1 of 6 10/31/2013 2:18 PM Math Model Unit 3.1 Algebra 2 East Bay Overview Algebra 2, Quarter 3, Unit 3.1 Solving Systems of Polynomial, Rational, Radical, Absolute Value, and Exponential Equations and Inequalities Both Graphically and Algebraically (10 days) Content to be Learned Exactly and approximately solve systems of equations with three unknowns. Solve a simple systems of equations consisting of a linear equation and a quadratic equation algebraically and graphically. Solve systems of equations and inequalities graphically, including polynomial, rational, radical, absolute value, linear, logarithmic and exponential functions using multiple representations and technology. Explain why the x-coordinates of the points where the graphs intersect are the solutions to the system. Explain why the intersection of two functions graphed on a coordinate plane (y = f(x) and y = g(x)) is the solution to the equation f(x) = g(x). Graph the solution set to a system of linear inequalities in two variables as the intersection of two half-planes. Represent constraints with equations and/or systems of equations. Interpret solutions as viable or not viable in a modeling context. Mathematical Practices to be Integrated Make sense of problems and persevere in solving them. Analyze relationships within a system and the meaning of its solution(s). Analytically evaluate a system to determine a feasible viewing window. Choose an appropriate method to obtain a solution. Construct viable arguments and critique the reasoning of others. Explain why the intersection of two functions is the solution to a system and justify why it makes sense to the given problem. Model with mathematics.
ath Model Unit 3.1 Algebra 2 East Bay of 6 10/31/2013 2:18 PM Create a representative model to solve a system. Use appropriate tools strategically. Utilize technology to explore and deepen understanding of systems. Use technology to help visualize systems and their solutions. Decide when and what tools are appropriate. Attend to precision. Obtain a solution that is accurate within the parameters of a system. Essential Questions What do all types of systems of equations have in common? What are the similarities and differences of roots, zeros, solutions, x-intercepts, and factors? Why would a system have no, one, or many solutions? How can you apply the solution of a system to a real-world context? How do you interpret the intersection of two graphs in the context of a problem? How do you determine the best method for solving a system of equations? Written Curriculum Common Core State Standards for Mathematical Content Algebra Reasoning with Equations and Inequalities A-REI Solve systems of equations A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. Represent and solve equations and inequalities graphically [Combine polynomial, rational, radical, absolute value, and exponential functions] A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Common Core State Standards for Mathematical Practice
3 of 6 10/31/2013 2:18 PM 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. 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. 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. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
ath Model Unit 3.1 Algebra 2 East Bay of 6 10/31/2013 2:18 PM 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In grade 1, students understood the meaning of the equals sign. In grades 2 and 3, students solved problem using the four operations. In grade 3, they developed an understanding of fractions as numbers. In grade 4, students extended their understanding of fractional equivalence. They gained familiarity of factors and multiples. In grade 5, students wrote and interpreted numerical expressions through the use of parentheses, brackets, or braces. In grades 5 and 6, they applied and extended previous understandings of multiplication and division to multiply and divide fractions. In grade 6, they computed fluently with multidigit numbers and found common factors and multiples. In grade 7, students applied properties of operations as strategies to add, subtract, multiply, and divide, rational numbers. In grade 8, students solved linear equations in one variable. Students analyzed and solved pairs of simultaneous linear equations. In algebra 1, students solved systems of linear equations in two variables. Students learned to graph points on the coordinate plane and interpreted the values of coordinates in the context of the situation in grade 5. In grade 6, students learned the process of finding a solution set for an equation, and they solved equations of the form x + p = q and px = q. They learned to write inequalities and represent solutions to a simple inequality on a number line. They also learned to write equations to describe relationships between two quantities. In grade 7, students used the
5 of 6 10/31/2013 2:18 PM properties of operations to generate equivalent linear expressions. They also learned to use algebraic equations and inequalities to solve word problems. In grade 8, students solved linear equations in two variables, including equations with 1, 0, and infinitely many solutions, and equations in which they applied the distributive property and collected like terms. Students began to develop techniques to solve and analyze systems of equations algebraically and graphically. They learned that the intersection of the graphs of two linear equations represented the solution to a system of equations. Finally, students learned the definition of a function, compared functions represented two different ways, and wrote functions to model linear relationships. They also examined and explained functions that were not linear. Current Learning As additional topics in algebra 2, students extend the process of solving systems of linear equations exactly and approximately in three variables. As additional content, they solve a simple system consisting of a linear equation and a quadratic equation algebraically and graphically. They solve systems including polynomial, rational, radical, absolute value, linear, logarithmic and exponential functions using technology, making tables or by finding successive approximations. As a major topic in algebra 2, they explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x) Students are expected to be fluent with the use of systems of linear equations and inequalities in two variables. In this unit, students solve systems of equations exactly and approximately after proving different methods for solving systems. Students represent and solve systems of equations and inequalities including linear linear and linear quadratic systems. They explain why the x-coordinates of the intersection points of two graphs are the solutions to the equation. Students use technology to graph two functions, make tables of values, and find successive approximations. Students graph the solution to a system of linear inequalities as the intersection of two half-planes. They represent constraints using equations, inequalities, and systems of equations, and they interpret solutions as viable or nonviable options in a modeling context. Future Learning Students will solve systems by using matrices in precalculus. They will see system solving in areas such as business, engineering, physics, astronomy, biology, and earthquake analysis. Further study of systems of equations will occur in precalculus, where students will represent a system of linear equations as a matrix equation on a vector. They will also use the inverse of a matrix to solve systems of linear equations, using technology for larger systems. In advanced mathematics courses, including linear algebra and differential equations, students will represent systems of multiple equations with matrices. Systems of equations, both linear and nonlinear, will be essential for student success in advanced courses in physics, economics, and chemistry. Additional Findings Students often struggle with seeing how a function can model a real-world problem. Using technology, teachers can help students model and understand mathematical concepts. With utilities for symbol manipulation, graphing, and curve fitting and with programmable software and spreadsheets to represent iterative processes, students can model and analyze a wide range of phenomena. These mathematical tools can help students develop a deeper understanding of
6 of 6 10/31/2013 2:18 PM real-world phenomena. At the same time, working in real contexts may help students make sense of the underlying mathematical concepts and may foster an appreciation of those concepts... In helping high school students learn about the characteristics of particular classes of functions, teachers may find it helpful to compare and contrast situations that are modeled by functions from various classes. (Principles and Standards for School Mathematics, p. 297) Students are challenged when they are asked to distinguish among different kinds of functions by looking at the functions equations: As high school students study several classes of functions, they should begin to see that classifying functions as linear, quadratic, or exponential makes sense because the functions in each of these classes share important attributes. (Principles and Standards for School Mathematics, p. 300) Resources (0) Standards (3) Other Info