Overview. Essential Questions. Grade 8 Mathematics, Quarter 4, Unit 4.2 Applying Systems of Equations. Number of instruction days: 8 10

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1 Applying Systems of Equations Number of instruction days: 8 10 Overview Content to Be Learned Solve real-world problems leading to two linear equations in two variables. Solve mathematical problems with two linear equations and two variables. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Write and solve a system of linear equations with two variables given real-world problems. 2 Reason abstractly and quantitatively. Make connections between the solution(s) to a system of linear equations and the context of the problem. Determine whether the lines through two pairs of points intersect or are parallel. 4 Model with mathematics. Use real-world situations to construct a system of linear equations. Essential Questions How do you know if a solution to a model of a system of equations is reasonable? How can systems of linear equations be used to represent real-world situations? 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? Providence Public Schools D-109

2 Applying Systems of Equations (8 10 days) Standards Common Core State Standards for Mathematical Content Expressions and Equations 8.EE Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.8 Analyze and solve pairs of simultaneous linear equations. c. Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. 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 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. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 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 D-110 Providence Public Schools

3 Applying Systems of Equations (8 10 days) Grade 8 Mathematics, Quarter 4, Unit 4.2 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 worked informally with one-variable linear equations as early as Kindergarten. In Grade 1, students began solving equations involving addition and subtraction. In Grade 3, students solved equations involving multiplication and division. 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 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 properties of operations to generate equivalent linear expressions. They also learned to use algebraic equations and inequalities to solve word problems and solved real-life and mathematical problems using numerical algebraic expressions and equations (px + q = r; p(x + q) = r) ; these are major clusters in Grade 7. Current Learning Solving pairs of simultaneous linear equations is a major cluster and must be mastered in Grade 8. Students solve systems of two linear equations in two variables algebraically. Students write and solve real-life problems involving systems of equations, deciding which method is best for solving that system. In Grade 8, students solve linear equations in two variables, including equations with 1, 0, and infinitely many solutions, and equations in which they apply the distributive property and collect like terms. Students begin to develop techniques to solve and analyze systems of equations algebraically and graphically. They learn that the intersection of the graphs of two linear equations represents the solution to a system of equations. Finally, students learn the definition of a function, compare functions, and write functions to model linear relationships. They also examine and explain functions that are not linear. Future Learning Algebra I students will be expected to be fluent in their work with systems of linear equations and inequalities in two variables. Students will solve systems of equations exactly and approximately after proving different methods for solving systems. Students will represent and solve systems of equations and inequalities including linear linear and linear quadratic. Students will use technology to graph two functions, make tables of values, and find successive approximations. Students will graph the solution to a system of linear inequalities as the intersection of two half-planes. They will represent constraints using equations, inequalities, and systems of equations, and interpret solutions as viable or nonviable options in a modeling context. In Algebra II, students will create equations and represent constraints with equations and inequalities using a variety of function types. They will graph and analyze a variety of function types, including radical, rational, polynomial, and trigonometric functions. They will also represent and solve systems of equations graphically, using a number of different function types. Further study of systems of Providence Public Schools D-111

4 Applying Systems of Equations (8 10 days) 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 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 first move from an equation in two variables to a function of one in order to enable use of the substitution algorithm; then they must move from an equation in two variables to an equation in one variable using the algorithm; then to generating equivalent expressions in solving the new equation. Chazan and Yerushalmy 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. (pp ) 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 for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise 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. Solve real-world and mathematical problems leading to two linear equations in two variables. D-112 Providence Public Schools

5 Applying Systems of Equations (8 10 days) Grade 8 Mathematics, Quarter 4, Unit 4.2 Learning Objectives Students will be able to: Instruction Write a system of linear equations given a real-world problem. Determine the best method for solving a system of linear equations. Apply systems of equations to new context. Resources Connected Mathematics 2, Pearson/Prentice Hall, 2008 The Shapes of Algebra Exam View Investigation 4: Solving Systems of Linear Equations Symbolically (Extensions); Student Materials (p. 65) Teacher s Guide Implementing and Teaching Guide Strategies for English Language Learners Teaching Transparencies Lab Sheet Resource Book Assessment Resource Book Additional Practice and Skill Workbook Special Needs Handbook Parent Guide Teacher CDs Algebra 1, Glencoe McGraw-Hill, Teacher Edition and Student Edition. Section 6-5 Applying Systems of Linear Equations (pp ) Glencoe McGraw-Hill Online Ancillary materials for Chapter 6 Section 5 Chapter 6 Resources Masters Study Notebook Interactive Classroom CD (PowerPoint Presentations) Teacher Works CD ROM Exam View Assessment Suite Providence Public Schools D-113

6 Applying Systems of Equations (8 10 days) 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 for specific recommendations. Materials No materials for this unit. Instructional Considerations Key Vocabulary There is no new vocabulary for this unit. Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: linear system, linear combination, elimination method, substitution method, systems of linear equations, solution, and linear equations in standard form. Living word walls will assist all students in developing content language. Word walls should be visible to all students, focus on the current unit s vocabulary, and have pictures, examples, and/or diagrams to accompany the definitions. Teachers should review the Mathematics of the Unit found on page 3 of all CMP2 teacher editions prior to planning. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the CMP resource. By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. 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. D-114 Providence Public Schools

7 Applying Systems of Equations (8 10 days) Grade 8 Mathematics, Quarter 4, Unit 4.2 Notes Providence Public Schools D-115

8 Applying Systems of Equations (8 10 days) D-116 Providence Public Schools