Algebra 7

AM108R Hayes® By Murney R. Bell Grade 7+ Mastering the Standards ALGEBRA By Murney R. Bell Illustrated by Reneé Yates © Copyright 2008, Hayes School Publishing Co., Inc., Printed in USA All rights reserved. The purchase of this book entitles the individual teacher to reproduce the activities in this book for use with children. No parts of these publications may be stored in a retrieval system or transmitted in any form by any means, electronic, mechanical, recorded, or otherwise, without prior written permission of Hayes School Publishing Co., Inc. TABLE OF CONTENTS Letter to Teachers and Parents...........................................................1 Number and Operations Teacher Overview...............................................................................2 Practice Assessment............................................................................3 Using Variables...................................................................................4 Exponents............................................................................................5 Order of Operations............................................................................6 Real Numbers.....................................................................................7 Adding and Subtracting Integers.......................................................8 Multiplying and Dividing Integers.....................................................9 Scientific Notation............................................................................10 Distributive Property.........................................................................11 Graphing Data on Coordinate Planes.............................................12 Linear Relations and Functions Practice Assessment..........................................................................13 Relations and Functions...................................................................14 Function Rules, Tables, and Graphs.................................................15 Writing Function Rules.....................................................................16 Solving One-step Equations..............................................................17 Solving Two-step Equations..............................................................18 Solving Multistep Equations ............................................................19 Solving Equations with Variables on Both Sides..............................20 Formulas...........................................................................................21 Absolute Values Inequalities............................................................22 Graphing Inequalities......................................................................23 Solving Inequalities Using Addition and Subtraction.....................24 Solving Inequalities Using Multiplication and Division.................25 Ratios and Proportions.....................................................................26 Percent of Change............................................................................27 Slope-Intercept Form.........................................................................28 Standard Form..................................................................................29 Point-Slope Form...............................................................................30 Mathematical Models Practice Assessment..........................................................................31 Scatter Plots and Equations of Lines................................................32 Graphing Absolute Value Equations................................................33 Zero and Negative Exponents..........................................................34 Multiplication Property of Exponents..............................................35 Power of Powers of Exponents..........................................................36 Division Property of Exponents........................................................37 Arithmetic Sequences.......................................................................38 Geometric Sequences........................................................................39 Exponential Functions......................................................................40 Symbol Manipulations Practice Assessment..........................................................................41 Solving a System Using Substitution................................................42 Solving a System Using Elimination................................................43 Solving a System Using Matrices......................................................44 Adding and Subtracting Polynomials. .............................................45 Multiplying and Factoring...............................................................46 Multiplying Binomials......................................................................47 Perfect Squares and Difference of Squares.......................................48 Factoring Trinomials.........................................................................49 Factoring Perfect Squares and Difference of Squares......................50 Factoring by Chunking.....................................................................51 Change Analysis Practice Assessment..........................................................................52 Quadratic Graphs.............................................................................53 Quadratic Functions.........................................................................54 Square Roots.....................................................................................55 Solving Quadratic Equations by Factoring......................................56 Using Quadratic Formulas...............................................................57 Simplifying Radicals.........................................................................58 Pythagorean Theorem......................................................................59 Distance and Midpoint Formulas....................................................60 Test-Taking Strategies.......................................................................61 INTRODUCTION This book contains standards-based problems similar to those students will find on mastery tests in mathematics. The problems are based on standards from the National Council of Teachers of Mathematics and state standards from across the nation. Practice pages include problems in Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability. Each section features a test for assessment and essential mathematical vocabulary terms for success. Problem solving is embedded throughout. One word problem on each page requires a written response on a separate piece of paper. The activities may be used at any time of the year to assess understanding, for additional practice, or for test preparation. MASTERING THE STANDARDS: ALGEBRA 1 Mastering the Standards AlgebraANSWER KEY Page 3 1. 5n – 8 2. 21 – n 3. 2(n + 27) 4. 56/n 5. n – 6 = 13 6. -12 + n = 32 7. 5n = 45 8. n/9 +3 = 42 9. 72 10. 100 11. n6 12. n2 13. -48 14. -42 15. -6 16. 9 17. -42 18. -72 19. 3.45 x 107 20. 9.56 x 10-4 21. 5.57 x 105 22. 2.68 x 10-2 23. 15 + 10x 24. 42 + 30x 25. -8x + 14 26. 6x – 12 Writing $0.71 per mile Page 4 1. n + 5 2. 15 – n 3. n + 15 4. n – 7 5. n/6 6. 8n 7. x – 4 = 6 8. x + -8 = 17 9. 28/x = 4 10. 7x = 56 11. 3x – 4 = 8 12. 1/4x + 3 = 10 Writing Answers may vary (seven dimes plus eight pennies). Page 5 1. 25 = 32 2. 36 = 729 3. 55 = 3,125 4. -25 = -32 5. 70 = 1 6. n7 7. n2 8. 34x4 = 81x4 9. 4 10. 64 11. 64 12. 1,024 13. 1/16 14. 8 15. 1,000 16. 1/256 Writing 200 x 24 = 3,200 Page 6 1. 22 2. 2 3. 18 4. 3 5. 11 6. -7 7. 47 8. 4 9. 50 10. 27 11. 41 12. 63 13. 32 14. 52 15. 24 16. 27 17. 22 18. 54 Writing 3(0.01) + 4(0.05) + 6(0.10) = $0.83 Page 7 1. real numbers 2. rational, real numbers 3. rational, real numbers 4. whole, integer, rational, real numbers 5. irrational, real numbers 6. integer, rational, real numbers 7. rational, real numbers 8. whole, integer, rational, real numbers 9. integer, rational, real numbers 10. rational, real numbers 11. irrational, real numbers 12. irrational, real numbers 13. answers will vary 14. answers will vary 15. answers will vary 16. answers will vary Writing a. whole numbers b. integers c. rational numbers d. irrational numbers Page 8 1. -9 2. 5 3. -9 4. -21 5. -21 6. 26 7. -16 8. -16 9. -0.3 10. 14.3 11. -6.5 12. -15 13. 2 14. -10 15. 10 16. -2 17. 10 18. -2 19. 2 20. -10 Writing 130 + d = 178, d = 48 Page 9 1. -12 2. -30 3. -3 4. -12 5. -10.3 6. -2.8 7. 36 8. 40 9. -15.6 10. -16.8 11. 18.67 12. 0 13. -44 14. 150 15. -125 16. 125 17. 5 18. -5 19. -112 20. 48 21. 7 22. -3 Writing 48° + (6)-2.2° + (4)1.8° = 42° Page 10 1. 5.6 x 105 2. 6.25 x 106 3. 1.65 x 10111 4. 7.83 x 108 5. 6.23 x 10-5 6. 3.78 x 10-3 7. 2.31 x 109 8. 4.0 x 104 9. 3.05 x 10-3 10. 26,700,000 11. 4,500 12. 5,003,000 13. 0.0045 14. 0.0408 15. 16,700 16. 0.0000751 17. 993 18. 26,800,000,000 Writing $0.50(125) – $41.25 = $21.25 Page 11 1. 2x + 10 2. 10x + 15 3. 21 + 49x 4. 63 + 27x 5. 16x + 24y 6. 4x – 2 7. -3x – 4 8. 2x – 6 9. -4x + 3 10. -12x + 4 11. 5x – 1 12. 6x + 4 13. 6x – 12 14. 14x – 35 15. 14x2 – 6xy Writing $7.50(40) + $7.50(1.5)(48 – 40) = $390.00 Page 12 1. (3, 1) 2. (-4, 3) 3. (0, -4) 4. (1, -2) 5. (3, -3) 6. (-4, -3) 7. (1, 3) 8. (-2, 1) 9. (-1, -2) 10. (3, 4) 11. 12. 13. 14. 15. 16. Writing (2,2) Page 13 1. -5 2. 18 3. sample values 4. sample values 5. 2.5 hours 6. $4.74 7. -8.2 8. 11 9. 7 10. 3 11. 6 12. 8 13. -26 14. 2 15. -5 16. 2 17. 3 18. 3 19. 27 20. 5 21. 13.5 22. 3 Writing 379 girls and 341 boys Page 14 1. domain: {-2, 0, 2, 5} range: {3, 1, 5, -7} 2. domain: {-7, -3, 2, 3} range: {-5, -2, 1, 2} 3. domain: {1, 2, 3, 4} range: {0, -3, 1} 4. -12 5. -9 6. 0 7. 8 8. 13 9. -10, 0, 10 10. 15, -3, 15 11. 0, -2, 4 12. 12, -3, -3 13. 51 14. 60 15. 62 Page 15 1. 2. MASTERING THE STANDARDS: ALGEBRA 2MASTERING THE STANDARDS: ALGEBRA 3 3. 4. 5. 6. Page 16 1. d = rt 2. A = lw 3. G = (t1 + t2+ t3)/3 4. mpg = m/g 5. W = Fd 6. V = lwh 7. 150 miles 8. 78 sq feet 9. 90% 10. 21.5 mpg 11. 3,300 foot-pounds 12. 960 cubic feet Writing Feet would be meters, pounds would be Newtons, but time does not change. The formulas would be the same. Page 17 1. 30 2. -131 3. -181 4. 16 5. -33 6. 57 7. 4 8. -9 9. -7 10. 60 11. 9 12. 12 13. -4 14. -3 2/3 15. 22 16. 4 17. -65 18. -19 Writing No, since 2 divided into 4, all numbers divisible by 4 is divisible by 2. Page 18 1. 6 2. 24 3. 3.5 4. 2 5. 4 6. 2 7. 26 2/3 8. 8 9. -3 10. 0 11. 3 12. 37 13. 8 14. 4 15. -3 16. 4 Writing Yes; answers will vary. Page 19 1. 4 2. 16 3. -21.5 4. 11.5 5. 8 6. -4 7. 9.5 8. 5 9. 0 10. -6 11. 2 2/3 12. -60 13. 1.5 14. -10 Writing $124 as caddy, $260 as clerk Page 20 1. 7 2. 4 3. -1/2 4. -9 5. -6 6. 1 1/7 7. 4/5 8. 1 2/5 9. -14.5 10. 1/4 11. -1/3 12. -20 Writing 74 to 54 Page 21 1. 440 miles 2. $202.50 3. 104°F 4. 21.4 mph 5. $921.30 6. 124 7. 54 sq ft 8. 17,784 in.3 9. 846 cm2 10. -40°C Writing same equations solved for °F or °C. Page 22 1. -8 < x < 8 2. x > 4 or x < -4 3. -5 < x < 1 4. x > 5 or x < -3 5. 0.5 < x < 3.5 6. x > -1 or x < -7/3 7. -1/2 ≤ x ≤ 7/2 8. x ≥ 2 or x ≤ -14 Writing x + 1201 = 67; 47°F or 87°F Page 23 1. t, t, t, f, f 2. t, t, t, f, f 3. f, f, f, f, f 4. f, f, f, f, f 5. f, f, f, t, t 6. t, t, t, t, f 7. 8. 9. 10. 11. -2 < 3 12. 5 > 1 13. -15 < -12 14. 24 > 6 Page 24 1. x < 5 2. x > 7 3. x ≤ 4 4. x ≥ 6 5. x ≤ -4 6. x ≤ 10 7. x > 5 8. x ≥ -3 9. x + 4 ≥ 12.5, x ≥ 8.5 10. x – 5 < 12, x < 17 Page 25 1. x < 5 2. x < -7 3. x < -10 4. x < -4 5. x < 15 6. x > -6 7. x > -9 8. x < -2 9. x > 2 10. x < -7 Writing 6 – 2n ≤ -1/2(n), n ≥ 4 Page 26 1. 12 2. 15 3. 3 4. 10.67 5. 25 6. 4 7. 16 8. 24 9. 18 10. 6 11. 9 12. 21 Writing 6.5 cm Page 27 1. increase: 20% 2. decrease: 20% 3. decrease: 20% 4. increase: 20% 5. increase: 50% 6. decrease: 40% 7. increase: 20% 8. decrease: 20% 9. decrease: 25% 10. decrease: 20% 11. increase: 30% 12. increase: 25% Writing true, true; answers will vary. Page 28 1. 1 2. 2 3. -1 4. -2 5. 2/3 6. -1/2 Writing positive slope, negative slope Page 29 1. 2, 2 2. 3, -3 3. 2, -4 4. -6, -2 5. 5, 2 6. 2, -3 Writing origin Page 30 1. y = x – 2 2. y = -1/2 x + 2 1/2 3. y = 2/5 x + 2 1/5 4. y = -2/5 x + 3 5. y = 1/3 x + 1 2/3 6. y = x – 1 Writing rectangle and m = -1 Page 31 1. 2. 3. 1/1000 = 0.001 4. 1/9 5. 1/25 = 0.04 6. 27 = 128 7. 58 = 390,625 8. 0.023 = 0.000008 9. 81 or 8 10. 38 = 6,561 11. a12 12. 210 = 1,024 13. 34x4 = 81x4 14. 9/16 15. 8/(x3) 16. (45x5)/(y5) = 1,024x5/y5 17. (34x4)/(24y4) = 81x4/16y4 18. 14, 17, 20 19. 11.5, 14, 16.5 20. 32, 64, 128 21. 25, 36, 49 Writing 20 pages Page 32 1. 2. 3. 4. Writing Answers will vary. Page 33 1. 2. 3. 4. 5. 6. Writing sample—The “V” shape becomes narrower than it was. Page 34 1. 1/1,000 = 0.001 2. 1/4 3. 1/54 = 1/625 4. 1 5. 2/3 6. 1/x2 7. 1 8. -1/x3 9. 52/22 = 25/4 = 6.25 10. 1/x4 11. 1/-42 = 1/16 12. 1/x3 13. 3/x2 14. 12/x 15. 1 16. 1 17. 1/x5 18. 1 19. 43(4-2) = 41 20. 2-3/25 = 1/28 = 1/256 Page 35 1. 29 = 512 2. 57 = 18,125 3. 0.023 = 0.0000038 4. 86 = 262,144 MASTERING THE STANDARDS: ALGEBRA 45. 1213 6. 49 = 262,144 7. 125 8. x5 9. x12 10. x11 11. x10 12. x2 13. 1 14. x10 15. 6x12 16. 18x7 17. 10x6 18. 48x4 Writing about 2.56 x 1013 miles Page 36 1. 4.096 2. x20 3. 212 = 4.096 4. 43 x3 = 64x3 5. 36 x8 6. x6 y12 7. 73 x3 y18 = 343x3 y18 8. x9 y12 9. 64 x8 y3 10. 27 x6 y15 11. 1 12. x12 y16 13. -64 x6 y9 14. 25 x10 y4 15. 6 x4 y4 16. x16 17. 4 x11 y5 18. 1/x2 Writing 2x5 is correct; since the expressions are being added, the coefficients are added not the exponents. Page 37 1. 9/25 2. 8/x3 3. 55 x5/y5 = 3,125x5/y5 4. 16x4/81y4 5. 26/27 = 64/27 6. 36/x4 7. x5 8. 81x8/16 9. 16 x4/y12 10. 34/24 = 81/16 11. 122 = 144 12. y2 13. x12 y9 z15 14. x5 y5/25 z5 15. x5 16. 1 17. x4/4y10 18. 9 x 8 = 72 Writing $1098 trillion or $1.098 x 1015 Page 38 1. 12, 14, 16 2. 33, 40, 47 3. 24, 29, 34 4. 12, 14.5, 17 5. 4, 7, 10 6. 5, 2, -1 7. 47, 60, 73 8. 3.9, 4.7, 5.5 9. 22, 32 10. 68, 103 11. 49, 74 12. 24.5, 37 13. 19, 34 14. -10, -25 15. 112, 177 16. 7.9, 11.9 Writing 21, 34, 55 Page 39 1. 32, 64, 128 2. 48, 96, 192 3. 2, 1, 0.5 4. 1/32, 1/64, 1/128 5. 3/4, 3/8, 3/16 6. 10,000, 100,000, 1,000,000 7. 112, -224, 448 8. 1, -1, 1 9. 2, 54, 4,374 10. -3, -24, -384 11. 0.5, 4, 64 12. 64, 8, 0.5 Writing 8.2 cm at the 4th bounce. Page 40 1. 2. 3. 4. Writing Page 41 1. (11, 22) 2. (3, 0) 3. (-4, -2) 4. (2, 2) 5. (1, -9/7) 6. (-4/9, 2/3) 7. 6x2 + 3x + 16 8. 8x2 + 8x – 7 9. 4x2 – 42x + 5 10. -x2 + 5x – 14 11. 5(2x – 3) 12. 4x(x2 – 2x + 4) 13. 3y2(2 + 3y2) 14. 3x(4x2 – x + 3) 15. x2 + 3x + 2 16. x2 + 2x – 15 17. x2 – 7x + 12 18. x2 – 49 19. x2 – 2x + 1 20. x2 + 4x + 4 21. x2 – 6x + 9 22. 4x2 – 20x + 25 23. (x + 4)(x + 2) 24. (x – 3)(x – 4) Writing x2 – 3x = 10, 5 or -2 Page 42 1. (11/7, 22/7) 2. (1, -1) 3. (4, 2) 4. (2, -8) 5. (3, 5) 6. (2, -2) 7. (-4, 3) 8. (2, 1) 9. (-1, 1) 10. (-1/2, 0) Writing 8 dimes, 4 pennies; 84¢ Page 43 1. (2, 5/3) 2. (3, -5) 3. (-7, 2) 4. (2, 2) 5. (1, -1) 6. (-1 1/3, 2) 7. (6, -4) 8. (-1, 0) 9. (-1, -2) 10. (-3, 1) Writing The width is 4 inches; the length is 8 inches. Page 44 1. (0, -2) 2. (3, -5) 3. (1, -4) 4. (1.4, -0.2) Writing 5 dimes and 6 quarters Page 45 1. 6x2 + 2x + 13 2. 5x2 + 12x – 7 3. 4x2 – 33x – 2 4. -2x2 + 4x – 15 5. 1.2x2 + 9.6x + 2.3 6. x2 – 8 7. 3x2 – 8x – 32 8. -5x2 + 4x – 4 9. -2x2 – 6x + 4 10. 5x2 + 9x – 11 Writing 50 cm, 9 cm, and 9 cm Page 46 1. 6x + 8y 2. 14x2 – 35y2 3. x2 + 2xy – 5xz 4. 12x – 9x2y 5. 42x2y – 72xy3 6. -30xy + 20xy2 + 40y 7. 7(3x – 4) 8. 5x(x2 – 2x + 3) 9. 3y2(1 + 2y2) 10. 2x(3x2 – 2x + 4) 11. 4x(2x2 + 3x + 1) 12. y(y2 – 2y + 8) 13. 12x2(x – 3) 14. 3x3y(2x2 – 4xy – 3y2) 15. 9x2y(2x3 + 5x2y – 8y3) 16. y3z(2z – 1) 17. 4x2(3 + 2y) 18. 2xy2(x – 5y + 12y2) Writing 18 inches and 8 shelves Page 47 1. x2 + 8x + 15 2. x2 + 5x – 24 3. x2 – 6x + 8 4. x2 – 9 5. x2 + 2x – 3 6. 2x2 + 12x + 16 7. 6x2 + 7x + 2 8. x2 + 10x + 25 9. x2 – 12x + 36 10. 4x2 + 12x + 9 11. 9x2 + 9x + 2 12. 8x2 – 18x + 9 13. 2x2 – 10x – 100 14. 15x2 – x – 6 15. 2x2 + 13x + 6 16. 15 + 8x + x2 Writing x3 + 3x2 + 3x + 1 Page 48 1. x2 – 6x + 9 2. x2 + 18x + 81 3. x2 – 4x + 4 4. 4x2 – 4x + 1 5. 25x2 + 60x + 36 6. 9x2 – 24x + 16 7. x2 + 2x + 1 8. 9x2 + 12x + 4 9. x2 – 4 10. x2 – 9 11. x2 – 25 12. 4x2 – 9 13. 16x2 – 25 14. 49x2 – 4 15. 81x2 – 64 16. 100x2 – 400 Writing 64 ft by 44 ft Page 49 1. (x + 1)(x + 1) 2. (x – 1)(x – 1) 3. (x + 3)(x – 1) 4. (x – 3)(x + 1) 5. (x + 2)(x + 2) 6. (x – 4)(x – 3) 7. (x – 6)(x + 4) 8. (x + 4)(x + 3) 9. (x + 3)(x + 3) 10. (x – 3)(x – 5) 11. (x + 2)(x + 8) 12. (x – 8)(x + 1) 13. (x + 9)(x – 4) 14. (x + 8)(x – 4) 15. (x – 24)(x – 2) 16. (x + 24)(x – 2) 17. (x – 5)(x – 1) 18. (x + 5)(x – 3) Writing (2x + 1), 31 ft Page 50 1. (x + 5)(x – 5) 2. (x + 7)(x – 7) 3. (x + 12)(x – 12) 4. (2x + 3)(2x – 3) 5. (2x + 9)(2x – 9) 6. (3x + 5)(3x – 5) 7. (x + 1)(x – 1) 8. (xy + 5)(xy – 5) 9. (x + 2)2 10. (x – 1)2 11. (x – 3)2 12. (x + 12)2 13. (x – 7)2 14. (x + 9)2 15. (2x + 1)2 16. (2x – 5)2 17. (4x + 3)2 18. (3x – 4)2 Writing 48 sq inches Page 51 1. (x + 3)2 2. 2x(2x + 2) 3. 9(x – 3)(x – 1) 4. (x + 9)(x – 2) 5. (x – 6)(x – 1) 6. (x + 6)(x + 3) 7. (x + 5)(x – 4) 8. (x + 30)(x + 14) 9. (x + 1)(x – 8) 10. (x + 10)(x – 9) 11. (x + 10)2 12. (x – 12)(x – 13) MASTERING THE STANDARDS: ALGEBRA 513. (x + 24)(x + 10) 14. (x – 11)2 15. (x – 3)(x + 7) 16. (x + 17)(x + 4) 17. 4, -10 18. 0, 5 19. 7, -1 20. 3, -6 Page 52 1. 2. 3. 4. 1. (0, 1) 2. (0, -3) 3. (-3, -7); x = -3 4. (-1, -3); x = -1 5. 10√ 3 6. 5√ 3 7. 3√ 2 8. 5√ 2 9. -2, -5 10. 6, -1 11. -3, -1 12. 7, -2 13. 20 14. 4.9 15. 5 16. 13 17. 5; (3.5,4) 18. 10; (1,0) Page 53 1. (0, 0) 2. (0, 2) 3. (0, -4) 4. (0, 3) Writing No, same shape translated y units up. Page 54 1. (-3, -11) 2. (-1, -1) x = -3 x = -1 3. (-1, 4) 4. (-1, 0) x = -1 x = -1 Writing (3, -5) Page 55 1. 4, 5, 20, 4/5 2. 10, 9, 90, 10 9 3. 12, 4, 48, 3 4. 16, 8, 128, 2 5. 9, 21, 189, 3/7 6. 7. 8. 9. 10. 11. 12. 13. Writing 2.5 sec Page 56 1. -2, -5 2. -1, 6 3. -1, -3 4. -2, 7 5. -2, -6 6. -1, 7 7. 2, -5 8. 1, -15 9. -6, 1 10. -3 11. -2, 4/3 12. -3, -12 Writing 6.7 sec Page 57 1. 8, 2 2. 11.7, 0.3 3. 8, 4 4. 0.6, -8.6 5. 2.5, -2 6. -2, 1.3 7. -1, 0.7 8. -15, 14 9. -0.5, 0.7 10. 1 11. -3, -4 12. 3, -3.5 Writing 0, 1,280 feet Page 58 1. 2/5 2. 3. /3 4. 6/5 5. 6. 9 7. 8. /7 9. /5 10. 11. /5 12. 13. 14. 6 15. 45 16. 3/2 Writing 100 Page 59 1. 13 2. 4.9 3. 13.9 4. 9.4 5. 8 6. 7.9 7. 15.0 8. 11.3 9. 21.9 10. 12 11. 11.2 12. 26.2 Writing The diagonals are hypotenuses of two right triangles that are congruent. Page 60 1. 4.5, (2, 3) 2. 11.3, (0, 0) 3. 5, (10.5, 12) 4. 8.1, (1, 1.5) 5. 10, (2, 4) 6. 7.1, (0.5, -1.5) 7. 2.8, (1, 5) 8. 6.3, (-4, 3) Writing 16 units MASTERING THE STANDARDS MATHEMATICS 6 10√—25√—3 2√—5√—34√—54√—5 2√——13 √——21 7 5√—2 10√—26√—22√—35√—53√—39√—2 6√—54√—7 2√——15Letter To Teacher and Parents Mastery benchmarks for Algebra in the middle grades include all topics of mathematics. These main areas are found in the National Council of Teachers Mathematic (NCTM) Standards and in state standards. These standards are Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability. The study of integers is included since middle school students often struggle with the concept. To help students master the Standard of Mathematics, there are several skills that students need to have. 1. Students need a working knowledge of computations. To improve in this area, use flash cards, drill games, and competitions for speed and accuracy. 2. Students need to be able to estimate and use number sense. When looking at problems, they need to determine if the answers make sense. The answer must be close to their estimate. Students can practice this on every problem they do. 3. Students need to practice and review previously learned concepts and principles, so that the skill becomes a part of their thinking. This gives them the ability to see past the basic skills and use new and creative ways of solving problems. 4. Students need to be familiar with the test format. Discuss the length of the test with them. Explain how to eliminate distracters that are not reasonable in multiple-choice questions. Point out the use of boldfaced words. In most tests, the writers will use bold words such as not, least, and most so as not to mislead students. 5. Students need to master mathematics test vocabulary. This vocabulary can be practiced in every lesson. Create vocabulary lists as spelling words and reinforce them on daily work to increase students’ familiarity and knowledge in this area. The use of calculators on state tests varies from state to state. If calculators are allowed, students need to practice with the type and model that is allowed on that test. Calculators are wonderful tools, but many students use them as an answer machine and lose sight of the above points. Calculators give many more wrong answers than correct answers. Students need to learn the limits of the calculator that they use. There are many strategies to help students maximize their efforts on standardized tests. These include: • Students need to be positive in their own minds and hear others being positive about them. • Students need the proper conditions for learning, including regular mental conditioning such as weekly reading and game play. • Students need a balanced, active life that includes daily physical exercise. • Students need a study schedule with review and practice skills to prevent them from cramming. • Students need to know and practice stress-relieving techniques such as controlled breathing and positive imaging. No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 1No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 2 Teacher Overview—Number and Operations NCTM Standard for Number and Operations The goal for students is to understand numbers, ways of representing numbers, relationships among numbers, number systems, meanings of operations, and how operations relate to one another. Mathematics is different from other subject areas. There are three characteristics that make math different. First, mathematics is a language to be learned and practiced. Second, mathematics is sequential; that is, there is an order of one concept building on another. Finally, mathematics is developed around skills such as addition, building to multiplication. The language of mathematics is concise but has multiple representations of numbers. Students need to build on their knowledge of fractions, ratios, and decimals to expand understanding of rational numbers as a foundation of real numbers to develop this area into algebra. Ratio, proportion, and percent are expanded upon and need to be mastered by the end of the middle school years. To help students develop number and operations sense, they need to use mathematical language for basic operations. The words sum, difference, product, and quotient are not used in normal conversation. Have students make up their own sentences using these words to practice and develop long-term memory for these terms. In discovering the pattern of exponents, students try to multiply the power and the base as if the exponent was a factor. Students should make tables of numbers to illustrate exponents increasing. By doing this for several different bases, they can see patterns develop. Students will need to practice this pattern several times until exponents become obviously repetitive multiplication. A student will need problem-solving strategies in order to achieve on most standards tests. Problem solving means (1) getting the solution, (2) knowing how to get the solution, and (3) showing the steps in the process. The third step represents a difficult aspect for students. They tend to think that writing is not part of mathematics. Having students explain what they are thinking can help this process. Practicing several of the problem-solving strategies will be beneficial for students. There are special strategies for getting the solution when the answer is not obvious: • Use a pattern to solve a problem. • Identify missing information that prevents a solution. • Match a problem to a number sentence. • Use objects to act out a solution. • Use guess and check to solve a problem. • Solve a simpler problem first, then solve the original problem. • Make a table to solve the problem. • Make an organized list to solve a problem. • Make a picture, map or diagram to solve a problem. • Use logical reasoning to solve a problem. • Solve a multistep problem. Some authors identify other problem-solving strategies such as estimate a reasonable answer and make an equation. When solving a problem, some people use one strategy while others will use another strategy. There is not one single correct strategy that is perfect for everyone. Be flexible and look for creativity. Remember the question is: Can you get a reasonable solution, and do you know how you were able to arrive at that solution? .31410,236.75 4,082 xy 32x=y+7n –7+4 –3 + ( 1 2 ) = –3 6 (–8 )No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 3 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Practice Assessment—Number and Operations Write the following expressions using n as the number. 1. The product of 5 and a number decreased by 8 ________ 2. The difference between 21 and a number ________ 3. Twice the total of a number and 27 ________ 4. The quotient of 56 and a number ________ Write the following equations using n as the number. 5. Six less than a number is 13. ________ 6. The sum of –12 and a number is 32. ________ 7. Five times a number is 45. ________ 8. A number divided by 9 plus 3 is 42. ________ Simplify the following expressions. 9. 23 × 32 ____ 10. 52 × (–2)2 ____ 11. n3 × n3 ____ 12. n-3n5 ____ 13. 12 × (–4) ____ 14. –7 × 6 ____ 15. –30 ÷ 5 ____ 16. –54 ÷ –6 ____ 17. (7 × –2) × 3 ____ 18. (–4 × 2) × 9 ____ Write each number in scientific notation. 19. 34,500,000 ____ 20. 0.000956 ____ 21. 557,000 ____ 22. 0.0268 ____ Simplify each expression by using the distributive property. 23. 5(3 + 2x) ____ 24. 6(7 + 5x) ____ 25. –2(4x – 7) ____ 26. 2(3x – 6) ____ You drove 336 miles on 12 gallons of gas. The cost of gas was $2.00 per gallon. How much did it cost to drive per mile?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 4 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Using Variables A variable is a letter or symbol that stands for a number. This number may change or may be a single number. When one of more variables are placed together, like a × b, this means a times b. A number with variables is referred to as a coefficient. You will find that variables will be used as a translation of sentences into algebraic expressions. The basic phrases are: + (addition) The sum of 7 and 3 is 10. as in 7 + 3 = 10 – (subtraction) The difference of 7 and 3 is 4. as in 7 – 3 = 4 × or • (multiplication) The product of 7 and 3 is 21. as in 7 × 3 = 21 ) or /(division) The quotient of 12 and 3 is 4. as in 3)12 or 12 = 4 4 3 The quotient of 3 divided into 12 is 4. as in 3)12 The quotient of 12 divided by 3 is 4. a s i n 12 = 4 3 Example: Write an expression for the following: A. the sum of a number and 12 B. the product of 5 and a number x + 12 5 × x Write the following expressions using n as the number. 1. a sum of a number and 5 ____ 2. the difference between 15 and a number ____ 3. a number increased by 15 ____ 4. seven less than a number ____ 5. the quotient of a number and 6 ____ 6. eight times a number ____ Example: Write equations and solve for the variable using n as the variable. A. Three more than a number is 10. B. The product of 7 and a number is 21. n + 3 = 10 7 × n = 21 n + 3 – 3 = 10 – 3 7 × n ÷ 7 = 21 ÷ 7 n = 7 n = 3 Write the following expressions using n as the number. 7. Four less than a number is 6. ____ 8. The sum of -8 and a number is 17. ____ 9. The quotient of 28 and a number is 4. ____ 10. Seven times a number is 56. ____ 11. The product of 3 and a number less 4 is 8. ____ 12. A quarter of a number plus 3 is 10. ____ Write an expression using pennies and dimes that shows 78 cents.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 5 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Exponents When working with variables, you will find that you need to understand exponents. Exponents are mathematical expressions for any real number b and any positive integer n, where b is the base and n is the exponent. bn = b × b × b × ... × b 34 = 3 × 3 × 3 × 3 = 81. 3 is used as a factor four times. A zero exponent: b0 = 1, if b is any nonzero real number. 50 = 1 Negative exponent: b-n = 1– bn 5-3 = 1– 53= 1– 125 Properties of Exponents Examples: Product Rule: bn × bm = bn+m 32 × 34 = 36 Quotient Rule: bn ÷ bm = b(n – m) or 57 ÷ 53 = 57 – 3 = 54 Power Rule: (ab)n = an × bn (3x)4 = 34x4 = 81x4 Simplify the following expressions. 1. 23 × 22= ______ 2. 32 × 34= ______ 3. 52 × 53= ______ 4. (–2)3 × (–2)2= ______ 5. 73 × 7-3= ______ 6. n2 × n5= ______ 7. n-4 × n6= ______ 8. (3x)4= ______ Evaluate each expression when x = -2 and y = 4. 9. x2= ______ 10. (xy)2= ______ 11. x4 × x2= ______ 12. y3 × y2= ______ 13. yx= ______ 14. (x + y)3= ______ 15. (-5x)3= ______ 16. (-4y)x= ______ A certain type of bacteria doubles in number every hour. At 8:00 a.m., there were 200 bacteria. How many bacteria will be present at noon?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 6 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Order of Operations In problems with more than one operation involved, you will need to determine which order of operations will be performed. The rule for order of operations has been established. Order of Operations 1. Perform operations inside parentheses or grouping symbols. 2. Simplify terms with exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. Example: Simplify (9 + 7) – 3(4 × 5 – 3 × 6) 2 (9 + 7) – 3(4 × 5 – 3 × 6) Work inside the parentheses first. 2 = 16 – 3(4 × 5 – 3 × 6) Multiply and divide within parentheses. 2 = 16 – 3(20 – 18) Complete subtraction within parentheses. 2 = 16 – 3 × 2 Divide and multiply, left to right. 2 = 8 – 6 Subtract. = 2 Simplify each expression. 1. 6 + 2 × 8 = _______________ 2. 18 /2 – 7 = _______________ 3. 15 + 9 /3 = _______________ 4. 5 – 32 /16 = _______________ 5. 7 + 3 × 5 – 11 = _______________ 6. 14 – 3(15 – 8) = _______________ 7. 8 × 4 + 3 × 5 = _______________ 8. 24 /4 – 18 /9 = _______________ 9. 4(4 + 5) + 7(7 – 5) = _______________ 10. (3 + 5)5 – 20 + (2 + 5) = _______________ 11. 3 × 9 – 2 × 8 + 5 × 6 = _______________ 12. 3(1 + 6 + 2(5 + 2)) = _______________ 13. 12 × 4 – 24 = _______________ 14. (22 + 32) × (1 + 1)2 = _______________ 15. 2(3(9 – 5)) = _______________ 16. 2 + 5 × 8 – 15 = _______________ 17. 2 × 5 + 15 – 3 = _______________ 18. 62 + 6 + 12 = _______________ Write an expression and simplify: You have 3 pennies, 4 nickels, and 6 dimes. How much money do you have?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 7 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Real Numbers You should review the properties of the classification of real numbers. Example: Look at the numbers –3.8, , 0, –π, –√–3, 57, –14. Natural: 57 numbers used to count Whole: 0, 57 natural numbers and zero Integers: 0, 57, –14 whole numbers and their opposites Rational: 0, 57, –14, –3.8, integers and terminating and nonrepeating decimals Irrational: –π, –√–3 infinite, nonrepeating decimals Real: 0, 57, –14, –3.8, –π, –√–3, rational and irrational numbers Name the set(s) of numbers to which each number belongs. 1. – 7– 9 _______________ 2. 41.25 _______________ 3. 51– 2 _______________ 4. 0 _______________ 5. √–7 _______________ 6. –40 _______________ 7. 1–2– 5 _______________ 8. √100 _______________ 9. √–4 _______________ 10. 6.25 _______________ 11. 2π _______________ 12. 0.121221222... _______________ Give an example of each kind of number and where you would use it. 13. A whole number _______________________________________________________________ 14. A negative integer _____________________________________________________________ 15. A rational number _____________________________________________________________ 16. A positive fractional number _____________________________________________________ The different number systems were invented to solve problems that could not be solved in their current number system. Identify a number system that would be needed to solve the equations below. a. x + 3 = 7 b. x + 12 = 5 c. 8x = 4 d. x2 = 15 ______________ ______________ ______________ ______________ 1–3– 19 1–3– 19 1–3– 19No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 8 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Adding and Subtracting Integers Review the rules for adding integers. • To add two numbers with the same sign, add their absolute values. The sum has the same sign as the numbers. • To add two numbers with opposite signs, subtract their absolute values. The sum has the same sign as the number with the greater absolute value. Review the rules for subtracting integers. • To subtract numbers, rewrite the problem to add the opposite of the numbers. • Follow the rules for adding integers. Example A: –9 + 4 9 – 4 Find the difference of the their absolute values. = 5 Difference –5 Since –9 has the greater absolute value, the answer takes the negative sign. Example B: 12 – 21 12 + (–21) Rewrite the problem to add the opposite of (+21). = 21 – 12 The difference of their absolute values = 9 Subtract –9 Since (–21) has the greater absolute value, the answer takes the negative sign. Simplify each expression. Check the sign of your answer. 1. –5 + (–4) ______________ 2. 12 + –7 ______________ 3. –15 + 6 ______________ 4. –15 + (–6) ______________ 5. –4 + (–17) ______________ 6. 17 – (–9) ______________ 7. –23 – –7 ______________ 8. –7 – 9 ______________ 9. –5.1 – (–4.8) ______________ 10. 5.6 – (–8.7) _____________ 11. 12.8 – 19.3 ______________ 12. –6.7 – 8.3 ______________ Evaluate each expression for a = 6 and b = -4. Check the sign of your answer. 13. a + b ______________ 14. –a + b ______________ 15. a + (–b) ______________ 16. (–a) + (–b) ______________ 17. a – b ______________ 18. –a – b ______________ 19. a – (–b) ______________ 20. –a – (–b) ______________ The veterinarian holds a dog. Without the dog, the veterinarian weighs 130 lb. With the dog, she weighs 178 lb. Let d represent the weight of the dog. Write an expression showing this situation, and find the weight of the dog.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 9 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Multiplying and Dividing Integers You should review the rules for multiplying and dividing integers. • The product or quotient of two numbers with the same signs is always positive. • The product or quotient of two numbers with different signs is always negative. Simplify each expression. Check the sign of your answer. 1. –3 × 4 ______ 2. –5 × 6 ______ 3. –36 ÷ 12 ______ 4. –84 ÷ –7 ______ 5. –82.4 ÷ 8 ______ 6. 11.2 ÷ –4 ______ 7. 18 ÷ .5 ______ 8. 10 ÷ 1/4 ______ 9. –3 × 5.2 ______ 10. (–0.6) × 28 ______ 11. –11.2 ÷ (–0.6) ______ 12. 0 ÷ –98 ______ 13. –4 × (17 + –6) ______ 14. –6 × (–12 + –13) ______ Evaluate each expression for a = -25 and b = 5. Check the sign of your answer. 15. a × b ______ 16. –a × b ______ 17. a ÷ (–b) ______ 18. –a ÷ (–b) ______ Evaluate each expression for t = -4. Check the sign of your answer. 19. 28 × t ______ 20. –12 × t ______ 21. 28 ÷ (–t) ______ 22. –12 ÷ (–t) ______ At the beginning of an experiment, the temperature of a solution was 48°C. During a 6-hour period, the temperature dropped 2.2 degrees each hour, then increased 1.8 degrees each hour for the next 4 hours. Write an expression to show the temperature change during this time period. What is the final temperature?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 10 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Scientific Notation To write a number in scientific notation, use the following steps: • Move the decimal to the right of the first integer that is not a zero. • If the original number is greater than 1, multiply by 10n, where n represents the number of places the decimal was moved to the left. • If the original number is less than 1, multiply by 10-n, where n represents the number of places the decimal was moved to the right. Examples: Write each number in scientific notation. A. 8,750,000 standard form Move the decimal to the left six places. 8.75 x 106 Drop all insignificant 0’s. Multiply by the power of 10. B. 0.000078 standard form Move the decimal to the right five places. 7.8 x 10-5 Drop all insignificant 0’s. Multiply by the power of 10. Write each number in scientific notation. 1. 560,000 ______ 2. 6,250,000 ______ 3. 165 billion ______ 4. 783 million ______ 5. 0.0000623 ______ 6. 0.00378 ______ 7. 2,310,000,000 ______ 8. 40,000 ______ 9. 0.00305 ______ Write each number in standard notation. 10. 2.67 x 107 ______ 11. 4.5 x 103 ______ 12. 5.003 x 106 ______ 13. 4.5 x 10-3 ______ 14. 4.08 x 10-2 ______ 15. 1.67 x 104 ______ 16. 7.051 x 10-5 ______ 17. 9.93 x 102 ______ 18. 2.68 x 1010 ______ Carol bought 125 pens for $41.25. She sold each pen for $0.50. Write an expression to show how much profit she made. How much profit did she make on the total sale of the pens?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 11 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Distributive Property You can think of the Distributive Property as passing out pencils to the class. Everyone in the classroom must receive a pencil. Likewise, you distribute the number in front of the parentheses to each term inside the parentheses by multiplying. Examples: Simplify. A. 4(2x + 5) 4(2x) + 4(5) Distribute 4 to 2x and 5 by multiplying. 8x + 20 Simplify. B. –5(4x + 3) –5(4x) + (–5)(3) Distribute –5 to 4x and 3 by multiplying. –20x + (–15) Simplify. –20x – 15 Subtraction expression. Simplify each expression by using the Distributive Property. 1. 2(x + 5) ______ 2. 5(2x + 3) ______ 3. 7(3 + 7x) ______ 4. 9(7 + 3x) ______ 5. 8(2x + 3y) ______ 6. 1/4 (16x – 8) ______ 7. –1(3x + 4) ______ 8. 0.2(10x – 30) ______ 9. –(4x – 3) ______ 10. –(12x – 4) ______ 11. –(–5x + 1) ______ 12. –(–6x – 4) ______ 13. (3x – 6)2 ______ 14. (2x – 5)7 ______ 15. 2x(7x – 3y) ______ You have a job that pays you $7.50 an hour for the first 40 hours and 1.5 times for all hours over 40. Write a formula that shows the total weekly pay. You worked 48 hours last week. How much should you be paid? No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 12 35.314 10,236.75 4,082 316 8926 43 10,236.75 Name Date Graphing Data on Coordinate Planes The coordinates of a point give it location on a coordinate plane. To locate a point on the plane, you need to give an order pair (x, y). Starting at the origin, (0, 0), you move x units to the right or left along the x-axis and then y units up or down parallel to the y-axis. Example: Given the coordinates of points A, B, C, and D: Point A is 2 units right of origin and 2 units up. The coordinates of A are (2, 2). Point B is 2 units left of origin and 3 units up. The coordinates of B are (–2, 3). Point C is 3 units left and 0 units up. The coordinates of C are (–3, 0). Point D is 3 units right and 2 units down. The coordinates of D are (3, –2). Name the coordinates of each point on the coordinate plane. 1. G ______ 2. H ______ 3. J ______ 4. K ______ 5. L ______ 6. M ______ 7. N ______ 8. R ______ 9. P ______ 10. Q ______ 11. S (3, 4) ______ 12. T (0, 5) ______ 13. U (–3, –4) ______ 14. V (–4, 5) ______ 15. W (–4, –5) ______ 16. Z (3, –3) ______ Graph the points on the same coordinate plane. Which of these points does not belong to the same straight line: (–4, –5), (0, –3), (2, 2), or (8, 1)?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 13 a = πr2 π mx + b = y Name Date Practice Assessment—Linear Relations and Functions Evaluate each function rule for x = –4. 1. f(x) = 2x + 3 ______ 2. g(x) = –5x – 2 ______ Use a table to graph each function. Choose at least four values for x. 3. y = x – 3 4. y = –2x + 5 Write an equation for each and solve. 5. You rode 150 miles at 60 miles per hour. How long did it take? ______ 6. A tennis racket is on sale for $79. With sales tax, a racket costs $83.74. How much is the sales tax? ______ Solve each equation. 7. x + 14.5 = 6.3 ______ 8. 4(x – 3) = 32 ______ 9. 5x – 3x + 1 = 15 ______ 10. –6x + 7 = –11 ______ 11. 3(x – 1) = 15 ______ 12. 6x – 12 = 36 ______ 13. –(2x + 4) = 48 ______ 14. 3– 4 x – 2– 3 = 5– 6 ______ 15. –4x – 6 = 19 + x ______ 16. 10x – 19 = 9 – 4x ______ 17. 3x + 2 = 14 – x ______ 18. 3 – 2x = 3x – 12 ______ Solve each proportion. 19. 3– 5 = x– 45 ______ 20. = ______ 21. = 9– x ______ 22. (x – 1 ) = 4 5 10 ______ 1–0– 7x 1–2– 42 1–2– 18 There are 720 students at your school. There are 38 fewer boys than girls. How many boys and girls are at your school?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 14 a = πr2 π mx + b = y Name Date Relations and Functions A function is a relation that assigns exactly one value in the range to each value in the domain. The domain includes all of the x values in the function. A function rule may be given as an equation. The function f(x) = 2x + 3 will take any value of x and change it into 2x + 3. You read this as “f of x equals two x plus three.” When x = 5, the function of 5 would be 2 • 5 + 3 or 13, or f(5) = 2 • 5 + 3. Evaluating a function means finding a value in the range for a given value from the domain. Example: Find the domain and range of this relation. {(–4,3), (–2,–5), (0,0), (1,4)} Domain: –4, –2, 0, 1 Range: 3, –5, 0, 4 Example: Evaluate f(x) = 2x + 3 for x = 0, 1, and 2 f(0) = 2(0) + 3 = 3 f(1) = 2(1) + 3 = 5 f(2) = 2(2) + 3 = 7 Find the domain and range of each relation. 1. {(–2,3), (0,1), (2,5), (5,–7)} __________________ __________________ 2. {(–7,–5), (–3,–2), (2,1), (3,2)} __________________ __________________ 3. {(1,0), (2,–3), (3,1), (4,–3)} __________________ __________________ Evaluate each function rule for x = –4 4. f(x) = 3x ______ 5. f(x) = x – 5 ______ 6. f(x) = 2x + 8 ______ 7. g(x) = –3x – 4 ______ 8. g(x) = x2 +2x + 5 ______ Find the range of each function with the given domain. 9. f(x) = 5x; {–2, 0, 2} f(–2) = ____ f(0) = ____ f(2) = ____ 10. g(x) = 2x2 – 3; {–3, 0, 3} g(–3) = ____ g(0) = ____ g(3) = ____ 11. h(x) = x2 + x – 2; {–2, 0, 2} h(–2) = ____ h(0) = ____ h(2) = ____ 12. f(x) = 3x2 – 15; {–3, –2, 2} f(–3) = ____ f(–2) = ____ f(2) = ____ Biologists have determined that the number of chirps made by a cricket in one minute is a function of the temperature (t) measured in degrees Fahrenheit. This relationship is modeled by the function c(t) = (1/4)t + 37. Determine the number of chirps for the given temperature. 13. 56°F ______ 14. 92°F ______ 15. 100°F ______No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 15 a = πr2 π mx + b = y Name Date Function Rules, Tables, and Graphs You may find that function rules will be easier to see if you model them with a table and a graph. Example: Graph the function y = 2x + 3 First: Pick at least four values for x. Choose some negative values for x. Place these values into a table. Second: Evaluate the function to find the value of y for each value of x. Third: Plot the ordered pairs to graph the data. Use a table to graph each function. Choose at least four values for x. Some x values need to be negative. 1. y = 3x + 1 2. y = –x + 2 3. y = x – 3 4. y = –2x + 5 5. y = x2 – 3 6. y = 2x2 – 3x – 5 Biologists have determined that the number of chirps made by a cricket in one minute is a function of the temperature (t) measured in degrees Fahrenheit. This relationship is modeled by the function c(t) = (1/4)t + 37. Graph this behavior for four temperatures between 80° and 100°.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 16 a = πr2 π mx + b = y Name Date Writing Function Rules You can write functions (which are equation rules from situations) by analyzing the sentence phrase by phrase. Example: Write a function for the following situation. The width of a sheet of plywood is one half the length. Let l represent the length and w represent the width. Then the function of width is 1/2l and w = 1/2l Write a function for each of the following situations. 1. The distance a car travels is rate times time. _______________________________ 2. The area of a rectangle is length times width. ______________________________ 3. Your grade is the sum of the percentages on three tests divided by three. ________________ 4. Gas mileage is miles driven divided by the number of gallons. _______________________ 5. Scientific work is determined by the product of force and distance. ______________________ 6. The volume of a rectangular prism is the length times the width times the height. __________ Use the functions above to determine the following of each. 7. What is the distance traveled by a car when its rate is 60 miles per hour and it is driven for 2 1/2 hours? ______________________________ 8. What is the area of a rectangle when the length is 12 feet and the width is 6.5 feet? ________ 9. What is your percentage grade when three tests were 95%, 92%, and 83%? ______________ 10. What is the gas mileage if you drive 258 miles on 12 gallons of gas? ____________________ 11. How much work was accomplished if a horse pulled 550 pounds for 6 feet? _______________ 12. What is the volume of a rectangular prism that is 10 feet by 12 feet by 8 feet? _____________ In the problems above, what would be the units of measures in the metric system? Would that change the function? No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 17 a = πr2 π mx + b = y Name Date Solving One-step Equations To solve an equation, you need to isolate the variable having a coefficient of 1 on one side of the equation. By using the inverse operation, you are able to next remove terms and the coefficient to the other side of the equation. For any number a, b, and c • Addition Property of Equality: If a = b, then a + c = b + c • Subtraction Property of Equality: If a = b, then a – c = b – c • Multiplication Property of Equality: If a = b, then a × c = b × c, with c ≠ 0. • Division Property of Equality: If a = b, then a/c = b/c, with c ≠ 0. Example: Solve each equation. A. x + 12 = 7 x + 12 – 12 = 7 – 12 Subtract 12 from both sides. x = –5 Simplify. B. 5x = 35 5x = 35 Divide both sides by 5. 5 5 x = 7 Simplify. Solve each equation. 1. x + 15 = 45 ______ 2. y + 42 = –89 ______ 3. 125 + x = –56 ______ 4. m – 31 = –15 ______ 5. –28 = k + 5 ______ 6. x – 39 = 18 ______ 7. 13x = 52 ______ 8. –7y = 63 ______ 9. 6m = –42 ______ 10. t– 3 = 20 ______ 11. 1 = x– 9 ______ 12. 3– 4 y = 9 ______ 13. –31– 2 x = 14 ______ 14. a + 2– 3 = –3 ______ 15. 11 = b– 2 ______ 16. –51– 2m = –22 ______ 17. 13 = – x– 5 ______ 18. –25 = y – 6 ______ Is it possible to find a number that is divisible by 4 and not divisible by 2? If so, give an example.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 18 a = πr2 π mx + b = y Name Date Solving Two-step Equations You use order of operations to determine if you do multiplication before addition. When solving two-step equations, you must reverse the order of operations and perform the inverse operation. The goal is to isolate the variable to one side of the equation. Example: Solve the equation. 3x + 5 = 11 Reverse the order of operation of adding 5. 3x + 5 – 5 = 11 – 5 Subtract 5. 3x = 6 Simplify. 3x = 6 Divide by 3. 3 3 x = 2 Simplify. Solve each equation. 1. 3x – 5 = 13 ______ 2. x– 4 + 3 = 9 ______ 3. 4x + 7 = 21 ______ 4. 3x + 4 = 10 ______ 5. 5x – 4 = 16 ______ 6. 7x + 16 = 30 ______ 7. 3– 4x – 4 = 16 ______ 8. 3x – 8 = 16 ______ 9. 4x + 9 = –3 ______ 10. 4(x + 3) = 12 ______ 11. 2– 3x + 4 = 6 ______ 12. (x + –5 ) = 8 ______ 4 13. 1.5(x – 3) = 7.5 ______ 14. 3– 4x + 15 = 18 ______ 15. 1.2x – 0.6 = –4.2 ______ 16. 15x + 12 = 72 ______ Your friend says that multiplying by the reciprocal of a number is the same as dividing by the number. Is this correct? Explain.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 19 a = πr2 π mx + b = y Name Date Solving Multistep Equations You will find that as problems become more complex, the procedure is to simplify the expressions on each side of the equation. Then isolate the variable to one side and the number to the other. Parentheses are removed with the distributive property. Example: Solve the equation. 3(x + 5) – 8 = 22 3x + 15 – 8 = 22 Distribute to remove parentheses. 3x + 7 = 22 Combine like terms. 3x + 7 – 7 = 22 – 7 Subtract 7 from both sides. 3x = 15 Simplify. 3x = 15 Divide both sides by 3. 3 3 x = 5 Simplify. Solve each equation. 1. 6x + 4 = 3x + 16 2. 12x – 8 – 8x = 56 3. 4(x – 2) – 6x = 35 4. 7x – 3(x + 7) = –67 5. (4x – 3) – (7x + 8) = –35 6. –x + 12 – 3x = 28 7. 15 = 2x – 4 8. 7 = 5x – 18 9. 3(4 – 6x) = 12 10. 2x + 5 – 6x = 29 11. 4(2 – 6x) = –56 12. 3(x + 7) – (x – 15) = –84 13. 18 = 3(10 – x) – 5x 14. 1– 2(4x – 6) – 3x = 7 You worked at two part-time jobs during the summer. As a clerk you earned twelve dollars more than twice as much as you did as a golf caddy. Your total earnings for the summer were $384. How much did you earn at each job? No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 20 a = πr2 π mx + b = y Name Date Solving Equations with Variables on Both Sides To solve equations with variables on both sides, rewrite the equation until all terms and constants are combined on opposite sides. As you rewrite the equation, use inverse operations and the equality properties. Remember to combine like terms on one side first. When you move terms to opposites sides, you perform the same operation on both sides. Example: Solve the equation. Solve 5x – 8 = 3x + 7 5x – 8 – 3x = 3x + 7 – 3x Remove 3x from the right side by subtracting 3x. 2x – 8 = 7 Combine like terms. 2x – 8 + 8 = 7 + 8 Remove -8 from the left side by adding 8. 2x = 15 Combine like terms. 2x = 15 Divide by 2. 2 2 x = 7.5 Simplify. Check 5(7.5) – 8 <=?> 3(7.5) + 7 37.5 – 8 <=?> 22.5 + 7 29.5 = 29.5 Solve and check each equation. 1. 3x – 2 = 2x + 5 ______ 2. 3 – 4x = 7 – 5x ______ 3. 8 + x = –3x + 6 ______ 4. 10 + 5x = –8 + 3x ______ 5. –4x – 6 = 12 – x ______ 6. 10x – 7 = 9 – 4x ______ 7. 4x + 2 = 6 – x ______ 8. 5 – 2x = 3x – 2 ______ 9. 3(4 – x) – 22 = x + 48 ______ 10. (x – 3) – (–2x + 5) = 2x – 7 – 3x ______ 11. (5x – 2) – (6 – 2x) = –x – 9 + 5x ______ 12. 31– 2x + 17 = 23– 4x +12 + 1– 2x ______ You received 20 more votes for class treasurer than your friend. One hundred twenty-eight votes were cast. How many votes did each of you receive?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 21 a = πr2 π mx + b = y Name Date Formulas Formulas are special relations and functions that give you answers to problems. Mathematics and science have many formulas. You will also find formulas in many other areas such as economics, engineering, taxes, payroll, and computers. Use the given formula and solve the problem. 1. You left your house at noon, traveling at 55 mph. How far will you be from your house in 8 hours? Distance equals rate times time (d = rt). ____________________________________ 2. You have $1,500 in a savings account at the bank. How much interest will you earn at the end of 3 years if the bank is paying an interest rate of 4.5% per year? I=prt (I: interest, p: principal, t: time in years, and r: rate) ________________________________________________ 3. What is the temperature outside in Fahrenheit if the temperature was reported as 40°C? F= 9– 5 x °C + 32 (°C: degrees Celsius and °F: degrees Fahrenheit) ____________________________ 4. How fast in miles per hour was the horse running if it ran 5/8 of a mile in 1.75 minutes? Rate equals distance divided by time (change time to hours). ______________________________ 5. How much money with interest would you have in a bank at the end of 5 years if the bank pays an interest rate of 4.2% per year and you made a deposit of $750? A = P(1+r)n (A: total amount, P: principal, r: rate, and n: number of years) ________________________________ 6. What is the perimeter of a square if one side is 31 meters? Perimeter of a square is four times one side. (P = 4s) ________________________________________________ 7. What is the area of a triangle if the base is 12 feet and the height is 9 feet? Area of a triangle is one half of base times height. (A = 1/2 bh) _________________________________________ 8. What is the volume of a block if the length is 38 inches, the width is 26 inches, and the height is 18 inches? Volume is length times width times height. (V = lwh) ______________________ 9. Find the surface area of a block with a length of 15 cm, a width of 12 cm, and a height of 9 cm. Surface area is found by adding the area of six faces. [SA = 2(lw + lh +wh)] _______________ 10. What is the temperature in Celsius if the temperature reported is –40°F? C= 5– 9(°F – 32) (°C: degrees Celsius and °F: degrees Fahrenheit) __________________________________ Examine the two formulas for changing temperature between Fahrenheit and Celsius. Are they different, or are they the same? Explain. No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 22 a = πr2 π mx + b = y Name Date Absolute Values Inequalities The absolute value of a real number x, written |x| , is the distance of x from 0 on the number line. In inequalities such as |x| < k, you will find two numbers that are k units from 0. These numbers are –k and k. So, x is a solution of |x| < k whenever –k < x < k. In inequalities such as |x| > k, you will find two numbers that are k units from 0. These numbers are –k and k. If x is greater than k, (meaning to the right of k, but to the left of –k), the distance becomes greater as you move to the left of –k. So, x is a solution of |x| > k whenever x > k or x < –k. Example A: Determine the two values. |x + 3| < 5 –5 < x + 3 < 5 Rewrite using the two values. –5 – 3 < x + 3 – 3 < 5 – 3 Subtract 3 from each part of the inequality. –8 < x < 2 Simplify. Example B: Determine the two values. |2x +1| ≥ 3 2x + 1≥ 3 or 2x + 1 ≤ –3 Rewrite using the two values. 2x ≥ 2 or 2x ≤ –4 Subtract 1 from each part of the inequality. 2x ≥ 2 2x ≤ –or 4 Divide each part of the inequality by 2. 2 2 x ≥ 1 or x ≤ –2 Solve each inequality and graph the solution. 1. |x| < 8 ______ 2. |x| > 4 ______ 3. |x + 2| < 3 ______ 4. |x – 1| > 4 ______ 5. |2x – 4| < 3 ______ 6. |3x + 5| > 2 ______ 7. |2x + 3| ≤ 4 ______ 8. |0.5x + 3| ≥ 4 ______ The weather forecaster stated that the temperature has changed 20°F over the past 6 hours. If the temperature is now 67°F, write an expression using absolute values showing the possible values of the temperature 6 hours ago. What are those values?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 23 a = πr2 π mx + b = y Name Date Graphing Inequalities An expression that contains the symbol >, <, ≥, or ≤ is called an inequality. An inequality expresses the relative order of two mathematical expressions. The sentence can contain numbers and/or variables. Symbol Meaning > is greater than < is less than ≥ is greater than or equal to ≤ is less than or equal to A solution of an inequality is a number that makes a true statement when it is substituted for the variable in the inequalities. Example: Write the set which makes the inequality true. Domain {1, 2, 3, 4} 2x + 5 < 10 2(1) + 5 = 7, which is less than 10; true 2(2) + 5 = 9, which is less than 10; true 2(3) + 5 = 11, which is not less than 10; false 2(4) + 5 = 13, which is not less than 10; false Write the set which makes the inequality true. Domain {–2, –1, 0, 1, 2} 1. x + 3 < 4 ______ 2. x – 3 < –2 ______ 3. 2x + 1 > 5 ______ 4. 2x – 3 > 7 ______ 5. x – 5 ≥ –4______ 6. x – 5 ≤ –4______ Graph the inequality on a number line. Domain {real number} 7. x +1 < 0 8. 3 > x + 3 9. –x < 5 10. –x – 2 > –3 When the signs on numbers in an inequality reverse, explain what happens to the inequality sign. Illustrate this property on the following pairs of numbers. 11. 2 > –3 12. –5 < –1 13. 15 > 12 14. –24 < 6No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 24 a = πr2 π mx + b = y Name Date Solving Inequalities Using Addition and Subtraction Finding all solutions of an inequality is solving the inequality. You can solve an inequality the same way you solve an equation. You can add or subtract the same value to each side of the inequality. Example: Solve and graph inequalities. x + 3 ≤ 5 x + 3 – 3 ≤ 5 – 3 Subtract 3 from each side. x ≤ 2 Simplify. Solve and graph inequalities. 1. x + 3 < 8 ______ 2. x – 5 > 2 ______ 3. x + 7 ≤ 11______ 4. 1.5x – 8 ≥ 1______ 5. x – 2 ≤ –6 ______ 6. 22 ≥ x + 12 ______ 7. x – 32 > –27 ______ 8. x – 15 ≥ –18 ______ Write an inequality that represents the sentence. Then solve the inequality. 9. x plus 4 is greater than or equal to 12.5. __________________________ 10. The difference of x and 5 is less than 12. __________________________No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 25 a = πr2 π mx + b = y Name Date Solving Inequalities Using Multiplication and Division You can solve inequalities the same way you solve equations. When multiplying or dividing with a negative number, the order (direction) of the inequality must change. Example: Solve and graph the inequality. 4x < 12 4x < 12 Divide by 4 (order does not change). 4 4 x < 3 Simplify. –3x < 12 –3x > 12 Divide by –3 (order does change). –3 –3 x > –4 Simplify. Solve and graph each inequality. 1. 3x < 15 ______ 2. –4x > 28 ______ 3. 1– 2x < –5 ______ 4. –5x > 20 ______ 5. –3– 5x > –9 ______ 6. –7x < 42 ______ 7. –2– 3x + 2 < 8 ______ 8. –1.5x – 0.8 > 2.2 ______ 9. –3 > –6x + 9 ______ 10. 15 < –7x – 34 ______ Six minus twice a number n is less than or equal to the opposite of one-half the number. Write an expression and solve the inequality.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 26 a = πr2 π mx + b = y Name Date Ratios and Proportions An equation that states that two ratios are equal is called a proportion. In a proportion, the cross products are equal. Example: Solve the proportion. 3– 8 = 9– x 3 × x = 8 × 9 Determine cross products. 3x = 72 Simplify. –3–x 3 = 72 –– 3 Divide by 3. x = 24 Simplify. Solve the proportions. 1. 2– 3 = 8– x ______ 2. 5– 6 = x –– 18 ______ 3. x– 5 = 3– 5 ______ 4. –x– 12 = 8– 9 ______ 5. 1–5– x = 9 –– 15 ______ 6. –6– 18 = x –– 12 ______ 7. 3–3– 22 = 24 –– x ______ 8. 3–0– 125 = x –– 100 ______ 9. 1–2– x = 16 –– 24 ______ 10. 3–5– 14 = 1–5– x ______ 11. (x– –+– –1 ) 15 = 2– 3 ______ 12. 5– 9 = 10 (x– ––– –3 ) ______ Small-scale cars are all in a constant 1/64 scale. This means for every centimeter on the model, the actual car would be 64 centimeters. How long would the model be if the car measures 4.16 meters? (Remember to convert meters to centimeters.)No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 27 a = πr2 π mx + b = y Name Date Percent of Change You find the percent of change from the original amount by using proportions. This could be a percent of increase or decrease. percent of change = amount of change 100 original amount Example: Determine if the quantity represents a percent of increase or decrease. Then find the percent. 2003 enrollment: 600 2004 enrollment: 780 From 2003 to 2004, the enrollment increased by 180 students. x% = 180 100 600 600 × x = 180 × 100 Determine cross products. 600 × x = 18,000 Divide both sides by 600. 600 600 x = 30% Simplify. Enrollment increased by 30%. Determine if the quantity represents a percent of increase or decrease. Then find the percent. 1. Before: 10, After: 12 _____________________ 2. Before: 15, After: 12 _____________________ 3. Before: 75, After: 60 _____________________ 4. Before: 110, After: 132 _____________________ 5. Before: 90, After: 135 _____________________ 6. Before: 260, After: 156 _____________________ 7. 2001: $20.50, 2002: $24.60 _____________________ 8. 2003: $200, 2004: $160 _____________________ 9. Regular Price: $49.00, Sale Price: $36.75 _____________________ 10. Regular Price: $80.00, Sale Price: $64.00 _____________________ 11. Old Price: $60, New Price: $78 _____________________ 12. Old Price: $20, New Price: $25 _____________________ Determine if the statement is true or false. Explain. Two times a number is a 100% increase of the number. ______ A 25% increase of 100 is 125. ______No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 28 a = πr2 π mx + b = y Name Date Slope-Intercept Form To determine the slope of a line on a graph, you must calculate the rise and run of that line. Example: First: Pick any two points on the line. Write their coordinates. You will find it helpful if you underline the x-coordinate and circle the y-coordinate. Point A (1, 2 ) and point B (3, 6 ) Second: Subtract the y-coordinates. This shows the vertical change or rise. 6 – 2 = 4 Third: Subtract the x-coordinates. This shows the horizontal change or run. 3 – 1 = 2 Fourth: Find the slope by writing the ratio of rise to run. Slope = rise = 4 or 2 run 2 Using the procedure above, determine the slope of the following lines. 1. 3. 5. 2. 4. 6. What do the slopes of problems 1, 2, and 5 have in common? Do the slopes of 3, 4, and 6 have the same fact in common?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 29 a = πr2 π mx + b = y Name Date Standard Form The standard form of a linear equation is Ax + By = C in which A, B, and C are integers. This form of a linear equation is a simple way of finding the x-and y-intercepts of an equation in order to graph the function. Example: Graph 2x + 3y = 12 First: Let x = 0 0 + 3y = 12 Replace 2x with the value of x. 3y = 12 Divide both sides by 3. 3 3 y = 4 Simplify. This gives us a point on the y-intercept: (0,4) Second: Let y = 0 2x + 0 = 12 Replace 3y with the value of y. 2x = 12 Divide both sides by 2. 2 2 x = 6 Simplify. This gives us a point on the x-intercept: (6,0) Third: Graph the two points. Draw the line. Determine the x-and y-intercepts and graph each equation. 1. x + y = 2 _____ _____ 2. x – y = 3 _____ _____ 3. 2x – y = 4 _____ _____ 4. x + 3y = –6 _____ _____ 5. 2x + 5y = 10 _____ _____ 6. 3x – 2y = 6 _____ _____ What is the name of the point where the x-and y-intercepts are the same value?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 30 a = πr2 π mx + b = y Name Date Point-Slope Form You can determine the equation for a line by using a method called point-slope form. Slope-intercept form, a form of a linear equation, is another name. y = mx + b The slope is m, and b is the y-intercept. Example: Find the equation of the line that passes through (2,5), (4,9). First: (9 – 5 )= 4 = 2 Compute the slope, m = rise. (4 – 2) 2 run Second: 5 = 2 • (2) + b Compute the y-intercept, substituting x and y values. 5 – 4 = 4 + b – 4 Subtract 4 from both sides. b = 1 Simplify. Third: Use the point-slope form to write the equation: y = 2x + 1 Graph and write the equation for the line through the given points in point-slope form. 1. (5,3), (3,1) __________ 2. (–1,3), (5,0) __________ 3. (–3,1), (2,3) __________ 4. (0,3), (5,1) __________ 5. (4,3), (–2,1) __________ 6. (5,4), (–3,–4) __________ Plot the four points, determine the slope of the four line segments, and name the shape of the figure. (–4,2), (2,–4), (6,0), and (0,6)No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 31 e = mc2 + a2r b3t Name Date Practice Assessment—Mathematical Models Make a scatter plot using the data in the table. 1. Make a table and graph each function. 2. f(x) = |x| + 1 Simplify each expression. 3. 10-3 __________ 4. 3-2 __________ 5. 5-2 __________ Simplify each expression. 6. 23 × 24 __________ 8. (0.02)2 × (0.02)1 __________ 7. 53 × 55 __________ 9. 83 × 82 × 8-4 __________ Simplify each expression. 10. (32)4 __________ 12. (25)2 __________ 11. (a3)4 __________ 13. (3x)4 __________ Simplify each expression. 14. (3– 4 )2 ______ 16. (4—x y )5 ______ 15. (2– x )3 ______ 17. (3—x 2y)4 ______ Find the next three terms in each sequence. 18. 2, 5, 8, 11, _____, _____, _____ 20. 2, 4, 8, 16, _____, _____, _____ 19. 1.5, 4, 6.5, 9, _____, _____, _____ 21. 1, 4, 9, 16, _____, _____, _____ Leon, Tracy, and Pat each wrote a report for history class. Leon’s report was twice as long as Pat’s. Tracy’s report was 4 pages shorter than Pat’s. If Tracy’s report was 6 pages, how long was Leon’s? No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 32 e = mc2 + a2r b3t Name Date Scatter Plots and Equations of Lines A graph in which the two sets of data shown as points on a coordinate plane is called a scatter plot. You can make a scatter plot by graphing ordered pairs of data. Example: Make a scatter plot using the data in the table. Make a scatter plot using the data in the table. 1. 3. 2. 4. Measure the arm span and the height of 10 students. Use the data and make a scatter plot.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 33 e = mc2 + a2r b3t Name Date Graphing Absolute Value Equations An absolute value equation has a graph that looks like a “V” and it points either upward or downward. The graph slides from the home position upward or downward when a number is added to or subtracted from the |x|. The graph is shifted left or right when the number is added to or subtracted from the x before the absolute value function is applied. Example: Graph f(x) = |x| + 1 First: Make a table. x f(x) = |x| + 1 (x, f(x)) -3 |–3| + 1 = 4 (–3, 4) -1 |–1| + 1 = 2 (–1, 2) 0 |0| + 1 = 1 (0, 1) 1 |1| + 1 = 2 (1, 2) 3 |3| + 1 = 4 (3, 4) Second: Graph the function. Hint: An equation in the form f(x) = |x + 1| would be shifted horizontally. This shift would be to the left, opposite of the sign. Make a table and graph each function. 1. f(x) = |x| + 2 2. f(x) = |x| – 3 3. f(x) = –|x| + 4 4. f(x) = |x| – 6.5 5. f(x) = |x + 3| 6. f(x) =|x – 4| What will happen if the x is multiplied with a number greater than one? Graph the following function: f(x) = |2x|No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 34 e = mc2 + a2r b3t Name Date Zero and Negative Exponents For any nonzero real number a, a0 = 1 For any nonzero real number a, if n is a positive integer, a-n. Example: A. 570 = 1 B. 5-2 = 1 –– 52 = 1 –– 25 Simplify each expression. 1. 10-3 ______ 2. 2-2 ______ 3. 5-4 ______ 4. 70 ______ 5. (3– 2)-1______ 6. x-2 ______ 7. 33.50 ______ 8. (-x)-3 ______ 9. (2/5)-2 ______ 10. x-4 ______ 11. (-4)-2 ______ 12. x-3______ 13. 3x-2 ______ 14. 12x-1______ 15. (1– 3)0 ______ 16. (-35)0 ______ 17. x-5 ______ 18. (25x)0 ______ Write an expression for each verbal phrase and simplify. 19. The product of four raised to the third power and four raised to the negative two power. 20. The quotient of two raised to the negative third power and two raised to the fifth power.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 35 e = mc2 + a2r b3t Name Date Multiplication Property of Exponents When you use exponents, remember that you are telling people the number of facts that are being multiplied together. For every nonzero number a and integer m and n, am • an = am + n. Examples: A. 94 • 93 = 97 B. 24 • 26 • 2-2 = 2(4 + 6 + -2) = 28 C. 125 • 12-5 = 120 = 1 Simplify each expression. 1. 24 • 25 ______ 2. 54 • 53 ______ 3. (0.02)3 • (0.02)0 ______ 4. 86 • 83 • 8-3 ______ 5. 125 • 128 ______ 6. 46 • 43 ______ 7. 1253 • 125-2 ______ 8. x2 • x3 ______ 9. x5 • x7 ______ 10. x 8 • x3 ______ 11. x2 • x3 • x5 ______ 12. x12 • x-10 ______ 13. x2 • x-2 ______ 14. x5 • x4 • x1 ______ 15. 2x3 • 3x9 ______ 16. 6x2 • 3x5 ______ 17. 2x3 • 5x3 ______ 18. 12x7 • 4x-3 ______ The distance light travels in one year (1 light-year) is about 5.88 • 1012 miles. The closest star to Earth, other than the sun, is Alpha Centauri, which is 4.35 light-years from Earth. About how many miles from Earth is Alpha Centauri?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 36 e = mc2 + a2r b3t Name Date Power of Powers of Exponents For every nonzero number a and integers m and n, (am)n = amn. For every nonzero number a and b and integer n, (ab)n = an × bn. Examples: A. (122)3 = 126 B. (4x3)2 = 42x6 C. (5x2y4)3 = 53x6y12 Simplify each expression. 1. (23)4 2. (x4)5 3. (24)3 4. (4x)3 5. (6x4)2 6. (x2 y4)3 7. (7x y6)3 8. (x3 y4)3 9. (4x2 y)3 10. (3x2 y5)3 11. (x2 y3)0 12. (x3 y4)4 13. (–4x2 y3)3 14. (5x5 y2)2 15. (2xy2) • (3x3 y2) 16. (x2)5 (x3)2 17. (2x4 y)2 (xy)3 18. (x2)-1 One student simplified x5 + x5 to x10. Another student simplified x5 + x5 to 2x5. Which student is correct? Explain.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 37 e = mc2 + a2r b3t Name Date Division Property of Exponents For every nonzero number a and b and integer n, (a– b)n = . Examples: A. (53 –x2)3 = 53 –x6 = 1251 —– x 6 B. (2—x 3 )-2 = ( 3— 2x )2 = = Simplify each expression. 1. (3– 5)2 2. (2– x )3 3. (5—yx )5 4. (2—x 3y)4 5. (2– 32)3 6. ( 6– x 2 )2 7. (x– x )5 8. (3—x 2 2)4 9. (2—x y 3)4 10. (32 • 70)2 —— 24 11. —(125 (1—20)2) 123 12. —(x0 —y3) y1 13. (x4 y3 z5)3 14. (x—y 2z)5 15. (x2 x3)2 —— x5 16. (25x5– 36y7)0 17. (—(2x—y6) x3y )-2 18. (1– 3)-2 (1– 2)3 Earth’s crust contains approximately 122 trillion tons of gold. One ton of gold is worth about $9 million. What is the approximate value of gold in Earth’s crust? Use scientific notation to solve this problem. an bn 9 4x2 32 22x2No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 38 e = mc2 + a2r b3t Name Date Arithmetic Sequences A number pattern in which you can predict the next term is called a sequence. Each number in a sequence is a term. You form an arithmetic sequence by adding a fixed number to each previous term. You can determine any term in an arithmetic sequence if you know the first term and the common difference by using this rule: A(n) = a + (n – 1)d, where A(n) is the nth term; a is the first term; n is the nth number; and d is the common difference. Example: This sequence has a common difference of 5: 2, 7, 12, 17, 22, etc. Find the 10th term. A(n) = a + (n – 1)d A(10) = 2 + (10 – 1)5 = 47. Find the next three numbers in each sequence. 1. 4, 6, 8, 10, _____, _____, _____ 2. 5, 12, 19, 26, _____, _____, _____ 3. 4, 9, 14, 19, _____, _____, _____ 4. 2, 4.5, 7, 9.5, _____, _____, _____ 5. –8, –5, –2, 1, _____, _____, _____ 6. 17, 14, 11, 8, _____, _____, _____ 7. –5, 8, 21, 34, _____, _____, _____ 8. 0.7, 1.5, 2.3, 3.1, _____, _____, _____ Determine the 10th and 15th terms in each of the above sequences. 9. ______________ ________________ 10. ______________ ________________ 11. ______________ ________________ 12. ______________ ________________ 13. ______________ ________________ 14. ______________ ________________ 15. ______________ ________________ 16. ______________ ________________ The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, . . . After the first two numbers, each number is the sum of the two previous numbers. Find the next three terms in this sequence. No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 39 e = mc2 + a2r b3t Name Date Geometric Sequences You form a geometric sequence by multiplying a term in the sequence by a fixed number to find the next term. A(n) = a × r(n – 1), where A(n) is the nth term; a is the first term; n is the nth number; and r is the common ratio (multiple). Example: The sequence has a common ratio of 2: 0.5, 1, 2, 4, 8, etc. Find the 10th term. A(n) = a × r(n – 1) A(10) = 0.5 × 2(10 – 1) = 256. Find the next three terms in each sequence. 1. 2, 4, 8, 16, _____, _____, _____ 2. 3, 6, 12, 24, _____, _____, _____ 3. 32, 16, 8, 4, _____, _____, _____ 4. 1/2, 1/4, 1/8, 1/16, _____, _____, _____ 5. 12, 6, 3, 3/2, _____, _____, _____ 6. 1, 10, 100, 1,000 _____, _____, _____ 7. 7, –14, 28, –56 _____, _____, _____ 8. 1, –1, 1, –1, _____, _____, _____ Find the first, fourth, and eighth terms in each sequence. 9. A(n) = 2 × 3(n – 1) _____ _____ _____ 10. A(n) = –3 × 2(n – 1) _____ _____ _____ 11. A(n) = 1/2 × 2(n – 1) _____ _____ _____ 12. A(n) = 64 × 0.5(n – 1) _____ _____ _____ You drop a basketball from a height of 2 meters. Each bounce of the ball is 45% of the previous height. What will the height of the basketball be on the fourth bounce? (Round to the nearest tenth of a centimeter.)No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 40 e = mc2 + a2r b3t Name Date Exponential Functions Two concepts of an exponential function are familiar. Compare these expressions: x2 and 2x. They have the form of base and exponent, but the positions of the variable and constant have been reversed. In x2, the exponent is constant. In 2x, the exponent is the variable x. An expression like 2x is used in defining the concept of exponential functions. Graph the following expressions with domain values given (x). 1. y = 2x 3. y = 3x 2. y = (1– 2)x 4. y = (1– 3)x A population of 100 insects triples in size every month. The expression y = 100 × 3x models the population after x months. Graph the expression.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 41 ax2 + bx + c Name Date Practice Assessment—Symbol Manipulations Solve each system of equations. 1. y = 2x 3x – 2y = –11 ______ 2. x = 2y + 3 x – y = 3 ______ 3. x = 2y y – 2 = x ______ 4. –2x + 3y = 2 2x + 7y = 18 ______ 5. 2x + 7y = –7 –5x + 7y = –14 ______ 6. 3x + 5y = 2 –3x + y = 2 ______ Simplify. 7. (x2 + 5x + 7) + (5x2 – 2x + 9) ______ 8. (5x2 + 2x – 15) + (3x2 + 6x + 8) ______ 9. (x2 – 11x – 7) + (3x2 – 31x + 12) ______ 10. (3x2 + 2x – 6) + (–4x2 + 3x – 8) ______ Factor the following expressions. 11. 10x – 15 ______ 12. 4x3 – 8x2 +16x ______ 13. 6y2 + 9y4 ______ 14. 12x3 – 3x2 + 9x ______ Simplify. 15. (x + 1) (x + 2) ______ 16. (x + 5) (x – 3) ______ 17. (x – 3) (x – 4) ______ 18. (x – 7) (x + 7) ______ 19. (x – 1)2 ______ 20. (x + 2)2 ______ 21. (x – 3)2 ______ 22. (2x – 5)2 ______ Factor the following trinomial. 23. x2 + 6x + 8 ______ 24. x2 – 7x + 12 ______ When 3 times a number is subtracted from its square, the remainder is 10. Find the number.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 42 ax2 + bx + c Name Date Solving a System Using Substitution When you have two linear equations with the same two variables, this forms a system of equations. The solution of a system of equations is the point (ordered pair) where the two equations intersect. This ordered pair will be true in both equations. Solving a system of equations using an algebraic method known as substitution is useful when one equation has been solved for one of the variables. Example: Find the solution to the system of equations: 2x – y = 5 and y = –x + 4 Solution Since y = –x + 4, substitute this expression for y in the first equation. 2x – (–x + 4) = 5 Distribute negative sign in parentheses. 2x + x – 4 = 5 Combine like terms. 3x – 4 = 5 Add 4 to both sides. 3x = 9 Divide by 3. x = 3 Substitute 3 for x in the second equation. y = –x + 4 y = –(3) + 4 Add. y = 1 The solution is (3, 1). Solve each system of equations. 1. y = 2x 3x – 5y = –11 ______ 2. x = 2y + 3 2x – y = 3 ______ 3. x = 2y y + 2 = x ______ 4. y = –4x x – y = 10 ______ 5. y = 8 – x 4x – 3y = –3 ______ 6. y = x – 4 5x + 3y = 4 ______ 7. x + 3y = 5 –3x + 2y = 18 ______ 8. 3x + y = 7 x – y = 1 ______ 9. y = –x y – 2 = x ______ 10. 2x + y = –1 2x= –3y – 1 ______ You have 12 coins in your pocket. There are twice as many dimes as pennies. How many of each type of coin do you have? How much money do you have?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 43 ax2 + bx + c Name Date Solving a System Using Elimination When you have two linear equations with the same two variables, this forms a system of equations. The solution of a system of equations is the point (ordered pair) where the two equations intersect. This ordered pair will be true in both equations. Solving a system of equations using an algebraic method known as linear combination is useful when one variable is a multiple of the other. You solve a system of equations by adding or subtracting the two equations to produce a new equation in only one variable. You may need to multiply by some factor before adding or subtracting the equations. Example: Solve –2x + 3y = 2 2x + 7y = 18 Solution: Since the coefficients of the x-terms are opposites, add the two equations. –2x + 3y = 2 2x + 7y = 18 Add the two equations. 10y = 20 Divide by 10. y = 2 Substitute y = 2 in one of the equations. –2x + 3y = 2 –2x + 3(2) = 2 Substitute 2. –2x + 6 = 2 Subtract 6 from both sides. –2x = –4 Divide by –2. x = 2 Solution: (2, 2) Check in both equations. Solve each system of equations. Check the solutions. 1. x – 3y = –3 2x + 3y = 9 ______ 2. 3x + y = 4 –x + y = –8 ______ 3. 2x + 9y = 4 5x + 9y = –17 ______ 4. –2x + 3y = 2 2x + 7y = 18 ______ 5. 2x + 7y = –5 –5x + 7y = –12 ______ 6. 3x + 5y = 6 –3x + y = 6 ______ 7. 2x + 4y = –4 2x + y = 8 ______ 8. 2x + y = –2 3x – y = –3 ______ 9. 3x – y = –1 x + y = –3 ______ 10. –3x – 5y = 4 2x + y = –5 ______ The perimeter of a rectangle is 24 inches. The length is twice the width. Determine the dimensions.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 44 ax2 + bx + c Name Date Solving a System Using Matrices You have a method of solving a system of linear equations that requires you to change an equation into matrices. A 2 x 2 determinant is a square array in the form a b consisting of two rows and 2 columns. c d The determinant of a system of equations, det A, is formed using the coefficients of the variables when the equations are written in standard form. ax + by = e a b det A = determinant of coefficients cx + dy = f c d To compute the value of the determinant, use det A = a × d – b × c, which is the difference of the product of the diagonals. You can find the solution to a system of equations by using the det A and another determinant formed by replacing the x column of the determinant with the constant column. These are the values of e and f. e b Replace the x-column a e Replace the y-column Ax = Ay = f d with the constant column. c f with the constant column. To find x, divide Ax by A, which is (ed – fd) . (ad – bc) To find y, divide Ay by A, which is (ae – bf) . (ad – bc) Example: Solve 4x + 3y = –4 –4 3 3x – y = –3 Ax –3 –1 (–4 × –1) – (3 × –3) 4 – –x = = = = 9 = 1 3 = –1 A 4 3 (4 × –1) – (3 × 3) –4 – 9 –13 3 –1 4 –4 Ay 3 –3 (4 × –3) – (3 × –4) –12 – –y = = = = 12 = 0 = 0 A 4 3 (4 × –1) – (3 × 3) –4 – 9 –13 3 –1 The solution is (–1, 0). The check is left to you. Solve the following with determinants. 1. 3x – y = 2 x + 2y = –4 ______ 2. x + 2y = –7 3x – y = 14 ______ 3. x + y = –3 x – y = 5 ______ 4. x + 2y = 1 2x – y = 3 ______ You have $2.00 in dimes and quarters. If the dimes were nickels and the quarters were dimes, you would have $1.15 less. Using determinants, how many dimes and quarters do you have? No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 45 ax2 + bx + c Name Date Adding and Subtracting Polynomials Polynomials are expressions with several terms. The simplest expression is called a monomial. A monomial is a single number or a product of a number and one or more variables. A monomial includes variables. The number part is called the coefficient of the term and is written first. Other polynomials are expressions that contain several monomial terms. If it has two terms, it is a binomial. With three terms, it is called a trinomial. Like terms are terms in which the variable or sets of variables are identical. The coefficient may be different. You simplify a polynomial when you group and combine all like terms by adding or subtracting the coefficient. Example: Simplify: (2x2 + 4x – 7) – (3x2 + 2x + 5) Solution: 2x2 + 4x – 7 Line up like terms. Write zero where there are no like terms. (3x2 + 2x + 5) 2x2 + 4x – 7 -3x2 -2x – 5 Change signs and add to simplify. -x2 + 2x – 12 Simplify. 1. (x2 + 5x + 7) + (5x2 – 3x + 6) ________________________________ 2. (2x2 + 4x – 15) + (3x2 + 8x + 8) ________________________________ 3. (x2 – 12x – 7) + (3x2 – 21x + 5) ________________________________ 4. (5x2 + x – 7) + (-7x2 + 3x – 8) ________________________________ 5. (0.5x2 + 3.4x – 2.7) + (0.7x2 + 6.2x + 5) ________________________________ 6. (4x2 + 2x – 3) – (3x2 + 2x + 5) ________________________________ 7. (8x2 – 4x – 11) – (5x2 + 4x + 21) ________________________________ 8. (–2x2 + 6x – 7) – (3x2 + 2x – 3) ________________________________ 9. (x2 + –4x + 9) – (3x2 + 2x + 5) ________________________________ 10. (8x2 + 7x – 16) – (3x2 – 2x – 5) ________________________________ The longest side of a triangle is 6x + 8, a second side is x + 2, and the third side is 2x – 5. How long is each side if the perimeter is 68 cm?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 46 ax2 + bx + c Name Date Multiplying and Factoring When you multiply polynomials, the main concept is the distributive principle. The factor must be applied to each term. Example: 5x • (2x + 3y – 8) = 10x2 + 15xy – 40x When you factor, begin by determining if a common number factor is in all the terms. Then determine if there is a common variable in all the terms. Extract these factors out. This is the greatest common factor, or GCF, because it includes all the common factors in the expression. Example: 15xy3 – 3x2y2 = 3 • 5 • x • y • y • y – 3 • x • x • y • y = 3 • x • y • y ( 5 • y – x) = 3xy2 (5y – x) Multiply each expression. 1. 2(3x + 4y) ________________ 2. 7(2x2 – 5y2) ________________ 3. x(x + 2y – 5z) ________________ 4. x(12 – 9xy) ________________ 5. 6xy(7x – 12y2) ________________ 6. -10y(3x – 2xy – 4) ________________ Factor the following expressions. 7. 21x – 28 ______ 8. 5x3 – 10x2 + 15x ______ 9. 3y2 + 6y4 ______ 10. 6x3 – 4x2 + 8x ______ 11. 8x3 + 12x2 + 4x ______ 12. y3 – 2y2 +8y ______ 13. 12x3 – 36x2 ______ 14. 6x5y – 12x4y2 – 9x3y3 ______ 15. 18x5y + 45x4y2 – 72x2y4 ______ 16. 2y3z2 – y3z ______ 17. 12x2 + 8x2y ______ 18. 2x2y2 – 10xy3 + 24xy4 ______ You have two pieces of lumber. One is 54 inches long and the other is 90 inches long. You need to determine the length of shelves from these boards. You must use the full length of each board, dividing them into equal pieces but making each piece as long as possible. How long is each piece, and how many shelves will you have?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 47 ax2 + bx + c Name Date Multiplying Binomials When multiplying two binomials, the following shortcut is used: F Multiply the first terms of each binomial. O Multiply the outer terms. I Multiply the inner terms. L Multiply the last terms. This procedure is called the FOIL (first, outer, inner, last) method for multiplying binomials. Example: Find the product of (3x + 2)(x – 4) Solution: Use the FOIL method and combine like terms. (3x + 2)(x – 4) = (3x × x) + (3x × –4) + (2 × x) + (2 × –4) = 3x2 + –12x + 2x + –8 = 3x2 – 10x – 8 Simplify. 1. (x + 3) (x + 5) ______ 2. (x + 8) (x – 3) ______ 3. (x – 2) (x – 4) ______ 4. (x – 3) (x + 3) ______ 5. (x + 3) (x – 1) ______ 6. (2x + 8) (x + 2) ______ 7. (3x + 2) (2x + 1) ______ 8. (x + 5) (x + 5) ______ 9. (x – 6) (x – 6) ______ 10. (2x + 3) (2x + 3) ______ 11. (3x + 1) (3x + 2) ______ 12. (4x – 3) (2x – 3) ______ 13. (x + –10) (2x + 10) ______ 14. (3x – 2) (5x + 3) ______ 15. (0.5x + 3) (4x + 2) ______ 16. (3 + x) (5+x) ______ Simplify the following binomial (x+1)3. Hint: Look at how you multiply 113. No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 48 ax2 + bx + c Name Date Perfect Squares and Difference of Squares You can discover a pattern with a perfect square. (x – 5)2 (x – 5)(x – 5) = x2 – 5x – 5x + 25 = x2 – 10x + 25 The first and last terms are perfect squares x2 and 52. The middle two terms are identical, so double the product of the first and last terms. When two binomials are identical but one is added and the other is subtracted, their products will be the difference of the two squares. Example: (5x + 3)(5x – 3) = (5x)2 – (3)2 = 25x2 – 9 Find each product. 1. (x – 3)2 _______ 2. (x + 9)2 _______ 3. (x – 2)2 _______ 4. (2x – 1)2 _______ 5. (5x + 6)2 _______ 6. (3x – 4)2 _______ 7. (x + 1)2 _______ 8. (3x + 2)2 _______ 9. (x + 2)(x – 2) _______ 10. (x + 3)(x – 3) _______ 11. (x – 5)(x + 5) _______ 12. (2x + 3)(2x – 3) _______ 13. (4x + 5)(4x – 5) _______ 14. (7x + 2)(7x – 2) _______ 15. (9x – 8)(9x + 8) _______ 16. (10x + 20)(10x – 20) _______ A garden is in the shape of a rectangle. The area is 36x2 – 25y2. Find the length and width of the rectangle if x = 9 ft and y = 2 ft.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 49 ax2 + bx + c Name Date Factoring Trinomials If you are able to find two numbers such that b = r + s and c = r × s in the trinomial, x2 + bx + c, you will be able to factor the expression into two binomials. Example: Factor x2 + 10x +16 10 = r + s 16 = r × s First: Write all the pairs of integers with a product of 16. 1 × 16 (–1) (–16) 2 × 8 (–2) (–8) 4 × 4 (–4) (–4) Second: Select the pair with a sum of 10. 2 + 8 = 10 Third: Write the expression in the form x2 + (r + s)x + r × s x2 + 2x + 8x + 2 × 8 Fourth: Write the expression as the sum of two binomials. (x2 + 2x) + (8x + 2 × 8) Fifth: Factor each binomial. x(x + 2) + 8(x + 2) Sixth: Apply the distributive property. (x + 8)(x + 2) Factor the following trinomials. 1. x2 + 2x + 1______ 2. x2 – 2x + 1______ 3. x2 + 2x – 3 ______ 4. x2 – 2x – 3 ______ 5. x2 + 4x + 4 ______ 6. x2 – 7x + 12 ______ 7. x2 – 2x + 24 ______ 8. x2 + 7x + 12 ______ 9. x2 + 6x + 9 ______ 10. x2 – 8x + 15 ______ 11. x2 + 10x + 16 ______ 12. x2 – 7x – 8 ______ 13. x2 + 5x – 36 ______ 14. x2 + 4x – 32 ______ 15. x2 – 26x + 48 ______ 16. x2 + 22x – 48 ______ 17. x2 – 6x + 5 ______ 18. x2 + 2x – 15 ______ The area of a rectangle is 2x2 + 7x + 3. The width is (x + 3). Find an expression for the length, and determine the length if x is 15 feet.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 50 ax2 + bx + c Name Date Factoring Perfect Squares and Difference of Squares A trinomial is a perfect square if the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms. Example: 4x2 + 20x + 25 First: 4x2 is a square of 2x, and 25 is a square of 5. Second: The double of (2x)(5) is 20x. Third: The factors of 4x2 + 20x + 25 are (2x + 5)2. Remember that a square root can be positive or negative. So 4x2 – 20x + 25 = (2x – 5)2, 20x is negative. The binomial x2 – 16 is called a difference of two squares. Notice that x2 and 16 are both perfect squares. Since the two factors of 16 that add to make 0 are –4 and 4, x2 – 16 factors into (x + 4)(x – 4). Factor. 1. x2 – 25 ______ 2. x2 – 49 ______ 3. x2 – 144 ______ 4. 4x2 – 9 ______ 5. 4x2 – 81 ______ 6. 9x2 – 25 ______ 7. x2 – 1 ______ 8. x2 y2 – 25 ______ 9. x2 + 4x + 4 ______ 10. x2 – 2x + 1 ______ 11. x2 – 6x + 9 ______ 12. x2 + 24x + 144 ______ 13. x2 – 14x + 49 ______ 14. x2 + 18x + 81 ______ 15. 4x2 + 4x + 1 ______ 16. 4x2 – 20x + 25 ______ 17. 16x2 + 24x + 9 ______ 18. 9x2 – 24x + 16 ______ You have a piece of metal that is 8 inches on each side. You cut a 4 inch by 4 inch square out of a corner. How many squares inches are left?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 51 ax2 + bx + c Name Date Factoring by Chunking You will find it easier to solve some expressions by first making a complicated expression easier. Look for a common expression, or chunk, which is repeated within the equation. To identify the chunk, you need to examine the expression carefully. Several ideas are reversal of signs, order, and multiples. Example: Factor (x + 3)2 – 3(x + 3) + 2 Let (x + 3) = u u2 – 3u + 2 Substitute u for (x + 3). u2 – u + –2u + 2 Factor the expression. u(u – 1) –2(u – 1) Factor the expression. (u – 2)(u –1) Distribute. (x + 3 – 2)(x + 3 – 1) = (x + 1)(x + 2) Substitute (x + 3) for u and simplify. Factor each expression. Look for chunks. 1. (x + 2)2 + 2(x + 2) + 1 ______ 2. (2x – 3)2 + 2(2x – 3) – 15 ______ 3. (3x – 5)2 – 2(3x – 5) – 8 ______ 4. (x + 2)2 + 3(x + 2) – 28 ______ 5. (3 – x)2 + (3 – x) – 6 ______ 6. (x + 1)2 + 7(x + 1) + 10 ______ 7. (x – 2)2 + 5(x + 2) + 6 ______ 8. (x + 15)2 + 14(x + 15) – 15 ______ 9. (x – 6)2 + 5(x – 6) – 14 ______ 10. (x – 8)2 + 17(x – 8) – 18 ______ 11. (x + 9)2 + 2(x + 9) + 1 ______ 12. (x – 9)2 – 7(x – 9) + 12 ______ 13. (x + 11)2 + 12(x + 11) – 13 ______ 14. (x – 10)2 + 2(10 – x) + 1 ______ 15. (x + 4)2 + 4(–x + –4) – 21 ______ 16. (x + 3)2 + 15(x + 3) + 14 ______ You can solve y = √49 . (Remember there are two solutions.) By using the strategy of chunking, solve the following problems. 17. (x + 3)2 = 49 18. (2x – 5)2 = 25 19. (6 – 2x)2 = 64 20. (3 + 2x)2 = 81No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 52 d = r t+ Name Date Practice Assessment—Change Analysis Graph each function and state the vertex for that function. 1. y = x2 + 1 ______ 2. y = –x2 – 3 ______ Find the vertex and the axis of symmetry for each equation. Graph each equation. 3. y = x2+ 6x + 2 ______ 4. y = 2x2 + 4x – 1 ______ Simplify. 5. √300 ______ 7. √18 ______ 6. √75 ______ 8. √50 ______ Solve by factoring. 9. x2 + 7x + 10 = 0 ______ 11. x2 + 4x + 3 = 0 ______ 10. x2 – 5x – 6 = 0 ______ 12. x2 – 5x – 14 = 0 ______ Sketch and label a triangle. Use the Pythagorean Theorem to find the unknown length. Round to the nearest tenth. 13. a = 12, b = 16, c = ______ 15. a = 3, b = 4, c = ______ 14. a = ______, b = 5, c = 7 16. a = 5, b = 12, c = ______ Find the distance between each pair of points. Determine the midpoint of each line segment. Round to the nearest tenth. 17. (2, 2), (5, 6) ______ 18. (–2, –4), (4, 4) ______No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 53 d = r t+ Name Date Quadratic Graphs When an equation has a second degree term, you have a quadratic equation. If the function is of the form y = ax2 + c, the shape of the graph is a parabola, opening up or down. The graph is symmetric about the y-axis. The vertex of a parabola is the minimum or maximum point of a parabola, depending upon whether it opens upward or downward. Example: Graph the equation y = x2 – 3 and state the vertex for that equation. Graph each equation and state the vertex for that equation. 1. y = x2 ______ 3. y = x2 – 4 _______ 2. y = –x2 + 2 ______ 4. y = –x2 + 3 _______ Will the graphs of y = x2 + 2x – 3 and y = x2 + 2x + 3 ever intersect? Graph and explain. No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 54 d = r t+ Name Date Quadratic Functions To graph a quadratic function of a general form y = ax2 + bx + c, you need to examine the axis of symmetry by substituting the values of a and b into the equation x = –b/2a. This is also the x-coordinate of the vertex. To find the y-coordinate of the vertex, you substitute this x-value into the quadratic equation, and solve for y. Example: Graph y = x2 – 2x – 4 First: Check to see if the equation is in standard form. Second: Find the axis of symmetry. x = –(–2)/(2 × 1) = 1 Third: Compute the vertex. y = 12 – 2(1) – 4 = –5 Vertex: (1, –5) Fourth: Find two values before and after x = 1. x x2 – 2x – 4 y -1 (-1)2 – 2(-1) – 4 -1 0 (0)2 – 2(0) – 4 -4 1 (1)2 – 2(1) – 4 -5 2 (2)2 – 2(2) – 4 -4 3 (3)2 – 2(3) – 4 -1 Find the vertex and the axis of symmetry for each equation. Graph each equation. 1. y = x2 + 6x – 2 ______ 3. y = –3x2 – 6x + 1______ 2. y = 2x2 + 4x + 1______ 4. y = –x2 – 2x – 1______ You are riding a roller coaster. The shape of the ride is y = x2 –6x + 4. At what point do you reach the lowest point?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 55 d = r t+ Name Date Square Roots You will find square roots to problems that are quadratic. The symbol √—x means the square root of a number x. The inverse process of squaring a number is to find its square root. Since 42 is 16, the square root of 16 is 4. But there is no square root of -16. Most square roots are irrational, a number that cannot be made into a rational number (fraction). There are two properties that are used with square roots: Product Property: √—ab = √—a × √—b, and the numbers a and b must be positive (zero is possible). Quotient Property: √—( ) = √—a —– √—b, and a and b must be positive (b can never be zero). Example: Complete the table. a b √—a √—b √—a × √—b √—( ) 4 9 2 3 2 × 3 (2– 3) Complete the table. a b √—a √—b √—a × √—b √—( ) 1. 16 25 2. 100 81 3. 144 16 4. 256 64 5. 81 441 You may find simplifying rational expressions will keep answers accurate. Calculators round irrational numbers. Example: Simplify √—32 A. √—32 = √—16 × √—2 = 4 × √—2 = 4√—2 B. √—300 = √—3 × √—100 = 10√—3 Simplify. 6. 7. √—72 8. √—12 9. 10. √—27 11. 12. 13. √—112 The formula T = 2π√—( ) represents the period of oscillation of a pendulum, where T is the period of oscillation, L is the length of the pendulum in centimeters and g = 980 —cm s2 , the force of gravity. Approximately, what is the period of oscillation for a pendulum with a length of 156.8 cm? Round to the nearest hundredth of a second. a– b a– b a– b L– g √300 √180 √125 √162No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 56 d = r t+ Name Date Solving Quadratic Equations by Factoring If the product of two numbers is zero, you must have one number that is zero. Example: Solve x2 – 3x = –2 x2 – 3x = –2 x2 – 3x + 2 = 0 Subtract –2 from each side. (x – 2)(x – 1) = 0 Factor. x – 2 = 0 or x – 1 = 0 Set each factor equal to zero. x = 2 or x = 1 Solve for x. Check each answer in the original problem. Solve by factoring. 1. x2 + 7x + 10 = 0 ______ 2. x2 – 5x – 6 = 0 ______ 3. x2 + 4x + 3 = 0 ______ 4. x2 – 5x – 14 = 0 ______ 5. x2 + 8x + 12 = 0 ______ 6. x2 – 6x – 7 = 0 ______ 7. x2 + 3x – 10 = 0 ______ 8. x2 + 14x – 15 = 0 ______ 9. x2 + 5x = 6 ______ 10. x2 + 6x = –9 ______ 11. 3x2 + 2x = 8 ______ 12. x2 + 15x = –36 ______ When an object is dropped on Earth the formula d = 1– 2 at 2 is used to calculate the distance, acceleration, and time. In the formula, d is distance in meters, a is the acceleration due to gravity on Earth (9.8 m— s2), and t is the elapsed time in seconds. You dropped a ball from a window, and you are 222 meters above the ground. How many seconds will it take the ball hit the ground?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 57 d = r t+ Name Date Using Quadratic Formulas You will find many quadratic equations cannot be factored. The Quadratic Formula will allow you to solve any quadratic equation. Quadratic Formula: For a quadratic equation of the form ax2 + bx + c = 0, where a, b, and c are real numbers and a is not zero, the formula is as follows: –b ± √ x = b2 – 4ac 2a Example: Solve x2 + 6x – 16 –6 ± √ 62– 4(1)(–16) –6 ± √ 36 + 64 –6 ± √——100 –6 ± 10 4 or –x = = = = = 16 = 2 or –8 2(1) 2 2 2 2 2 Check each solution. Solve using the Quadratic Formula. Round decimal answers to the nearest tenth. 1. x2 – 10x + 16 = 0 ______ 2. x2 – 12x + 3 = 0 ______ 3. x2 – 12x + 32 = 0 ______ 4. x2 + 8x – 5 = 0 ______ 5. 2x2 – x – 10 = 0 ______ 6. 3x2 + 2x – 8 = 0 ______ 7. 6x2 + 2x – 4 = 0 ______ 8. 6x2 + x – 35 = 0 ______ 9. 14x2 – 3x – 5 = 0 ______ 10. –3x2 + 6x – 3 = 0 ______ 11. x2 + 7x + 12 = 0 ______ 12. 2x2 + x – 21 = 0 ______ The formula d = vt – 16t 2 describes the path of a projectile, where d = distance (feet), v = velocity (ft/sec), and t = time (seconds). A model rocket has a velocity of 80 ft/sec. How much time will it take for the rocket to fall back to the earth? (Hint: Let t be x and the distance is zero.)No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 58 d = r t+ Name Date Simplifying Radicals You will find two basic rules on simplifying radical expressions. Example: Remove all perfect square roots from the radical: A. √—20 = √—4 × √—5 = 2√—5 B. √——1–6 √—25 = 4– 5 Remove the radical from denominator: C. 3 3√—2 3√—2 = = √—2 √—2√—2 2 Simplify each radical expression. 1. √——1–2 √—75 ______ 2. √(26 x 2) ______ 3. √—(5—0 9 ) ______ 4. √—(7—2 50) ______ 5. √200 ______ 6. √—27 × √—3 ______ 7. √——2–1 √—49 ______ 8. √——7–5 √—49 ______ 9. —2– √—5 ______ 10. —3– √—3______ 11. —4– √—5 ______ 12. √—18—0 √—3 ______ 13. √—8 × √—10 ______ 14. √—3 × √—12 ______ 15. √—75 × √—27 ______ 16. √——5–4 √—24 ______ The expression x2 – 20x can be made into a perfect square by adding what number?No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 59 d = r t+ Name Date Pythagorean Theorem In using the Pythagorean Theorem, you need to do the following: First: Sketch a picture of the triangle. Second: Label the legs and hypotenuse of the right triangle. Third: Solve for the missing side. The Pythagorean Theorem is a2 + b2 = c2, where a and b are legs of a right triangle, and c is the hypotenuse. Example: Use the Pythagorean Theorem to find the unknown length. Round your answers to the nearest tenth. A. a = 3 cm, b = 4 cm, and c = ? B. a = 8 ft and c = 10 ft 32 + 42 = c2 82 + b2 = 102 9 + 16 = c2 64 + b2 = 100 25 = c2 b2 = 100 – 64 √—25 = 5 cm = c b2 = 36 b = √—36 = 6 ft Sketch and label a triangle. Use the Pythagorean Theorem to find the unknown length. Round to the nearest tenth. 1. a = 12, b = 5, c = ____ 2. a = ____, b = 5, c = 7 3. a = 7, b = 12, c = ____ 4. a = 5, b = 8, c = ____ 5. a = 6, b = ____, c = 10 6. a = ____, b = 9, c = 12 7. a = ____, b = 10, c = 18 8. a = 8, b = 8, c = ____ 9. a = 12, b = ____, c = 25 10. a = ____, b = 9, c = 15 11. a = ____, b = 10, c = 15 12. a = 25, b = 8, c = ____ To check for a 90° angle, a carpenter measures the two diagonals. Explain what principle is being used.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 60 d = r t+ Name Date Distance and Midpoint Formulas You can determine the length of a line by using the distance formula. You will need to know the coordinate of the two end points. Distance formula: d = √(x2 – x1)2 + (y2 – y1)2 Example: Find the distance between (2,5) and (–1, –3). Round to the nearest tenth. Identify x and y values. d = √(–1 – 2)2+ (–3 – 5)2 d = √–32 + –82 d = √9 + 64 d = √73 d ≈ 8 • 5 The midpoint of a line segment is determined by using the (x 1 + x2,) (y1 + y2) 2 2 Example: Find the midpoint of (2, 5) and (–1, –3) Identify x and y values. 2 +–1 5 + –( , 3 ) = ( 1 , 2 ) = ( 1, 1) 2 2 2 2 2 Find the distance between each pair of points. Determine the midpoint of each line segment. Round to the nearest tenth. 1. (4, 2), (0, 4) ______ 2. (–4, –4), (4, 4) ______ 3. (9, 10), (12, 14) ______ 4. (3, –2), (–1, 5) ______ 5. (–2, 7), (6, 1) ______ 6. (–3, –2), (4, –1)______ 7. (2, 4), (0, 6) ______ 8. (–5, 0), (–3, 6) ______ Find the perimeter of a triangle with coordinates A(2, 2), B(8, 2), and C(5, 6). Round to the nearest tenth.No. AM108R © Copyright 2008, Hayes School Publishing Co., Inc. 61 Test-Taking Strategies The following are suggestions that will help you improve your score on standardized mathematics tests. Remember, when you are preparing for a test, a good diet, regular exercise, and rest are the basic requirements to success. A Strategy for Better Test-Taking for Students I. As soon as you receive your test, write down ideas, formulas, facts, relationships, and concepts that you might forget under the pressure and confusion of the test. II. Preview the test. Write your name on the test and look over the entire test to determine length. Mark easy problems. III. Add to your fact list ideas that you may have recalled during your preview. IV. Determine how much time you should be spending on easy to hard problems. These problems may have different values. Work easy problems first—quickly, and accurately. V. Review your answers. “Does it make sense?” Write your work clearly so you can make this review work for you. VI. You may see a problem that you don’t know how to solve. Mark it not finished, and go to the next problem. Come back to this problem at the end. Depending on how the test is graded, you may not want to leave it blank. VII. Return to marked problems when you reach the end of the test. VIII. Bring a highlighter to the test so when you are reading a problem, you can highlight important information. IX. Construct tables, graphs, or concept maps to organize information. Making sense out of the confusion will help you answer even the hardest problems. X. At the end of the test, review to see that your name is on the test, that your answers are clearly marked, and that all answers are completed.HAYES® Published by Hayes School Publishing Co., Inc. 321 Pennwood Avenue Pittsburgh, PA 15221 7 3 4 6 7 5 0 1 3 2 3 1 Other titles available from Hayes Mathematics: Drill & Practice H-A279R Mathematics: Drill & Practice • Grade 1 H-A280R Mathematics: Drill & Practice • Grade 2 H-A281R Mathematics: Drill & Practice • Grade 3 H-A282R Mathematics: Drill & Practice • Grade 4 H-A283R Mathematics: Drill & Practice • Grade 5 H-A284R Mathematics: Drill & Practice • Grade 6 Mathematics Puzzles and Games H-A292R Mathematics Puzzles and Games • Grade 2 H-A294R Mathematics Puzzles and Games • Grade 3-4 H-A296R Mathematics Puzzles and Games • Grade 5-6 Arithmetic Drills H-AD211R Mastery Arithmetic Drills • Grade 1 H-AD212R Mastery Arithmetic Drills • Grade 2 H-AD213R Mastery Arithmetic Drills • Grade 3 H-AD214R Mastery Arithmetic Drills • Grade 4 H-AD215R Mastery Arithmetic Drills • Grade 5 H-AD216R Mastery Arithmetic Drills • Grade 6 H-AD217R Mastery Arithmetic Drills • Grade 7-9 Mastering the Standards: Mathematics H-AM100R Mastering the Standards: Mathematics • Kindergarten H-AM101R Mastering the Standards: Mathematics • Grade 1 H-AM102R Mastering the Standards: Mathematics • Grade 2 H-AM103R Mastering the Standards: Mathematics • Grade 3 H-AM104R Mastering the Standards: Mathematics • Grade 4 H-AM105R Mastering the Standards: Mathematics • Grade 5 H-AM106R Mastering the Standards: Mathematics • Grade 6 H-AM107R Mastering the Standards: Pre-Algebra • Grade 7+ H-AM108R Mastering the Standards: Algebra • Grade 7+ H-AM109R Mastering the Standards: Geometry • Grade 7+ Know the Essentials Mathematics: Drills & Tests H-HS500R Know the Essentials of Pre-Algebra • Grade 7+ H-HS501R Solving Algebra Mysteries thru Puzzles • Grade 7+ H-HS503R Essentials of Algebra • Grade 7+ H-HS504R Essentials of Geometry • Grade 7+ H-HS506R Know the Essentials of Algebra 2 • Grade 7+ H-HS510R Algebra for Everyday • Grade 7+ H-HS531R Know the Essentials of Fractions • Grade 6+ H-HS532R Know the Essentials of Decimals & Percents • Grade 6+ Standardized Testing H-ST202R Achieving Proficiency on Standardized Tests • Grade 2 H-ST203R Achieving Proficiency on Standardized Tests • Grade 3 H-ST204R Achieving Proficiency on Standardized Tests • Grade 4 H-ST205R Achieving Proficiency on Standardized Tests • Grade 5 H-ST206R Achieving Proficiency on Standardized Tests • Grade 6 H-ST207R Achieving Proficiency on Standardized Tests • Grade 7 H-ST208R Achieving Proficiency on Standardized Tests • Grade 8 ISBN 1-55767-565-1