Common Core State Standards The Wisconsin State Standards High School Mathematics Conceptual Categories • Number and Quantities • Algebra • Functions • Modeling • Geometry • Statistics and Probability High School Mathematics Conceptual Category: Number and Quantity The Real Number System • Extend the properties of exponents to rational exponents. • Use properties of rational and irrational numbers. Quantities★ • Reason quantitatively and use units to solve problems. The Complex number System • Perform arithmetic operations with complex numbers. • Represent complex numbers and their operations on the complex plane. • Use complex numbers in polynomial identities and equations. Vector and matrix Quantities • Represent and model with vector quantities. • Perform operations on vectors. • Perform operations on matrices and use matrices in applications. High School Mathematics Conceptual Category: Algebra Seeing Structure in Expressions • Interpret the structure of expressions •Write expressions in equivalent forms to solve problems. Arithmetic with Polynomials and Rational Expressions • Perform arithmetic operations on polynomials • Understand the relationship between zeros and factors of polynomials. • Use polynomial identities to solve problems • Rewrite rational expressions. Creating Equations • Create equations that describe numbers or relationships Reasoning with Equations and Inequalities • Understand solving equations as a process of reasoning and explain the reasoning. • Solve equations and inequalities in one variable. • Solve systems of equations. • Represent and solve equations and inequalities graphically. High School Mathematics Conceptual Category: Functions Interpreting Functions • Understand the concept of a function and use function notation. • Interpret functions that arise in applications in terms of the context. • Analyze functions using different representations. Building Functions • Build a function that models a relationship between two quantities. • Build new functions from existing functions. Linear, Quadratic, and Exponential Models • Construct and compare linear, quadratic, and exponential models and solve problems. • Interpret expressions for functions in terms of the situation they model. Trigonometric Functions • Extend the domain of trigonometric functions using the using circle. • Model periodic phenomena with trigonometric functions. • Prove and apply trigonometric identities. High School Mathematics Conceptual Category: Modeling Examples: • Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed. • Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player. • Designing a layout of the stalls in a school fair so as to raise as much money as possible. • Analyzing stopping distance for a car. • Modeling savings account balance, bacterial colony growth, or investment growth. • Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport. • Analyzing risk in situations such as extreme sports, pandemics, and terrorism. • Relating population statistics to individual predictions. The basic modeling cycle is: (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle. High School Mathematics Conceptual Category: Geometry Congruence • Experiment with transformations in the plane. • Understand congruence in terms of rigid motions. • Prove geometric theorems. • Make geometric constructions. Similarity, Right Triangles, and Trigonometry • Understand similarity in terms of similarity transformations. • Prove theorems involving similarity. * Define trigonometric ratios and solve problems involving right triangles. • Apply trigonometry to general triangles. Circles • Understand and apply theorems about circles. • Find arc lengths and areas of sectors of circles. Expressing Geometric Properties with Equations • Translate between the geometric description and the equation for a conic section. • Use coordinates to prove simple geometric theorems algebraically. Geometric Measurement and Dimension • Explain volume formulas and use them to solve problems. • Visualize relationships between two-dimensional and three-dimensional objects Modeling with Geometry • Apply geometric concepts in modeling situations. High School Mathematics Conceptual Category: Statistics and Probability Interpreting Categorical and Quantitative Data • Summarize, represent, and interpret data on a single count or measurement variable. • Summarize, represent, and interpret data on two categorical and quantitative variables. • Interpret linear models. Making Inferences and Justifying Conclusions • Understand and evaluate random processes underlying statistical experiments. • Make inferences and justify conclusions from sample surveys, experiments and observational studies. Conditional Probability and the Rules of Probability • Understand independence and conditional probability and use them to interpret data. • Use the rules of probability to compute probabilities of compound events in a uniform probability model. Using Probability to Make Decisions • Calculate expected values and use them to solve problems. • Use probability to evaluate outcomes of decisions. High School Mathematics Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. 2 Reason abstractly and quantitatively 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 5 Use appropriate tools strategically. 6 Attend to precision. 7 Look for and make use of structure. 8 Look for and express regularity in repeated reasoning. Key CCSS Summary These Standards are not intended to be new names for old ways of doing business. They are a call to take the next step. It is time for states to work together to build on lessons learned from two decades of standards based reforms. It is time to recognize that standards are not just promises to our children, but promises we intend to keep. (page 5) 1 Make sense of problems and persevere in solving them. • Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. • They analyze givens, constraints, relationships, and goals. • They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. • They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. … more • They monitor and evaluate their progress and change course if necessary. • Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. • Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. … more • Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. • Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” • They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. !! Challenge Consider the Olympic data set, and the research question: Will women ever run faster than men in the 200-meter race, and if yes, when? Design a task (lesson) that involves the above data and question. Include in the task connections to a conceptual category, cluster and important standards. More importantly for our current discussion, connect the task Practice #1. Common Core State Standards Alignment Task: Textbook or curriculum resource: Grade Level ______ Part 1: Planning Description of Task(s): For high school, identify the Conceptual Category Domain(s): Big Mathematics Ideas: Cluster(s): Standards for Mathematical Practice: Description of expected or intended student learning and behaviors regarding the practice: Learning Intention: Success Criteria: Vertical Connections Prerequisite Concepts: Future Standards Where are students coming from? Where are students headed?