The Mathematical Practice Standards are integrated throughout the text. From the Activity through the Exercises, students see where the Mathematical Practice Standards can be applied. In the Activities, there is always a callout in the minor column that relates to the Practice Standards. Students can tell from the title of the callout which Practice Standard it refers to, and forces students to look at the math and make connections to the Practice Standards. The same is true during the Lessons. These callouts appear in the minor column of the Lesson pages as well. Then during the Exercises, students and teachers will notice inline heads relating directly to the Practice Standards, like Structure, Reasoning, Modeling, Number Sense, and Precision.
Mathematical Practice Standards are called out In Laurie’s Notes in the Teaching Edition as well. Author Laurie Boswell has provided teachers with an invaluable resource to use on a daily basis. She provides ways to motivate the students each day, questions during the lesson, common errors students might make, various connections, tips for teachers, and suggestions for closure. Throughout these notes, teachers will see the Mathematical Practices in boldface text, so they will quickly see when and where the standards are being used each day.
Students also must persevere through the Activities while working with a partner and without direct instruction. Students develop their own mathematical thinking through the activities, and frequently critique the reasoning of others in the class. Students are repeatedly required to explain their reasoning and communicate mathematical ideas precisely. Students must make sense of real-life application and modeling exercises. They must plan a solution pathway and model the mathematics while justifying their reasoning. Students learn when to create diagrams, tables, and graphs, and know when these tools are useful. Students are continually exposed to the Mathematical Practices throughout each course and the entire program. For instance:
Make sense of problems and persevere in solving them.
• Every section begins with an Essential Question helping students focus throughout lessons.
• Clear stepped-out Examples encourage students to plan a solution pathway rather than jumping into a solution attempt. Guided questions and scaffolding support students' perseverance.
• Multiple representations are presented to help students move from concrete to representative and into abstract thinking.
• In Your Own Words questions provide opportunities for students to look for meaning and entry points to problems.
Reason abstractly and quantitatively.
• Throughout the series students are expected to model, deduce, and conjecture.
• Opportunities for students to decontextualize and contextualize problems are presented in every lesson.
• Essential Questions, Error Analysis exercises, and Reasoning exercises provide opportunities for students to make assumptions, examine results, and explain their reasoning.
• Specialized questions including What Is Your Answer?, In Your Own Words, You Be The Teacher, and Which One Doesn't Belong? encourage debate and sense making.
• Visual problem-solving models help students create coherent representations of problems.
Construct viable arguments and critique the reasoning of others.
• In each lesson, students work with the mathematics of everyday life. Students are asked to write stories involving math, such as using percents to help them improve their grades.
• Students use graphs, tables, charts, number lines, diagrams, flowcharts, and formulas to organize, make sense of, and identify plausible solutions to real-life situations.
• Visual representations, such as integer tiles, fraction models, tape diagrams, double number lines, and ratio tables are used to help students make sense of numeric operations.
• Error Analysis, Which One Doesn’t Belong?, and Different Words, Same Question features provide students the opportunity to construct arguments and critique the reasoning of others.
• Inductive Reasoning activities help students make conjectures and build a logical progression of statements to explore their conjecture.
Model with mathematics.
• Students learn to represent problems by consistently using verbal models and paying close attention to units and properties. This helps students manipulate the representative symbols and represent problems symbolically.
• Students are taught to contextualize by thinking about the referents and symbols involved in each problem.
• Real-life situations are translated into diagrams, tables, equations, and graphs to help students analyze relationships and to draw conclusions.
Use appropriate tools strategically.
• Opportunities for students to select and use appropriate tools such as graphing calculators, protractors, measuring devices, websites, and other external resources are provided throughout the program.
• Graphic Organizers support the thought process of what, when, and how to solve problems.
• A variety of tool papers, such as graph paper, number lines, and manipulatives, are available as students consider how to approach a problem.
Attend to precision.
• Students have the opportunity to communicate mathematically daily. Students work through Activities, Examples, and Exercises to understand and use the language of mathematics, paying careful attention to the importance of units, labeling, and quantities.
• On Your Own questions encourage students to formulate consistent and appropriate reasoning.
• Cooperative learning opportunities support precise communication.
Look for and make use of structure.
• Real-world problems are posed to encourage students to "see" these problems as being composed of several objects. Students see that some mathematical representations share common structures and learn to look for these relationships when discerning inherent patterns.
• Inductive Reasoning activities provide students the opportunity to see patterns and structure in mathematics.
Look for and express regularity in repeated reasoning.
• The program helps students see that mathematics is well-structured and predictable.
• Students are encouraged to work through a problem, not through the numbers. They consider factors such as what the problem is asking, if the intermediate steps are reasonable, and if the solution is realistic.
• Students are continually encouraged to check for reasonableness in their solutions.