Linear Systems of Equations

Transcription

1 Mathematical Models with Applications, Quarter 2, Unit 2.1 Linear Systems of Equations Overview Number of instruction days: 5-7 (1 day = 53 minutes) Content to Be Learned Mathematical Practices to Be Integrated Solve systems of equations in two variables using a table, graphically, substitution, and linear combination. 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. Essential Questions 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? How do you know if a solution of a system of equations is viable or non-viable? Providence Public Schools D-1

2 Math Models, Quarter 2, Unit 2.1 Linear Systems of Equations (5-7 Days) Standards Common Core State Standards for Mathematical Content Algebra 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. Reasoning with Equations and Inequalities A-REI Solve systems of equations [Linear-linear and linear-quadratic] A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically [Linear and exponential; learn as general principle] A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 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 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 D-2 Providence Public Schools

3 Linear Systems of Equations (5-7 Days) Math Models, Quarter 2, Unit 2.1 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. 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. 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 Providence Public Schools D-3

4 Math Models, Quarter 2, Unit 2.1 Linear Systems of Equations (5-7 Days) 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 non-viable options in a modeling context. Current Learning In this unit, students solve systems of equations using a table, graphically, substitution, and linear combination 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, make tables of values, and find successive approximations. They represent constraints using equations, and systems of equations, and interpret solutions as viable or non-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. 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 ( ). They also indicate that graphing technology can assist students in making sense of equivalent expressions (130). Assessment D-4 Providence Public Schools

5 Linear Systems of Equations (5-7 Days) Math Models, Quarter 2, Unit 2.1 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 such as performance assessments, verbal responses, tests, or quizzes to express their understandings of the concepts that follow: Solve systems of two equations in two variables using various methods. Determine the best method for solving systems of equations in real a real world context. Model systems of equations from real-world situations. Interpret the intersection of two graphs in a real world context. Instruction Learning Objectives Students will be able to: Solve systems of linear equations by using tables. Solve systems of linear equations by graphing. Solve systems of equations using substitution. Solve systems of equations using linear combination. Determine the best method for solving systems of equations. Providence Public Schools D-5

6 Math Models, Quarter 2, Unit 2.1 Linear Systems of Equations (5-7 Days) Review and demonstrate knowledge of important concepts and procedures related to linear systems of equations. Resources Modeling with Mathematics: A Bridge to Algebra II, W.H. Freeman and Company, 2006 Sections 3.1 through 3.5 (pp ) Online Companion Website: TI-Nspire Teacher Software Additional Resources located in the Supplementary Unit Materials Section of the Binder: o education.ti.com Boats in Motion (ID: 11298) What is a solution to a system? Exploration 26: Solving a Pair of Linear Systems by Graphing 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 and Assessment sections below for specific recommendations. Materials Graphing calculators, grid paper, colored pencils, algebra tiles Instructional Considerations Key Vocabulary consistent dependent elimination independent point of intersection system of equations inconsistent Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: substitution, and equation. A critical resource to make Math Models effective is the use of tables, handouts, and assessments provided by the publisher at (or google: Math D-6 Providence Public Schools

7 Linear Systems of Equations (5-7 Days) Math Models, Quarter 2, Unit 2.1 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 address as the UserId. You will also be prompted for the following companion website code: BFW41INST. Select the Lesson Plan Activities tab on the companion website to access related PowerPoint presentations. In this unit, there are the following PowerPoint presentations aligned to the textbook: Day 1 and 2 Solving Systems Notes (Sections ) You may provide students with graphic organizers. One example would be using an organizer to assist students with identifying similarities and differences by comparing the different methods of solving a system of equations. Identifying the best method to use when solving a particular system of equations is a critical skill in this unit. Foldables can be used for making study guides from notes, examples, graphs, or other representations regarding systems of equations. They can also serve as an additional tool for identifying similarities and differences through comparing the different methods for solving a system, thereby organizing their knowledge. Foldables also offer students a sense of ownership in their work and learning. Have students work in pairs for graphing systems of equations, taking turns between graphing the equation and checking the solution set. Discuss with students when a system of equations has one solution, no solution, or infinitely many solutions. Write examples of systems on the board and have students answer: one, none, or infinite solutions. In the classroom, have a blog where students can write entries, explaining how they decide when to use each of the different methods for solving systems of equations. One thing they can do is write a pros and cons list for each method. If algebra tiles are available, have students use these physical models (nonlinguistic representations) to solve systems of equations using elimination. When students begin to use elimination using multiplication, they sometimes forget to multiply each term on both sides of the equation. You may ask students to show this step when they are learning to apply it to help them remember to multiply each term. If not, be sure to ask questions about this step to ensure that students understand the method. Graphing technology will assist students with modeling real-world problems involving systems of equations and for identifying the solution to a system of equations. Several examples supporting the integration of technology in this unit are provided below. The teacher and student pages for the activities are provided in the supplementary materials section of this curriculum frameworks binder. The activities can also be found by going to education.ti.com and searching for the activity titles. Boats in Motion: This lesson involves substituting values for variables, evaluating expressions, and solving equations. The emphasis is on helping students understand that an expression and an equation are two distinctly different mathematical objects. As a result, Providence Public Schools D-7

8 Math Models, Quarter 2, Unit 2.1 Linear Systems of Equations (5-7 Days) students will make conjectures about the connection between the values being substituted in the expression or equation and the outcomes. You will need to download the tns file to students calculators for this activity. What is a solution to a system?: Students solve a system of linear equations and begin to explore the infinite numbers of solutions to one equation. You will need to download the tns file to students calculators for this activity. Exploration 26 Solving a Systems of Linear Equations by Graphing: Students solve a system of linear equations and begin to explore the infinite numbers of solutions to one equation. There is no tns file for this activity. Additional TI-Nspire resources can be found using the TI-Nspire Teacher Software. Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions should be included in the unit assessment. 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