The Math Teacher's Toolbox

Math teachers will find the classroom-tested lessons and strategies in this book to be accessible and easily implemented in the classroom   The Teacher’s Toolbox series is an innovative, research-based resource providing teachers with instructional strategies for students of all levels and abilities. Each book in the collection focuses on a specific content area. Clear, concise guidance enables teachers to quickly integrate low-prep, high-value lessons and strategies in their middle school and high school classrooms. Every strategy follows a practical, how-to format established by the series editors. The Math Teacher's Toolbox contains hundreds of student-friendly classroom lessons and teaching strategies. Clear and concise chapters, fully aligned to Common Core math standards, cover the underlying research, required technology, practical classroom use, and modification of each high-value lesson and strategy. This book employs a hands-on approach to help educators quickly learn and apply proven methods and techniques in their mathematics courses. Topics range from the planning of units, lessons, tests, and homework to conducting formative assessments, differentiating instruction, motivating students, dealing with “math anxiety,” and culturally responsive teaching. Easy-to-read content shows how and why math should be taught as a language and how to make connections across mathematical units. Designed to reduce instructor preparation time and increase student engagement and comprehension, this book: Explains the usefulness, application, and potential drawbacks of each instructional strategy Provides fresh activities for all classrooms Helps math teachers work with ELLs, advanced students, and students with learning differences Offers real-world guidance for working with parents, guardians, and co-teachers The Math Teacher's Toolbox: Hundreds of Practical ideas to Support Your Students is an invaluable source of real-world lessons, strategies, and techniques for general education teachers and math specialists, as well as resource specialists/special education teachers, elementary and secondary educators, and teacher educators.

List of Tables xix About the Authors xxi About the Editors xxiii Acknowledgments xxv Letter from the Editors xxvii Introduction 1 Our Beliefs about Teaching Math 2 Structure of This Book 3 Why Good Math Teaching Matters 4 I Basic Strategies 5 1. Motivating Students 7 What is It? 7 Why We Like It 8 Supporting Research 8 Common Core Connections 9 Application 10 Nurturing Student Confidence 10 Motivating Through Math 11 Rewards 14 Motivating Through Popular Culture 15 Motivating English Language Learners and Students with Learning Differences 16 Student Handouts and Examples 18 What Could Go Wrong 18 Using Fear to Motivate 18 Stereotype Threat 19 “Why Do We Need to Know This?” 19 Misreading Students 20 Limitations to Motivation 21 Technology Connections 21 Figures 22 Figure 1.1 Pattern Blocks 22 Figure 1.2 Rotational Symmetry 23 Figure 1.3 Exponential Growth 24 Figure 1.4 Identify a Void 26 2. Culturally Responsive Teaching 27 What is It? 27 Why We Like It 28 Supporting Research 28 Common Core Connections 29 Application 30 Self-Reflection 30 Building a Collaborative Learning Partnership 32 What Could Go Wrong 36 “Color-Blind” Teaching 36 Good Intentions 37 Finding the Right Time or Place 38 Technology Connections 38 3. Teaching Math as a Language 41 What is it? 41 Why We Like It 41 Supporting Research 42 Common Core Connections 42 Application 42 Eliciting the Need for Mathematical Language 42 Introducing Symbols and Terms 43 Translating Between Symbols and Words 45 Making Connections Between Math and English 46 Examples of Confusing Mathematical Language 46 Encouraging Mathematical Precision 48 Vocabulary Charts and Flash Cards 49 Visual and Verbal Aids 51 Word Walls and Anchor Charts 52 Student Handouts and Examples 53 What Could Go Wrong 53 Not Treating Math as a Language 53 Math as a “Bag of Tricks” 54 Technology Connections 55 Figures 57 Figure 3.1 Concept Attainment 57 Figure 3.2 Words and Symbols Chart 58 Figure 3.3 Why the Word “Height” is Confusing 58 Figure 3.4 Draw a Picture 59 Figure 3.5 Functions Anchor Chart 60 Figure 3.6 Polynomials Anchor Chart 61 Figure 3.7 Why the Formula a2 + b2 = c2 is Confusing 61 4. Promoting Mathematical Communication 63 What is It? 63 Why We Like It 63 Supporting Research 64 Common Core Connections 64 Application 64 Open-Ended Questions 64 Guiding Students in Conversation 71 Four-Step Thinking Process 74 Mathematical Writing 79 Differentiating for ELLs and Students with Learning Differences 87 What Could Go Wrong 87 Dealing with Student Mistakes 87 Dealing with Teacher Mistakes 88 Problems in Discourse 88 Finding the Time 89 Student Handouts and Examples 89 Technology Connections 89 Attribution 90 Figures 91 Figure 4.1 Algebra Tiles Activity 91 Figure 4.2 Which One Doesn’t Belong? 92 Figure 4.3 Error Analysis 93 Figure 4.4 Lesson Summary 95 5. Making Mathematical Connections 97 What is It? 97 Why We Like It 97 Supporting Research 98 Common Core Connections 98 Application 98 Equivalence 99 Proportionality 101 Functions 102 Variability 104 Differentiating for ELLs and Students with Learning Differences 107 Student Handouts and Examples 108 What Could Go Wrong 108 Technology Connections 109 Figures 111 Figure 5.1 Addition and Subtraction of Polynomials 111 Figure 5.2 Multiplication with the Area Model 112 Figure 5.3 Division with the Area Model 114 Figure 5.4 Completing the Square 115 Figure 5.5 Determining the Center and Radius of a Circle 115 Figure 5.6 Why (a + b)2 ≠ a2 + b2 115 Figure 5.7 Ratios and Similarity 116 Figure 5.8 Areas of Similar Polygons 117 Figure 5.9 Volumes of Similar Solids 118 Figure 5.10 Arc Length and Sector 119 Figure 5.11 Proportional Reasoning in Circles 120 Figure 5.12 Four Views of a Function 120 Figure 5.13 Rate of Change 121 Figure 5.14 Characteristics of Polynomial Functions 123 Figure 5.15 Even and Odd Polynomial Functions 124 Figure 5.16 Why f(x) = sin (x) is Odd and g(x) = cos (x) is Even 126 Figure 5.17 Linear Regression 127 Figure 5.18 Long-Run Relative Frequency 129 Figure 5.19 Two-Way Tables 131 Figure 5.20 Conditional Probability 133 II How to Plan 135 6. How to Plan Units 137 What is It? 137 Why We Like It 137 Supporting Research 138 Common Core Connections 138 Application 139 Getting Started 139 Making Connections Between Big Ideas 139 Developing a Logical Sequence 140 Organizing Topics and Problems 141 Summarizing the Unit Plan 141 Being Flexible 141 Developing Students’ Social and Emotional Learning 141 Incorporating Students’ Cultures 142 Differentiating for ELLs and Students with Learning Differences 143 Student Handouts and Examples 143 What Could Go Wrong 143 Technology Connections 145 Figures 145 Figure 6.1 Unit Plan: List of Skills 146 Figure 6.2 Unit Plan: Concept Map 147 Figure 6.3 Unit Plan: Sequence of Lessons 148 Figure 6.4 Sample Unit Plan 149 7. How to Plan Lessons 151 What is It? 151 Why We Like It 151 Supporting Research 152 Common Core Connections 152 Application 152 Defining the Lesson’s Scope 152 Introductory Activity 153 Presenting New Material Through Guided Questions 154 Practice 155 Differentiating for ELLs and Students with Learning Differences 155 Summary Activity 156 Student Handouts and Examples 157 What Could Go Wrong 157 Technology Connections 159 Figures 162 Figure 7.1 Do Now Problem 162 Figure 7.2 Lesson Plan: Standard Deviation 162 Figure 7.3 Lesson Plan: Slope-Intercept Form 166 Figure 7.4 Revised Baseball Field Word Problem 168 8. How to Plan Homework 169 What is It? 169 Why We Like It 169 Supporting Research 169 Common Core Connections 170 Application 170 Sources 171 Homework Format 171 Homework as Practice 172 Homework as Discovery 173 Homework as Transfer 173 Discussing Homework 174 Collecting Homework 175 Grading Homework 176 Differentiating for ELLs and Students with Learning Differences 177 Student Handouts and Examples 178 What Could Go Wrong 178 Students Who Don’t Do Homework 178 Mismanaging Class Time 179 Homework Review Challenges 179 Choosing the Wrong Problems 180 Technology Connections 180 Figures 183 Figure 8.1 Homework as Practice 183 Figure 8.2 Homework as Discovery—Ratios 184 Figure 8.3 Homework as Discovery—Mean Proportional Theorem 185 Figure 8.4 Homework as Discovery—Parabolas 186 Figure 8.5 Homework as Transfer—Similarity 187 Figure 8.6 Homework as Transfer—Bank Accounts 188 9. How to Plan Tests and Quizzes 189 What is It? 189 Why We Like It 189 Supporting Research 190 Common Core Connections 190 Application 190 Types of Questions 190 Test Format 193 Quiz Format 196 Reviewing for Assessments 196 Creating Scoring Guidelines for Assessments 199 Grading Assessments 202 Analyzing Test Results 203 Returning Tests 204 Differentiating for ELLs and Students with Learning Differences 207 Alternate Forms of Assessment 208 Student Handouts and Examples 208 What Could Go Wrong 208 Poor Scheduling and Preparation 209 Assessments as Classroom Management 210 Poorly Chosen Questions 210 Mistakes on Assessments 211 Student Cheating 212 Different Versions of Tests 213 Grading and Returning Assessments 214 Test Retakes and Test Corrections 215 Technology Connections 215 Test Questions, Answers, and Scoring Guidelines 215 Test Review 216 Test Analysis 216 Figures 217 Figure 9.1 Algebra I Test 217 Figure 9.2 Precalculus Test 220 Figure 9.3 Quiz 224 Figure 9.4 Creating Scoring Guidelines 225 Figure 9.5 Blank Test Corrections Sheet 226 Figure 9.6 Completed Test Corrections Sheet 228 Figure 9.7 Test Reflection Form 229 10. How to Develop an Effective Grading Policy 231 What is It? 231 Why We Like It 232 Supporting Research 232 Common Core Connections 232 Application 232 Standards-Based Grading 232 Minimum Grading Policy 234 Point Accumulation System for Grading 236 Differentiating for ELLs and Students with Learning Differences 237 More Than Just a Grade 238 What Could Go Wrong 239 Student Handouts and Examples 240 Technology Connections 240 Figures 241 Figure 10.1 Grade Calculation Sheet 241 Figure 10.2 Completed Grade Calculation Sheet 242 III Building Relationships 243 11. Building a Productive Classroom Environment 245 What is It? 245 Why We Like It 245 Supporting Research 245 Common Core Connections 246 Application 246 Making a Good First Impression 246 Learning Names 248 Getting to Know Students 248 Classroom Organization 249 Classroom Rules and Routines 250 Course Descriptions 252 Soliciting Student Opinion 253 Taking Notes 254 What Could Go Wrong 257 Classroom Tone 257 Mishandling the Teacher–Student Relationship 258 Taking Notes 259 Student Handouts and Examples 259 Technology Connections 259 Classroom Environment 259 Student Surveys 260 Note-Taking 260 Figures 261 Figure 11.1 Student Information Sheet 261 Figure 11.2 Course Description 263 Figure 11.3 Brief Handout 265 Figure 11.4 Full-Page Handout 266 Figure 11.5 Annotated Work 268 Figure 11.6 Double-Entry Journal 269 12. Building Relationships with Parents 271 What is It? 271 Why We Like It 271 Supporting Research 272 Common Core Connections 272 Application 272 Communicating with Parents 272 Addressing Parents’ Math Anxiety 273 Parent–Teacher Conferences 277 Home Visits 277 Working with Parents of Culturally Diverse Students 278 Working with Parents of Students with Learning Differences 279 What Could Go Wrong 280 Student Handouts and Examples 281 Technology Connections 281 Figures 282 Figure 12.1 Parent Communication Script 282 Figure 12.2 Parent Communication Log 283 13. Collaborating with Other Teachers 285 What is It? 285 Why We Like It 285 Supporting Research 286 Common Core Connections 286 Application 286 Discussing Values 287 Planning with Other Math Teachers 288 Interdisciplinary Collaboration 288 Observing Other Teachers 289 Co-Teaching 291 Mentoring 294 Lesson Study 294 Professional Learning Community 295 What Could Go Wrong 297 Lack of Trust 297 Reinforcing Negative Stereotypes 297 Lack of Colleagues 297 Lack of Time 298 Technology Connections 298 IV Enhancing Lessons 301 14. Differentiating Instruction 303 What is It? 303 Why We Like It 303 Supporting Research 304 Common Core Connections 305 Application 305 Differentiation by Content 305 Differentiation by Process 313 Differentiation by Product 315 Differentiation by Affect 320 What Could Go Wrong 320 Student Handouts and Examples 321 Technology Connections 321 Figures 323 Figure 14.1 Tiered Lesson—Literal Equations 323 Figure 14.2 Tiered Lesson—Midpoint 325 Figure 14.3 Curriculum Compacting—Coordinate Geometry 328 Figure 14.4 Tiered Test Questions 331 Figure 14.5 Review Sheet 331 Figure 14.6 Fill-In Review Sheet 332 Figure 14.7 Review Booklet 333 15. Differentiating for Students with Unique Needs 335 What is It? 335 Why We Like It 336 Supporting Research 336 Common Core Connections 337 Application 337 Strengths and Challenges of Students with Unique Needs 337 Techniques to Support Students with Unique Needs 340 What Could Go Wrong 348 Student Handouts and Examples 349 Technology Connections 349 Figures 351 Figure 15.1 Frayer Model (Blank) 351 Figure 15.2 Frayer Model—Perpendicular Bisector 352 Figure 15.3 Concept Map 352 16. Project-Based Learning 353 What is It? 353 Why We Like It 353 Supporting Research 354 Common Core Connections 355 Application 355 Open-Ended Classwork Problems 355 Open-Ended Homework Problems 357 Projects 358 What Could Go Wrong 367 Student Handouts and Examples 368 Technology Connections 368 Figures 369 Figure 16.1 Discovering Pi 369 Figure 16.2 Area of a Circle 370 Figure 16.3 Point Lattice Assignment 371 Figure 16.4 Paint a Room 374 Figure 16.5 Project—Bus Redesign Plan 375 17. Cooperative Learning 379 What is It? 379 Why We Like It 380 Supporting Research 380 Common Core Connections 381 Application 381 General Techniques 381 Differentiating for Students with Unique Needs 384 Examples 387 What Could Go Wrong 398 Student Handouts and Examples 399 Technology Connections 400 Figures 401 Figure 17.1 Jigsaw as Practice 401 Figure 17.2 Jigsaw as Discovery 402 Figure 17.3 Factoring Station 403 Figure 17.4 Peer Editing 404 18. Formative Assessment 405 What is It? 405 Why We Like It 405 Supporting Research 406 Common Core Connections 406 Application 406 Asking the Right Questions 407 Eliciting Student Responses 409 Responding to Student Answers 412 Other Methods of Formative Assessment 412 Differentiating Formative Assessment 413 What Could Go Wrong 414 Technology Connections 415 19. Using Technology 417 What is It? 417 Why We Like It 417 Supporting Research 418 Common Core Connections 418 Application 418 Classroom Organization 418 Mathematical Content 422 Using Technology for Culturally Responsive Teaching 425 Using Technology to Differentiate Instruction 425 What Could Go Wrong 425 Student Handouts and Examples 427 Technology Connections 428 Figures 429 Figure 19.1 Simulation of 1,000 Coin Flips 429 Figure 19.2 Transformations of Functions 429 Figure 19.3 Centroid of a Triangle 431 Figure 19.4 Two Views of a Graph Using Technology 432 20. Ending the School Year 433 What is It? 433 Why We Like It 433 Supporting Research 433 Common Core Connections 434 Application 434 Review 434 Reflection 438 Recognition 439 Maintaining Relationships with Students 440 Differentiating Year-End Activities 440 What Could Go Wrong 441 Year-End Fatigue 441 “What Can I Do to Pass?” 441 Running Out of Time 442 Technology Connections 443 Appendix A: The Math Teacher’s Toolbox Technology Links 445 References 461 Index 515