Table of Contents Teacher Resources Research to Support Algebra Intervention.... 5 Correlation to the NCTM Standards......... 9 How to Use This Program................ 12 Planning Your Intervention Program........ 15 Algebra Intervention Pacing Plan.......... 18 Differentiation Strategies................ 19 Data-Driven Instruction.................. 27 Manipulatives......................... 33 Teacher Resource CD................... 34 Professional Development DVD............ 39 References Cited....................... 40 Lesson Index.......................... 41 Lesson Plans Week 1............................ 43 65 Game of the Week: Math Bee............. 44 Lesson 1: Diagnostic Pre-test............. 45 Lesson 2: Adding Integers................ 47 Lesson 3: Multiplying and Dividing Integers..................... 54 Lesson 4: Writing One-Variable Equations.......................... 58 Lesson 5: Week 1 Review................ 64 Week 2............................ 66 89 Game of the Week: Who Has?............ 67 Lesson 6: Collecting Like Terms........... 68 Lesson 7: Distributing and Collecting....... 73 Lesson 8: Solving Linear Equations 1....... 76 Lesson 9: Solving Linear Equations 2....... 84 Lesson 10: Week 2 Review............... 88 Week 3........................... 90 108 Game of the Week: Coordinate Plane Game......................... 91 Lesson 11: Solving Equations............. 92 Lesson 12: Coordinate Planes............. 95 Lesson 13: Relations and Functions........ 99 Lesson 14: Equations as Relations........ 104 Lesson 15: Week 3 Review.............. 107 Week 4.......................... 109 125 Game of the Week: Algebra Aces 1....... 110 Lesson 16: Slope-Intercept Form.......... 111 Lesson 17: Point-Slope Form............. 115 Lesson 18: Graphing Linear Equations..... 118 Lesson 19: Graphing Inequalities......... 121 Lesson 20: Week 4 Review.............. 124 Week 5.......................... 126 148 Game of the Week: Algebra Aces 2....... 127 Lesson 21: Solving Multistep Inequalities... 128 Lesson 22: Compound Inequalities........ 132 Lesson 23: Absolute Value Inequalities..... 138 Lesson 24: Graphing Two-Variable Inequalities........................ 143 Lesson 25: Week 5 Review.............. 147 Week 6.......................... 149 164 Game of the Week: Algebra Aces 3....... 150 Lesson 26: Comparing Systems........... 151 Lesson 27: Substitution Method for Solving Systems.................... 154 Lesson 28: Addition Method for Solving Systems.................... 158 Lesson 29: Systems of Equations......... 162 Lesson 30: Week 6 Review.............. 163
Table of Contents (cont.) Lesson Plans (cont.) Week 7.......................... 165 181 Game of the Week: Trash ketball........ 166 Lesson 31: Proportions................. 167 Lesson 32: Scientific Notation............ 169 Lesson 33: Mixed Operations with Monomials........................ 172 Lesson 34: Adding Polynomials........... 177 Lesson 35: Week 7 Review.............. 180 Week 8.......................... 182 205 Game of the Week: Factoring MATHO..... 183 Lesson 36: Factoring the Greatest Common Factor..................... 184 Lesson 37: Factoring Trinomials.......... 188 Lesson 38: Factoring Special Types of Problems.......................... 195 Lesson 39: Radical Expressions........... 199 Lesson 40: Week 8 Review.............. 204 Week 9.......................... 206 230 Game of the Week: Radicals MATHO...... 207 Lesson 41: Solving Radical Equations...... 208 Lesson 42: Parabolas and Quadratic Equations......................... 213 Lesson 43: Quadratic Formula............ 218 Lesson 44: Rational Expressions and Equations......................... 224 Lesson 45: Diagnostic Post-test........... 230 Appendices Appendix A: Diagnostic Test....... 232 240 Appendix B: Assessments.......... 241 253 Working with Integers Quiz............. 241 Mixed Integers Quiz................... 243 Distributing and Collecting Quiz.......... 244 Writing Equations of Lines Quiz.......... 245 Solving Compound Inequalities Quiz....... 247 Solving Systems of Equations Quiz........ 249 Factoring Special Types of Problems Quiz... 250 Solving Quadratic Equations Quiz......... 252 Appendix C: Games............... 254 267 Math Bee Cards....................... 254 Who Has? Cards...................... 255 Algebra Aces 1 Cards.................. 257 Algebra Aces 2 Cards.................. 260 Algebra Aces 3 Cards.................. 263 Factoring MATHO Answer Sheet.......... 266 Radicals MATHO Answer Sheet.......... 267 Appendix D: Answer Keys......... 268 280 Guided Practice Book Answer Key........ 268 Assessments Answer Key................276 Transparencies Answer Key.............. 279 Transparencies #11002 (i2567) Algebra I Crash Course Teacher Created Materials Publishing
Research to Support Algebra Intervention By Sharon S. Whitehead, Ph.D. Secondary Math Specialist, Mesa, Arizona Teacher Resources High school students study algebra more than any other course in mathematics (Usiskin 1980). Perhaps because of the abstractness of the material, the ages of the students, the pedagogical styles of the teachers, or the complexity of the curriculum, algebra can be very difficult and stressful for many students. However, students will have difficulty in all mathematical courses subsequent to Algebra I if they do not master the material in this pivotal course. And, almost half of the states now require students to pass Algebra I before graduation. So, the question becomes, How can we best help these students who are having difficulty with the concepts presented in Algebra I? The first thing to do is to remind teachers why algebraic thinking is so important. Algebra helps students engage in problem solving and develop rich analytical skills that will serve them throughout their lifetimes. The type of thinking encouraged in an Algebra I course helps students seek solutions, not just memorize procedures; explore patterns, not just memorize formulas; and formulate conjectures, not just do exercises. The National Research Council (1989) encourages this type of thinking to be developed in mathematics courses. Active Algebra: Algebra I Crash Course models all of these practices. The National Council of Teachers of Mathematics (NCTM) (2000) challenges teachers to imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. Algebra I Crash Course is a high-quality, engaging curriculum designed for grades 8 through 12. It can be used for standardized test preparation, credit recovery, Algebra I intervention, and summer school. This program is ideally suited for students having trouble with Algebra I. Not every student learns the same way, and the flexibility of this program allows teachers to engage students in different ways. The program is set up in such a way that it helps students grasp complex algebra concepts with a conceptual understanding that ensures deep procedural knowledge. Conceptual knowledge can be defined as knowledge that involves relations or connections (but not necessarily rich ones) (Star 2005). Star defines deep procedural knowledge as involving comprehension, flexibility, and critical judgment [as] distinct from (but possibly related to) knowledge of concepts (p. 116). Too often, mathematical knowledge is placed only into one category or the other. This eliminates the necessity of the dialogue between both conceptual and procedural knowledge. Crash Course moves students between these two types of knowledge fluidly.
Correlation to the NCTM Standards Teacher Resources NCTM Standard Lesson Title and Page Number Understand the meaning and effects of arithmetic operations with fractions, decimals, and integers. Adding Integers (pp. 47 53) Multiplying and Dividing Integers (pp. 54 57) Write equivalent forms of equations, inequalities, and systems of equations and solve them with fluency mentally or with paper and pencil in simple cases and using technology in all cases. Writing One-Variable Equations (pp. 58 63) Point-Slope Form (pp. 115 117) Graphing Inequalities (pp. 121 123) Solving Multistep Inequalities (pp. 128 131) Compound Inequalities (pp. 132 137) Absolute Value Inequalities (pp. 138 142) Graphing Two-Variable Inequalities (pp. 143 146) Comparing Systems (pp. 151 153) Substitution Method for Solving Systems (pp. 154 157) Addition Method for Solving Systems (pp. 158 161) Systems of Equations (p. 162) Rational Equations (pp. 224 229) Develop an initial conceptual understanding of different uses of variables. Collecting Like Terms (pp. 68 72) Distributing and Collecting (pp. 73 75) Understand the meaning of equivalent forms of expressions, equations, inequalities, and relations. Solving Linear Equations 1 (pp. 76 82) Solving Linear Equations 2 (pp. 84 87) Solving Equations (pp. 92 94) Equations as Relations (pp. 104 106) Factoring the Greatest Common Factor (pp. 184 187) Factoring Trinomials (pp. 188 194) Factoring Special Types of Problems (pp. 195 198) Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments.
Teacher Resources Correlation to the NCTM Standards (cont.) NCTM Standard Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. Lesson Title and Page Number Coordinate Planes (pp. 95 98) Make and use coordinate systems to specify locations and to describe paths. Coordinate Planes (pp. 95 98) Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations. Relations and Functions (pp. 99 103) Explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope. Slope-Intercept Form (pp. 111 114) Graphing Linear Equations (pp. 118 120) Develop an understanding of large numbers and recognize and appropriately use exponential, scientific, and calculator notation. Scientific Notation (pp. 169 171) Understand and use ratios and proportions to represent quantitative relationships. Proportions (pp. 167 168) Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments. 10 #11002 (i2567) Algebra I Crash Course Teacher Created Materials Publishing
Teacher Resources Correlation to the NCTM Standards (cont.) NCTM Standard Understand and compare the properties of classes of functions, including exponential, polynomial, rational, logarithmic, and periodic functions. Lesson Title and Page Number Mixed Operations with Monomials (pp. 172 175) Adding Polynomials (pp. 177 179) Radical Expressions (pp. 199 203) Solving Radical Equations (pp. 208 212) Parabolas and Quadratic Equations (pp. 213 217) Quadratic Formula (pp. 218 223) Rational Equations (pp. 224 229) Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship. Solving Radical Equations (pp. 208 212) Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments. 11
Appendix A Name Directions: Solve each equation. 1. 5 6 = A. 11 C. 11 B. 1 D. 1 2. 4 + 7 = A. 12 C. 12 B. 3 D. 3 3. 7 15 = A. 8 C. 8 B. 22 D. 22 4. ( 4)(3) = A. 12 C. 7 B. 9 D. 12 5. 24 4 = A. 20 C. 6 B. 6 D. 28 6. (6)( 3)( 2) = A. 11 C. 1 B. 36 D. 36 Directions: Read each question and circle the correct answer. 7. Find the expression for the given problem. A chicken sandwich costs $1.50 more than a hamburger. A. h + $1.50 C. $1.50h B. h $1.50 D. $1.50 h Diagnostic Test 8. Find an equation for the given problem. The width of a rectangle is 9 meters. The width is 3 less than twice the length. What is the equation for the length of the rectangle? A. 3 2x = 9 C. 2x 3 = 9 B. 2x + 3 = 9 D. 3 + 2x = 9 Directions: Simplify each given expression. 9. 4x 7 + 3x 2 x + 3 x 2 + 7x A. 4x 2 + 8x + 4 C. 2x 2 10x + 4 B. 2x 2 10x + 4 D. 2x 2 + 10x 4 10. 10 m m 2 13 4m + 5 + 3m 2 m A. 4m 2 6m + 2 C. 2m 2 6m + 2 B. 2m 2 + 6m 2 D. 4m 2 + 6m 2 11. m + 10 (6m + 4) 3 A. 5m + 3 C. 5m + 11 B. 5m + 3 D. 5m 11 12. 4x 3(2x + 3) + 7x + 12 A. 3x + 3 C. 3x + 3 B. 3x + 21 D. 3x 21 Directions: Solve for each variable. 13. 3x 5 = 4x + 3 A. x = 8 7 C. x = 8 B. x = 8 D. x = 8 7 232 #11002 (i2567) Algebra I Crash Course Teacher Created Materials Publishing
Data-Driven Instruction Teacher Resources Standards-based instruction must begin with the goal that all students can reach the standards if given appropriate instruction, materials, and support. Therefore, it is the responsibility of teachers to use various assessment and data-collection strategies to help identify when a student has misconceptions that will affect further learning of mathematical skills and concepts. Frequent formal and informal assessments provide teachers with the data needed to make informed decisions about what to teach and how to teach it. This is the only way that teachers will know who is struggling with various concepts and how to address the difficulties that students are experiencing with the curriculum. Formal Assessments The following formal assessments are included throughout the Algebra I Crash Course program: Diagnostic Test This assessment is given at the beginning of the program and at the end of the program. When students are retested, they should have the same classroom conditions and directions. This assessment is meant to be given within the increments of one school year and not stretched out beyond a single school year of instruction. In this way, the increments between testing are not too stretched out to detect growth. The final post-test use of the assessment becomes the ultimate summative evaluation that measures whether students have mastered what they were taught throughout the whole program. Quizzes The quizzes are integrated into the lessons throughout the program. They measure the learned objectives for a specific lesson or series of lessons. These assessments can be used by teachers to determine whether reteaching is necessary. There are two versions of each quiz. Form A is in the appendix and Form B is only on the Teacher Resource CD. Use both versions at the same time to discourage copying, or use one version as the initial assessment and the other version as the retest. Informal Assessments The following informal assessments are included throughout the program. These allow teachers to frequently watch for markers of comprehension in student responses. Problems in the Lessons Within each lesson, practice problems are provided for modeling. In the lessons, teachers first model the problem while the students watch. Then, the teachers model the problems while the students help. Next, the students model the problems while the teacher helps. Finally, the students model the problems while the teacher watches. Guided Practice Book This student consumable contains activity sheets to be used in correlation with lessons. These activities should be used to practice skills and concepts, and it is recommended that these activities are only given a completion grade versus a formal grade. 27
Game of the Week: Radicals MATHO Week 9 Mathematics Game Description The object of the game is to be the first student to have a straight line of answers covered on his or her MATHO card. Materials Transparencies folder Radicals MATHO Problems (trans19.pdf) Appendix C: Games Radicals MATHO Answer Sheet (page 267; radmatho.pdf) Games folder on the CD MATHO cards (mathocrd.pdf) Games folder on the CD MATHO chips (mathochp.pdf) Procedure Step 1 Step 2 Cut the Radicals MATHO Problems transparency into strips with one problem and corresponding letter on each strip. Give each student a MATHO card and a handful of small game markers/chips. Students also need copies of the Radicals MATHO Answer Sheet. Explain the directions to the student: This game is played like Bingo. Put one problem strip at a time on the overhead. Have the students solve each problem. Note: The problem is printed on one end of the strip and the answer on the other for easy location of winning answers. When placing a problem on the overhead, make sure the answer is not visible. It should be hanging off the side. Require all students to play. They should write the problems and answers in an organized manner on notebook paper. If necessary, give students grades for this work. Once students have solved each problem, they should look on their MATHO cards for the answer. If a student s card has the right answer, he or she covers it with a marker. Tell students to call out, MATHO when they get a straight line of covered answers. You can also play for diagonal lines or blackouts (the whole card). Have the winning student(s) call out their answers and check them on the transparency strips that have been played. 207
Lesson 41 Solving Radical Equations Step 1 20 min. Step 2 20 min. Step 1 Lesson Description Understands basic concepts, applications, and solution methods of radical equations. Students will understand the Pythagorean theorem and learn how to solve radical equations. Materials PowerPoint folder on the CD Solving Radical Equations (lesson41.ppt) (optional) Guided Practice Book Solving Radical Equations (page 134; page134.pdf) Procedure Step 3 25 min. Step 4 25 min. Present the Notes on the Pythagorean Theorem (pages 208 209). These notes are provided as part of the lesson s PowerPoint slide show on the CD (lesson41.ppt). Discuss the bolded questions with students. You may want to share some of the history of the Pythagorean theorem with students. Notes on the Pythagorean Theorem What is the Pythagorean theorem? a 2 + b 2 = c 2 a 2 c2 What is the hypotenuse? It is represented by the variable c. It is the longest side. It is always the side across from the right angle. What are the legs? They are represented by the variables a and b. When solving for one of the variables, the order of a and b can be switched because of the commutative property. b 2 208 #11002 (i2567) Algebra I Crash Course Teacher Created Materials Publishing
Solving Radical Equations Lesson 41 Notes on the Pythagorean Theorem (cont.) When is the Pythagorean theorem used? It is used to find the length of the third side of a right triangle when the other two sides are known. It is used to determine if a right triangle can be made from three given line segments. How could the Pythagorean theorem help carpenters? Discuss with students how buildings are constructed with right angles. Examples Complete these examples with the students by working out the formula for each triangle. Round the answers to the nearest hundredth. a. 3 5 b. x 24 c. 13 c x 25 3 2 + x 2 = 5 2 9 + x 2 = 25 x 2 = 25 9 x 2 = 16 x 2 = 16 x = 4 Teacher Tip 24 2 + x 2 = 25 2 576 + x 2 = 625 x 2 = 625 576 x 2 = 49 x 2 = 49 x = 7 If available, you may want students to check their solutions on graphing calculators using the Home Screen method. Store a value for a variable, input the equation, and press enter. The calculator will use Boolean logic to determine if the value stored is true (1) or false (0). 17 13 2 + 17 2 = c 2 169 + 289 = c 2 458 = c 2 458 = c 2 c 21.40 209
Lesson 41 Solving Radical Equations Step 2 Practice Procedure (cont.) Review radical operations. d. 6 4 24 + 5 6 = 2 6 e. 2 9 = 3 2 f. 3 3(2 6 2 2 + 5 8) = 18 2 + 24 6 g. (4 2 + 5 3)(4 2 5 3) = 43 h. (2 6 4)(2 6 + 4) = 8 i. ( 2 5)( 2 + 5) = 3 Step 3 Present the Notes on Solving Radical Equations (pages 210 212). These notes are given as part of the lesson s PowerPoint slide show on the CD (lesson41.ppt). Notes on Solving Radical Equations Examples j. m + 3 = 8 3 3 ( m) 2 = (5) 2 m = 25 210 #11002 (i2567) Algebra I Crash Course Teacher Created Materials Publishing
Solving Radical Equations Lesson 41 Notes on Solving Radical Equations (cont.) Examples k. p 2 + 5 = 10 5 5 ( p 2) 2 = (5) 2 p 2 = 25 +2 +2 p = 27 l. 3y = 5 ø Discuss the principle square root in the above example. Tell students that the symbol represents the principle square root and that it is a positive root. Ask students, Can a principle square root equal a negative value? No; therefore, the answer to the problem is ø. m. y + 3 = y + 1 ( y + 3) 2 = (y + 1) 2 y + 3 = y 2 + 2y + 1 y 3 y 3 0 = y 2 + y 2 0 = (y + 2)(y 1) y + 2 = 0 y 1 = 0 2 2 + 1 +1 y = 1 Remind students to substitute solutions into the original equation to find the correct answer. 211
Lesson 41 Solving Radical Equations Procedure (cont.) Notes on Solving Radical Equations (cont.) n. 3 x 3 = x 3 x = x + 3 ( 3 x ) 2 = (x + 3) 2 3 x = x 2 + 6x + 9 0 = x 2 + 7x + 6 0 = (x + 6)(x + 1) x = 6 x = 1 x = 1 o. 2 a 4 = 2 a = 54 3 p. x = 3 2 x = 18 q. 4n 2 = 8n n = 0, 2 r. 3 2y = 1 y = 2 s. 4z + 6 = 0 z = t. 2y 2 + 8 = 6 ø 3 2 Remind students that a principle square root cannot equal a negative value. Step 4 Assignment Assign the Solving Radical Equations activity sheet. 212 #11002 (i2567) Algebra I Crash Course Teacher Created Materials Publishing
Radicals MATHO Answer Sheet Appendix C: Games A (5x + 1)(2x + 3) B (7x 2)(x 1) C (x 1)(x 1) D 2(5x 1)(5x + 1) E x(x + 1) N (5x + 1)(2x 1) O (2x + 3)(x + 1) P 2x(2x 3) Q (3x + 1)(x + 2) R (8x 1)(x 1) F 2x 2 y(5 6xy) S (3x 5)(x + 1) G x 2 (x + 10)(x 2) H (1 2x)(1 + 2x) I (4x + 3)(4x 3) J (x + 3)(x 7) K prime L (5x + 1)(5x 1) M (5x + 4)(x 1) T 3x 3 (x 2 + 2) U 2(x + 9)(x + 2) V 5x(x 4)(x + 2) W (x + 3)(x 3) X 2xy 2 (7x 2 4y) Y (3x 1)(x + 1) Z (2x 1)(x + 3) 267
Name Solving Radical Equations Directions: Solve. 1. 2x = 7 2. p 10 = 0 3. 3 9y = 12 4. 2a + 4 = 0 5. 6 = 2 6. x 2 = x 4 7. 4x = 8 8. a = 3 5 9. 3 + m + 3 = m 134 #10361 Algebra I Crash Course, Guided Practice Book
Appendix D: Answer Keys Standardized Test Practice 7 (cont.) 17. A 18. D 19. D 20. C Dimensions of Rectangles GPB (pages 120 121) 1. (x + 3)(x + 1) A = x 2 + 4x + 3 2. (x 5)(x + 2) A = x 2 3x 10 3. (2x + 3)(x + 1) A = 2x 2 + 5x + 3 4. (x + 1)(x + 1) A = x 2 + 2x + 1 5. (x + 2)(x 2) A = x 2 4 6. (2x + 3)(x + 5) A = 2x 2 + 13x + 15 7. (2x + 1)(x + 2) A = 2x 2 + 5x + 2 8. (x + 5)(x 1) A = x 2 + 4x 5 9. (a + 5)(a 3) 10. (m 7)(m + 7) 11. (3m 5)(2m + 1) 12. (c 6)(c + 6) 13. (a 7b)(a 3b) 14. (3e + 7)(e 3) 15. (2a 3)(a 4) 16. e(e 5) 17. w(w 5) Area and Perimeter with Radicals GPB (pages 122 123) 1. 8 2 2. 5 3 + 6 5 3. 11 7 + 7 5 4. 6 2 + 2 10 5. 7 5 + 5 6. 20 2 7. 8 + 2 3 8. 20 3 9. 18 2 10. 15 2 11. 144 12. 24 13. 2 14. 2 30 15. 40 16. 8 Guided Practice Book Answer Key (cont.) 17. 48 5 18. 24 3 19. 150 20. 10 21. 48 22. 120 2 Algebra I Review 8 GPB (pages 124 127) 1. 4 hours 2. 40 people 3. C 4. D 5. C 6. y = 3x + 2 7. m = slope = rise run b = y-intercept 8. 3,4,5; 5,12,13; 7,24,25; 8,15,17; 9,40,41 9. C 10. check graph 11. check graph 12. Perimeter add all sides; 2l + 2w Area length x width 13. D = set of all x s or independent variable or 1 st coordinate R = set of all y s or dependent variables or 2 nd coordinate 14. t = 2s r = 3b b = f + 8 s = 1 3 r 15. n = 2 m = 8 16. use one variable 17. 1 variable: 4w + 8 = 38 or 2w + 2(w + 4) = 38; 2 variables: 2w + 2l = 38, l = w + 4 18. D 19. A Standardized Test Practice 8 GPB (pages 128 133) 1. B 2. C 3. C 4. A 5. B 6. D 7. A 8. A 9. B 10. D 11. B 12. D 13. C 14. A 15. B 16. A 17. B 18. B 19. D 20. D Solving Radical Equations GPB (page 134) 1. x = 49 2 2. p = 100 3. y = 25 4. Ø 5. x = 256 3 6. x = 4 7. x = 16 8. a = 45 9. m = 2, m = 3 Identifying the Axis of Symmetry and the Vertex GPB (page 135) 1. opens downward 2. (2, 64) 3. The x-axis represents time in seconds, and the y-axis represents the position of the ball in feet. 4. It tells us that at the time of two seconds, the ball is at its maximum height of 64 feet. 5. x = 2 Simplifying Rational Expressions GPB (page 136) 1. x + 2 6x 3 2. x 3 3. mn 2 y 2 4. 3x 2 5. 2(x 3) 2x + 7 275