Mathematical Models with Applications, Quarter 3, Unit 3.1 Quadratic and Linear Systems Overview Number of instruction days: 5-7 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Solve systems of quadratic and linear equations in two variables using tables, graphs, and technology. Explain the meaning of the intersection of two functions graphed on a coordinate plane. Write a system of equations to determine the solution to a real world problem. 1 Make sense of problems and persevere in solving them. Construct equations to represent a real world problem and solve the resulting equations. 4 Model with mathematics. Model with mathematics to solve real world problems involving system of equations. 5 Use appropriate tools strategically. Use graphing calculator to verify algebraic solutions to systems of equations. Essential Questions How do you interpret the intersection of two graphs in the context of a problem? How do you know if a solution of a system of equations is viable or not viable? Providence Public Schools D-1
Math Models, Quarter 3, Unit 3.1 Quadratic and Linear Systems (5-7 Days) Standards Common Core State Standards for Mathematical Content Algebra Reasoning with Equations and Inequalities A-REI Solve systems of equations [Linear-linear and linear-quadratic] 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 [Linear and exponential; learn as general principle] 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. Creating Equations A-CED Create equations that describe numbers or relationships [Linear, quadratic, and exponential (integer inputs only); for A.CED.3 linear only] A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Common Core State Standards for Mathematical Practice 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 D-2 Providence Public Schools
Quadratic and Linear Systems (5-7 Days) Math Models, Quarter 3, Unit 3.1 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. 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. 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. Providence Public Schools D-3
Math Models, Quarter 3, Unit 3.1 Quadratic and Linear Systems (5-7 Days) Clarifying the Standards Prior Learning Students learned to graph points on the coordinate plane in Grade 5, and interpreted the values of coordinates in the context of the situation. In Grade 6, students learned the process of finding a solution set for an equation, and 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 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. By the end of Algebra 1, students were expected to be fluent with their use of systems of linear equations and inequalities in two variables. They solved systems of equations exactly and approximately after proving different methods for solving systems. Students represented and solved systems of equations and inequalities including linear-linear and linear-quadratic systems. They explained why the x-coordinates of the intersection point of two graphs were the solutions to the equation. Students used technology to graph two functions, make tables of values, and found successive approximations. Students graphed the solution to a system of linear inequalities as the intersection of two half-planes. They represented constraints using equations, inequalities, and systems of equations, and interpreted solutions as viable or not viable options in a modeling context. Current Learning In this unit, students solve a quadratic-linear system of equations using tables, graphs, and technology in the context of real-world problems. They explain why the coordinates of the intersection point of two graphs are the solutions to the equation. Students use technology to graph two functions and make tables of values. They represent constraints using equations, and systems of equations, and interpret solutions as viable or not viable options in a modeling context. Future Learning In Algebra II, students will graph, analyze, and create equations and represent constraints with equations and inequalities using a variety of function types, including radical, rational, polynomial and trigonometric functions. They will also represent and solve systems of equations graphically, using multiple function types. 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 non-linear, will be essential for student success in advanced courses in physics, economics, and chemistry. D-4 Providence Public Schools
Quadratic and Linear Systems (5-7 Days) Math Models, Quarter 3, Unit 3.1 Additional Findings In A Research Companion to Principles and Standards for School Mathematics, Chazan and Yerushalmy discuss the cognitive difficulties that many students have in working with the complex relationships embedded in systems of equations. As an example, they describe the methods that students must use to solve a system of equations consisting of a linear equation in standard form and a circle in standard form. As students work through the solving of such a system, they must move from an equation in two variables to a function of one to enable use of the substitution algorithm, from an equation in two variables to an equation in one variable using the algorithm, to generating equivalent expressions in solving the new equation. They indicate that this complexity is common in learning about equivalence in school algebra, and that this cognitive complexity must be taken into account when approaching topics involving equivalence (129-131). They also indicate that graphing technology can assist students in making sense of equivalent expressions (130). Assessment When constructing an end of unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS Content Standards and the CCSS Practice Standards should be considered when designing assessments. Standards based mathematics assessment items should vary in difficulty, content and type. The assessment should include a mix of items which could include multiple choice items, short and extended response items and performance based tasks. When creating your assessment you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Math Models students should be provided with multiple, alternative methods to express their understandings of the concepts that follow: Solve quadratic-linear system of equations in two variables using various methods. Model quadratic-linear system of equations involving real-world situations. Interpret the intersection of the graph of a linear equation with a quadratic equation in a real world context. Providence Public Schools D-5
Math Models, Quarter 3, Unit 3.1 Quadratic and Linear Systems (5-7 Days) Instruction Learning Objectives Students will be able to: Solve a quadratic-linear system of equations using tables. Solve a quadratic-linear system of equations by graphing. Solve a quadratic-linear system of equations using technology. Review and demonstrate knowledge of important concepts and procedures related to a quadraticlinear system of equations. Resources Modeling with Mathematics: A Bridge to Algebra II, W.H. Freeman and Company, 2006 Section 5.6 (pp. 282 283) Section 5.6 Assignment (p. 284) Section 5.7 (pp. 285 287) Online Companion Website: http://bcs.whfreeman.com/bridgetoalgebra2/ TI-Nspire Teacher Software Additional Resources located in the Supplementary Unit Materials Section of the Binder: o Regents Exam Questions A2.A3: Quadratic-Linear Systems 2 o o Graphing Technology Lab Systems of Linear and Quadratic Equations education.ti.com: How many Solutions 2? Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery section below for specific recommendations. Materials Graphing calculators, grid paper, colored pencils D-6 Providence Public Schools
Quadratic and Linear Systems (5-7 Days) Math Models, Quarter 3, Unit 3.1 Instructional Considerations Key Vocabulary No new vocabulary Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: points of intersection, linear, quadratic, and system of equations. A critical resource to make Math Models effective is the use of tables, handouts, and assessments provided by the publisher at http://www.whfreeman.com/catalog/static/whf/mma/ (or google: Math Models: A Bridge to Algebra 2 ). Also available on this website are power point presentations, lesson plans, assessments and activities. For initial use, you will be prompted to set an an instructors account using an e-mail address as the UserId. You will also be prompted for the following companion website code: BFW41INST. There are numerous resources available on the internet to support solving quadratic-linear systems of equations. For example, the Khan Academy (khanacademy.org) has video tutorials for solving systems of non-linear equations. The supplementary resources in the binder provided additional opportunities for practice using real-world scenarios. Consider having students use a Venn diagram to identify the similarities and differences between solving quadratic-linear systems and linear-linear systems. The use of graphing technology is a nonlinguistic strategy which may benefit students with modeling real-world problems involving systems of quadratic and linear equations and identifying the solution to a system of equations. The Graphing Technology Lab Systems of Linear and Quadratic Equations provides instructions for using the TI-Nspire calculator to graph and solve systems of linear and quadratic equations. You should test the provided keystrokes in the activities in order to verify that they are aligned with the current operating systems on the calculators. An additional opportunity for solving systems of equations using technology to explore systems of quadratic and linear equations is provided below. The teacher and student pages for the activities are provided in the supplementary materials section of this curriculum frameworks binder. The activity can also be found by going to education.ti.com and searching for activity titles. However, you will need to download the tns file to the students calculators. How Many Solutions 2: This activity lets students manipulate two graphs to find the number of possible intersections of a system. Students will manipulate graphs created by linear and quadratic equations and recognize that a system of two equations in two variables can have no solution, one or more solutions, or infinitely many solutions. Providence Public Schools D-7
Math Models, Quarter 3, Unit 3.1 Quadratic and Linear Systems (5-7 Days) Additional TI-Nspire resources can be found using the TI-Nspire Teacher Software. As you formatively and summatively assess students, a cues, questions, and advance organizers strategy can be used, since students are answering questions about content that is important. Notes D-8 Providence Public Schools