions and Equations When a number is followed directly by a variable, the operation of multiplication is always implied. Evaluate the expression. a. 16n when n 5 4 b. 25 } when k 5 5 k c. h 2 8 when h d. 4 } 3 1 h when h 5 } 1 3 e. x 3 when x 5 4 f. a 2 when a a. 16n b. 25 } 5 } 25 c. k 5 h d } 3 1 h 5 } } 1 3 e. x f. a } (1.2)(1.2) Evaluate the expression. 1. 5b when b } h when h b when b v 1 7 } 6 when v 5 1 } 3 5. y 4 when y q 2 when q Order of Operations Order of operations Established rule for evaluating an expression involving more than one operation: Step 1: Evaluate expressions inside grouping symbols. Step 2: Evaluate powers. Step 3: Multiply and divide from left to right. Step 4: Add and subtract from left to right. Benchmark 1 Chapters 1 and 2 1

4 A. Expressions and Equations The multiplication that could be written in two steps (3 16 evaluated first, followed by 5 6) is combined as one step. Evaluate the expression. a b. 4( ) c. 5[12 2 (4 1 5)] a Evaluate power Multiply Subtract. b. 4( ) 5 4(9 1 5) Evaluate power. 5 4(14) Add within parentheses Multiply. c. 5[12 2 (4 1 5)] 5 5(12 2 9) Add within parentheses. 5 5(3) Subtract within brackets Multiply. Evaluate the expression. 7. 4(10 2 3) ( ) 9. 2[42 4 (9 2 3)] 3. Write Expressions Translate verbal phrases into expressions. Keep a glossary of a. The product of 8 and m increased by 5 terms that describe each of the four b. The quotient of 8 and the difference of a number x and 2 basic operations. c. The sum of 20 and the square of a number n a. 8m 1 5 b. 8 } x 2 2 c n 2 Translate the verbal phrases into expressions. 10. The quotient when the quantity of a number y increased by 4 is divided by less than twice the square of a number q more than the product of a number w and 6 4. Write Equations and Inequalities Open sentence A mathematical statement that contains two expressions and a symbol that compares them. Equation An open sentence that contains the symbol 5. Inequality An open sentence that contains one of the symbols,,, or. 2 Benchmark 1 Chapters 1 and 2

5 No less than () and no greater than () are opposites of less than () and greater than (), respectively. Write an equation or an inequality. a. The difference of a number p and 12 is at most 15. b. The product of 5 and a number m is 14. c. A number x is at least 6 and less than 9. a. p b. 5m 5 14 c. 6 x 9 Write an equation or inequality. 13. The quotient of 12 and a number q is at most 5. A. Expressions and Equations 14. The sum of twice a number h and 5 is the same as The difference of a number w and 4 is greater than 12 and no more than 20. Quiz Evaluate the expression. 1. h } } 3 when h } b 2 when b a } 4 when a 5 4 Evaluate the expression. 4. ( ) (2 1 3) [( ) 1 1] 6. [54 4 (6 2 3) 2 ] 2 }} Translate the verbal phrases into expressions. 7. The product of twice the number y and 4 increased by 8 8. The difference of 6 times the square of a number x and 15 Write an equation or inequality. 9. The sum of the number b and 12 is twice the number b. 10. The product of a number q and 3 is no less than 10 and no more than 15. Benchmark 1 Chapters 1 and 2 3

6 B. Problem Solving B. Problem Solving (pp. 4 5) One way to try solving a math problem is to use an organized strategy, or problem-solving plan. Read the problem to find what information is given and what you need to find out. Decide on the strategy you will use, and apply it to solve the problem. Finally, check that your solution makes sense. 1. Check Possible Solutions Solution of an equation or inequality A number that can be substituted for the variable in an equation or inequality to make a true statement. Check whether the given number is a solution of the equation or inequality. x a. 2x ; 3 b. } ; 6 c. x 2 5 3; 2 a. 2(3) b. 6 } c Þ 7 3 is a solution. 6 is not a solution. 2 is a solution. There may be more than one method that can be used to solve a problem. Check whether the given number is a solution of the equation or inequality. n 2 3 r a 10; } ; 4 3. } ; p 2 6 0; d ; 7 6. m ; Read and Understand a Problem Read the problem below. Identify what you know and what you need to find out. You do not need to solve the problem. You run in a city where the short blocks on north-south 0.2 mi streets are 0.03 miles long. The long blocks on east-west streets are 0.2 mile long. You will run 2 long blocks east, 0.03 mi a number of short blocks south, 2 long blocks west, then back to your starting point. You want to run 1.1 miles. How many short blocks should you run? What do you know? Each short block is 0.03 miles long. Each long block is 0.2 miles long. You will run 4 long blocks total (2 east 1 2 west). You will run s short blocks total (south and north). You want to run a total of 2 miles. What do you want to fi nd out? How many short blocks should you run so that the distance you run on short blocks and the distance you run on 4 long blocks makes a total of 1.1 miles? 4 Benchmark 1 Chapters 1 and 2

7 7. Read the problem below. Identify what you know and what you need to find out. You do not need to solve the problem. start/finish A bicycle park has a long trail and a short trail. The long trail is 5 km long. The short trail is 2 km long. You will ride 3 laps on the short trail and some number of laps on the long trail. You want to ride 21 km. How many laps should you ride on the long trail? 3. Make a Plan Write a verbal model of the statement below. How many short blocks should you run so that the distance you run on short blocks and the distance you run on 4 long blocks makes a total of 1.1 miles? Distance run on short blocks 1 Distance run on long blocks 5 Total distance (miles) (miles) (miles) 2 km 5 km B. Problem Solving Length of a short block Number of short blocks 1 Length of a long block Number of long blocks 5 Total distance (miles/block) (block) (miles/block) (block) (miles) 8. Write a verbal model for the problem in Exercise 7. Quiz Check whether the given number is a solution of the equation or inequality. n 1 5 m j 4; } 5 6; 7 3. } ; y 1 3 0; g ; b ; 213 Read the problem below. Identify what you know and what you need to find out. Then, write a verbal model of the problem. You do not need to solve the problem. start 7. Jim s grandmother exercises by walking the main rectangular hall of a local shopping mall. She walks 90 yards down the length of the hall, turns right, and walks 20 yards across the width of the hall. Then, she turns right and walks 90 yd up the length of the hall again. Finally, she turns right one more time, and walks 20 yards across the width of the hall and ends up at her starting point. Jim s grandmother wants to walk 970 yards. She will walk the length of the hall 20 yd 9 times. How many times will she walk across the width of the hall? Benchmark 1 Chapters 1 and 2 5

8 C. Representations of Functions C. Representations of Functions (pp. 6 9) Functions can be represented by mapping diagrams, tables, verbal or algebraic function rules, and graphs. Each input value and its corresponding output value make up an ordered pair. An ordered pair can be written as (input, output) or plotted as a point on a coordinate grid. 1. Identify the Domain and Range of a Function Function A pairing of input values to output values, where the value of each output depends on the value of the corresponding input, and each input corresponds to exactly one output. Domain The set of input values for a function. Range The set of output values for a function. Identify the domain and range of the function. a. Input Output b c. Input Output a. Domain: 0, 1, 2, 3 b. Domain: 3, 6, 9, 12 c. Domain: 28, 4, 6, 10 Range: 10, 11, 12, 13 Range: 21, 22, 23, 24 Range: 24, 2, 3, 5 Identify the domain and range of each function. 1. Input Output 2. Input Output Benchmark 1 Chapters 1 and 2

9 Tell whether the pairing is a function. a. Input Output b c. Input Output a. Yes b. No; 0 maps to c. No; 40 maps to two two outputs. outputs. C. Representations of Functions Tell whether the pairing is a function. 4. Input Output Input Output A function rule usually states the dependent variable y as a function of the independent variable x, such as y 5 x Write a Function Rule Independent variable A function s input variable. Dependent variable A function s output variable. Write a rule for the function. Input, x Output, y Each value of y is 3 times the corresponding x value. The function rule is y 5 3x. Write a rule for the function. 7. Input, x Input, x Output, y Output, y Input, x Input, x Output, y Output, y Benchmark 1 Chapters 1 and 2 7

10 C. Representations of Functions 3. Make a Table for a Function Make a table for the function and identify the range of the function. y 5 x Domain: 2, 3, 4, 5, 6 Input, x Output, y Range: 4.6, 5.6, 6.6, 7.6, 8.6 Make a table for the function and identify the range of the function. 11. y 5 2 } 3 x 12. y 5 x y 522x 1 5 Domain: 3, 6, 9, 12, 15 Domain: 25, 24, 23, 22, 21 Domain: 1, 2, 4, 7, y 5 } x y 5 x y 525x Domain: 20, 30, 40, 50, 60 Domain: 2, 5, 6, 8, 9 Domain: 23, 21, 4, 8, Graph a Function Graph the function y 5 x 2 2 with domain 4, 5, 6, 7, and 8. Step 1: Make an input-output table. Input, x Output, y Step 2: List the ordered pairs (x, y). (4, 2), (5, 3), (6, 4), (7, 5), (8, 6) Step 3: Plot a point for each ordered pair (x, y). y x Graph the function. 17. y 5 x y 5 1 } 3 x 19. y 5 3x 2 3 Domain: 5, 7, 10, 13, 15 Domain: 3, 9, 15, 21, 27 Domain: 1, 2, 3, 4, 5 8 Benchmark 1 Chapters 1 and 2

11 20. y 5 1.5x y 5 1 } 2 x y 5 x 1 2 } 3 Domain: 0, 3, 4, 6, 9 Domain: 4, 5, 7, 8, 10 Domain: 1, 4, 7, 10, 13 Quiz Identify the domain and range of each function. 1. Input Output Input Output } } } C. Representations of Functions Tell whether each pairing is a function. 4. Input Output Input Output Write a rule for the function. 7. Input, x Output, y Input, x Output, y Input, x Output, y Input, x Output, y Make a table for the function and identify the range of the function. 11. y 524x y 5 } 3 4 x y 5 } 2x Domain: 23, 21, 2, 5, 6 Domain: 10, 12, 14, 16, 18 Domain: 1, 9, 13, 19, 23 Graph the function. 14. y 5 x y 5 5x 16. y 52x 1 4 Domain: 6, 7, 8, 9, 10 Domain: 0, 2, 4, 5, 9 Domain: 0, 1, 2, 3, 4 Benchmark 1 Chapters 1 and 2 9

12 D. Operations Note that 2a is postive when a is negative. D. Operations (pp ) Whole numbers, integers, and rational numbers are part of the set of real numbers. The following examples describe different operations with real numbers. 1. Find Opposites of Real Numbers Opposite of a real number a 2a (read the opposite of a ) is the same distance from 0 on a number line as a, but it is on the opposite side of 0. For the given value of a, find 2a. a. a 5 3 b. a } 5 c. a a. 2a 52(3) b. 3 2a 5217 } 5 2 c. 2a 52(25.4) 2a a 527 } 5 2a For the given value of the variable, find the opposite. 1. x u m w 5 } k 52 } c 528 } 7 The absolute value of a number is always positive. 2. Find Absolute Values of Real Numbers Absolute value of a real number a ZaZ (read the absolute value of a ) is the distance between a and 0 on a number line. If a is greater than or equal to 0, ZaZ is a. If a is less than zero, ZaZ is the opposite of a. For the given value of a, find ZaZ. a. a 5 8 b. a 52 4 } 9 c. a a. ZaZ 5 Z8Z 5 8 b. 4 4 ZaZ 5 )2} 9 ) 5212 } } 4 9 c. ZaZ 5 Z11.5Z For the given value of the variable, find the absolute value. 7. b y p v n 5 } h 5 10 } Add Real Numbers Sum The result of adding two or more real numbers. To use a number line to find the sum of a 1 b: Start at a. If b 0, you will move to the right. If b 0, you will move to the left. Find ZbZ and move that many units. The number you stop on is the sum. 10 Benchmark 1 Chapters 1 and 2

13 Use a number line to find the sum. a. 4 1 (27) b a. End at 3. Start at 4. Move 7 units to the left (27) Use a number line to find the sum. b. Move 3 units to the right. Start at 2. End at (24) (28) (22) (26) Use the rules of real number addition to find the sum. a (221) b. 2 1 } } 2 c D. Operations You can find the sum of three or more numbers together by first adding two of the numbers and then adding the result to the third. Use grouping symbols around negative numbers in your work to keep track of signs as you simplify expressions. If two numbers have the same sign, add their absolute values. The sum has the same sign as the numbers added. a (221) 52(Z19Z 1 Z21Z) 52( ) 5240 If two numbers have different signs, subtract the absolute value of the smaller number from the absolute value of the larger number. The sum has the same sign as the number with the larger absolute value. 1 b. 2 } 2 1 } ) } ) 2 )2} 2 ) 5 } } c (Z22.8Z 2 Z1.5Z) 52( ) Use the rules of real number addition to find the sum (20.3) (210) (234) s 1 2 } } } (247.7) 4. Subtract Real Numbers Difference The result of subtracting one real number from another real number. Find the difference. a b c (23) To subtract b from a, add a and the opposite of b. a (218) b (29) c (23) Benchmark 1 Chapters 1 and 2 11

14 D. Operations To multiply three or more real numbers, first multiply two of the numbers, then multiply the result with the third number. Find the difference } (24) } } (282) 5. Multiply Real Numbers Product The result of multiplying two or more real numbers. Find the product. a. 21.7(4) b. 2 4 } 5 ( 210) c. 2(23)(28) The product of two numbers with the same sign is positive and the product of two numbers with different signs is negative. a. 21.7(4) b. 2 } 4 5 ( 210) 5 } 40 5 c. 2(23)(28) 5 [2(23)](28) (28) 5 48 Find the product (23) } 4 (6) } 8 12 } } (4)(23) (10.4)(27) (29.1) You can check your answer by multiplying the original number by its inverse and making sure the product is 1. Multiplicative inverse of a real number a The reciprocal of a, or 1 } a. The product of a and its multiplicative inverse is 1. Find the multiplicative inverse of a. a. a 5 9 b. a 524 c. a 52 1 } 8 1 a. } a 5 } 1 b. 9 1 } a 5 1 } } 4 c. Find the multiplicative inverse of the number } } } } 32 1 } a 5 1 } 2 1 } } Benchmark 1 Chapters 1 and 2

15 Division by 0 is undefined, because 0 does not have a multiplicative inverse. 6. Divide Real Numbers Quotient The result dividing a real number by another real number. Find the quotient. a (27) b (213) c } } a (27) } 7 2 b (213) } 13 2 c. 52 } } } } } } } Find the quotient (24) } 4 4 } } } } (28) } } 4 9 D. Operations The symbol 6 in front of a number refers to the number and its opposite. For example, 66 is the same as 6 and Find Square Roots Square root of a If b 2 5 a, then b is the square root of a. Every positive nonzero real number a has two square roots, 2Ï } a and Ï } a. Radicand The number or expression inside a radical symbol. Evaluate the expression. a. 6Ï } 49 b. Ï } 1 c. 2Ï } 144 a. 67 b. 1 c. 212 Evaluate the expression Ï } Ï } Ï } Ï } Ï } Ï } 900 Benchmark 1 Chapters 1 and 2 13

16 D. Operations Quiz For the given value of the variable, find the opposite, absolute value, and multiplicative inverse. 1. a y } r Evaluate the expression (265) } 7 } } } (215) 8. 22(235) 9. Ï } } Ï } Benchmark 1 Chapters 1 and 2

17 Use a Venn diagram to help remember which numbers are part of other numbers. E. Properties and Real Numbers (pp ) Taken together, the rational and irrational numbers make up the set of real numbers. The following examples illustrate some characteristics and properties of real numbers. 1. Classify Real Numbers Whole numbers A subset of the real numbers; A whole number is either 0 or one of the counting numbers, 1, 2, 3,... Integers A subset of the real numbers; The integers are the set of whole numbers and their opposites,... 23, 22, 21, 0, 1, 2, 3,... Rational numbers A subset of the real numbers; A rational number can be expressed as the ratio of two integers, and its decimal form terminates or repeats. Irrational numbers A subset of the real numbers; An irrational number cannot be expressed as the ratio of two integers, and its decimal form neither terminates nor repeats. E. Properties and Real Numbers Choose the word that best describes each: whole, integer, rational, or irrational. a. 28 b. 3 } 4 c. 2Ï } 7 d e. Ï } f. 4 } 3 a. Integer; opposite of a b. Rational; ratio of two c. Irrational; cannot be whole number integers written as ratio of two integers nor as a terminating or repeating decimal d. Rational; terminating e. Whole; Ï } f. Rational; } , decimal a repeating decimal Choose the word that best describes each: whole, integer, rational, or irrational } } Ï } Benchmark 1 Chapters 1 and 2 15

18 E. Properties and Real Numbers To order numbers, it is sometimes helpful to write decimal approximations of rational and irrational numbers. 2. Order Real Numbers To order real numbers from least to greatest, graph them first. Then read the numbers from left to right. 4 Order the numbers from least to greatest: 22, 24.5, 25, In order from least to greatest, the numbers are 25, 24.5, 2Ï } 4, and 2 4 } 5. Order each group of numbers from least to greatest , 1 } 2 3,} 3 2, Ï } , 21.9, 2Ï } 0.04,Ï } , Ï } 1 6, 6 } 6, 6.1 Remember that the additive inverse of a, 2a, is not necessarily a negative number. 3. Identify Properties of Addition Additive identity The number 0 is the additive identity. When 0 is added to a real number a, the sum equals a. Additive inverse The opposite of a real number a is its additive inverse. The sum of a real number and its additive inverse always equals 0. Identify the property of addition being illustrated. a. (24 1 b) (b 1 3) b. 7 1 x 5 x 1 7 c } } 8 d (217.3) 5 0 a. Associative; b. Commutative; c. Identity; d. Inverse; a changing the changing the adding 0 to a number plus grouping does order does not number does its opposite not change change the sum not change equals 0 the sum the number Identify the property of addition being illustrated k k } } } } (21 1 n) 5 (0 1 (21)) 1 n r 1 (s 1 t) 5 (r 1 s) 1 t Benchmark 1 Chapters 1 and 2

19 4. Identify Properties of Multiplication Multiplicative identity The number 1 is the multiplicative identity. The product of a real number a and 1 equals a. Identify the property of multiplication being illustrated. a. 0 w 5 0 b. 56 (92 11) 5 (56 92) 11 c. g h 5 h g d. 24, ,978 e. 277 (21) 5 77 f. (r s) t 5 r (s t) E. Properties and Real Numbers. Remember that associative is related to grouping and commutative is related to order. a. Zero; a number b. Associative; changing c. Commutative; changing times 0 equals 0 the grouping does not the order does not change the product change the product d. Identity; multiplying e. Negative one; the f. Associative; changing a number by 1 does product of a number the grouping does not not change the number and 21 is the opposite change the product of the number Identify the property of multiplication being illustrated p 5 p (21) 18. (25a)b 525(ab) (235) (29) Apply the Distributive Property Distributive property The product of two factors, where one factor is a sum, is the sum of the product of the first factor times the first addend plus the product of the first factor times the second addend. For example, a(b 1 c) 5 ab 1 ac. Equivalent expressions Expressions that are equal in value, for any value of the variable. Use the distributive property to write an equivalent expression. a. 4(9 1 2) b. 8(b 2 3) c. 22n(n 1 6) a. 4(9 1 2) 5 (4)(9) 1 (4)(2) b. 8(b 2 3) 5 8[b 1 (23)] b 1 8(23) b 1 (224) 5 8b 2 24 c. 22n(n 1 6) 5 (22n)(n) 1 (22n)(6) 522n 2 1 (212n) 522n n Use the distributive property to write an equivalent expression (4 1 v) 23. c(2c 1) 24. 5m(3 m) 25. 9(t 8) } 5 (15 20r) 27. 3p(5 7p) Benchmark 1 Chapters 1 and 2 17

20 BENCHMARK 2 1 E. Properties and Real Numbers. Quiz Choose the word that best describes each number in the list. Then, write the numbers in order from least to greatest. 1. Ï } 8, 4.1, 3, } , 29.02, 2} 2 9, 2Ï } } 1, 20.01, 21.1, 2Ï } 1 Identify the property being illustrated. 4. (8 3b) 2 8 (3b 2) 5. 7q 7q(1) 6. } 3 7 t } 3 7 t (3x 2 ) 3x (6 s) 90 15s h h y(2) 2(6y) (jk) 18j(k) 12. 6c(1 3c) 6c 18c m(0) (2z 1) 6z w 0 37w Use the distributive property to write an equivalent expression (h 2) } 2 d(8 6d) 18. 8f(4f 10) 18 Benchmark 1 Chapters 1 and 2

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