Possible Lesson 01 (7 days) State Resources: Algebra 1 End of Course Success CCRS: Objective 4 Lesson 2 Line Dancing, Systems of Equations Card Match, On Your Own: Systems of Equations, Using Technology: Systems of Equations, Challenge Activity Online: Objective 2 Lesson 2 For What It s Worth, Not 1 But 2, Looking at the Process, Multiple Representation Cards Fostering Algebraic Thinking: Introduction and Analyzing Written Student Work, Introductory Session Postage Stamp; Asking Questions of Students, Session 4 More Postage Stamps; Documenting Patterns of Student Thinking, Session 4 More Postage Stamps; Listening to Students, Session 4 More Postage Stamps Mathematics TEKS Toolkit: Clarifying Activity/Lesson/Assessment http://www.utdanacenter.org/mathtoolkit/instruction/alg1.php MTC Grades 9-12: Explore 1, Algebra 1: Hay Bale Problem http://www.txar.org/professional_dev/archived/training/materials/mtc/9-12/mtcalgebra1ee1participantpages.pdf TEA STAAR Mathematics Resources: http://www.tea.state.tx.us/student.assessment/staar/math/ TEA STAAR Released Test Questions: http://www.tea.state.tx.us/student.assessment/staar/testquestions/ TEXTEAMS: Algebra 1: 2000 and Beyond II Linear Functions; 3. Linear Equations and Inequalities, 3.4 Systems of Linear Equations and Inequalities, Act. 1 (Using a Table), Act. 2 (Solve the System Graphically), Act. 3 (Solve the System Symbolically) RATIONALE: This unit bundles student expectations that address formulating 2 x 2 linear systems of equations for a problem situation, solving the system, and justifying the reasonableness of the solution in terms of the problem. The use of graphing technologies is also incorporated to help students explore the relationships within systems of equations. Prior to this unit, in Algebra 1 Unit 06, students solved 2 x 2 systems of linear equations for the first time by applying various methods, including models, tables, graphs, and algebraic properties. Students also analyzed the solutions found from the different methods. During this unit, students encounter problem situations which apply methods for solving systems of linear equations, including tables, graphs, and algebraic methods (substitution and elimination). Linear systems are formulated for problem situations, solved by a method of choice, and the reasonableness of the solution justified in terms of the problem situation. Graphing calculator technologies continue to be utilized to explore the relationships within systems of equations. After this unit, in Algebra 2 Unit 04, students will extend their study of systems of equations to include systems of equations in two or more unknowns and to include connections between linear and non-linear systems. These skills are necessary not only in Algebra 1, but in subsequent courses in mathematics. 2012, TESCCC 05/07/12 page 1 of 8
Focusing on methods for solving linear systems including models, tables, graphs, and the properties of equality is emphasized in the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning B1, C1; II. Algebraic Reasoning A1, B1, C1, C2, D1, D2; VII. Functions B2, C2; IX. Communication and Representation A1, A3, B1, B2; and STAAR Readiness Standards: A.4A, A.8B. Writing equations to represent functions, formulating systems of equations, and justifying the reasonableness of the solutions are addressed in STAAR Supporting Standards: A.1C, A.3A, A.8A, and A.8C. According to Navigating through Algebra in Grades 9-12 from the National Council of Teachers of Mathematics (NCTM), High school students continue to develop fluency with mathematical symbols and become proficient in operating on algebraic expressions in solving problems. Their facility with representation expands to include equations, inequalities, systems of equations, graphs, matrices, and functions, and they recognize and describe the advantages and disadvantages of various representations for a particular situation. Such facility with symbols and alternative representations enables them to analyze a mathematical situation, choose an appropriate model, select an appropriate solution method, and evaluate the plausibility of their solutions (2002, p. 3). According to Algebra in a Technological World (NCTM, 1995, p. 2), graphing tools influence the content of algebra in a technological world in the following ways: They allow a visualization of relationships. They allow the accurate solution of equations and inequalities not possible through symbolic manipulation alone. They provide numerical and graphical solutions that support solutions found using algebraic manipulation. They promote exploration by students and their understanding of the effect of change in one representation on another representation. They encourage the exploration of relationships and mathematical concepts. They promote what if modeling of realistic situations. National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc. National Council of Teachers of Mathematics. (1995). Curriculum and evaluation standards for school mathematics: Algebra in a technological world. Reston, VA: National Council of Teachers of Mathematics, Inc. Texas Education Agency. (2009). Texas college and career readiness standards. Austin, TX: Author. MISCONCEPTIONS/UNDERDEVELOPED CONCEPTS: None Identified PERFORMANCE INDICATORS CONCEPTS KEY UNDERSTANDINGS FOR LEARNERS Analyze problem situations that can be represented algebraically by linear systems of equations with two unknowns such as the following: Unleaded gas and supreme gas sell for different prices per liter. Suppose that 2 liters of unleaded and 3 liters of supreme cost a total of $2.52. Five liters of unleaded and 4 liters of supreme cost a total of $4.48. What is the price per liter of each kind of gas? If Ted filled his Foundations for Functions Relationships; Representations; Symbols and Variables; Transforming and Solving Equations Linear Functions Systems of Equations Linear systems of equations can be written to represent problem situations and solved using a selected method. The reasonableness of the solution can be justified in terms of the problem situation. 2012, TESCCC 05/07/12 page 2 of 8
PERFORMANCE INDICATORS CONCEPTS KEY UNDERSTANDINGS FOR LEARNERS pickup with 30 liters of unleaded and also filled his car with 25 liters of supreme, how much would he have to pay? Assume prices have not changed. Green Grass, Inc. charges an initial fee of $15 plus $10 per hour for lawn services. Their main competitor S & G Lawn Care charges an initial fee of $35 plus $7 per hour. When would the charges for each company be equal? Explain how to determine when it would be better to use Green Grass, Inc. and when it would be better to use their competitor, S & G Lawn Care. Create a graphic organizer for each problem situation that includes formulating a representative linear system of equations, selecting two different solution methods (concrete model, table, graph, or algebraic method), solving the linear system of equations by each method, and interpreting the reasonableness of the solution in terms of the problem situation. (A.1C; A.3A; A.4A; A.8A, A.8B, A.8C) 1C; 3H KEY ACADEMIC VOCABULARY SUPPORTING CONCEPTUAL DEVELOPMENT 2 x 2 System of linear equations two linear equations with two unknowns that are analyzed to determine common solutions. The solutions to 2 x 2 systems of linear equations will be one of the following: (1) two lines that intersect at one point and the solution is represented by a single ordered pair (2) two lines whose equations are equivalent, represent the same line, and have an infinite number of ordered pairs that satisfy both equations (3) two parallel lines that do not intersect and no solutions (ordered pairs) satisfy both equations simultaneously A.1 Foundations for functions. The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student is expected to: A.1C Describe functional relationships for given problem situations and write equations or inequalities to answer questions arising from the situations. Describe 2012, TESCCC 05/07/12 page 3 of 8
Supporting Standard FUNCTIONAL RELATIONSHIPS FOR PROBLEM SITUATIONS Real-world problem situations Representations of functional relationships Verbal descriptions Algebraic generalizations (including equation and function notation) Equations Connections between representations of functional relationships Write EQUATIONS AND INEQUALITIES IN PROBLEM SITUATIONS Real-world problem situations Representations of functional relationships Verbal descriptions Algebraic generalizations (including equation and function notation) Equations Answer QUESTIONS ARISING FROM PROBLEM SITUATIONS Analysis of solutions for predictions and conclusions arising from problem situations Justification of reasonableness of solution Note: In middle school, students formulate equations from problem situations. A.3 Foundations for functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. The student is expected to: II. Algebraic Reasoning C2 Explain the difference between the solution set of an equation and the solution set of an inequality. II. Algebraic Reasoning D1 Interpret multiple representations of equations and relationships. II. Algebraic Reasoning D2 Translate among multiple representations of equations and relationships. VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C2 Develop a function to model a situation. 2012, TESCCC 05/07/12 page 4 of 8
A.3A Use symbols to represent unknowns and variables. Supporting Standard Use SYMBOLS Representation of an unknown quantity in a problem situation Representation of a quantity that varies in a data table process column Representations of patterns in data collections Representations of data collections or problem situations with and without technology Models Pictorial (e.g., diagrams of the problem situation) Verbal descriptions Algebraic generalizations Verbal expressions translated to algebraic expressions using symbols Problem situation as an expression or equation Express, Represent UNKNOWNS AND VARIABLES Representation of an unknown quantity in a problem situation Representation of a quantity that varies in a data table process column Representations of data collections or problem situations with and without technology Models Pictorial (e.g., diagrams of the problem situation) Verbal descriptions Algebraic generalizations Verbal expressions translated to algebraic expressions using symbols Problem situation as an expression, equation, or inequality Note: In middle school, students use symbols to represent arithmetic sequences. A.4 Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and VII. Functions B2 Algebraically construct and analyze new functions. VII. Functions C2 Develop a function to model a situation. IX. Communication and Representation A1 Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem. IX. Communication and Representation A3 Use mathematics as a language for reasoning, problem solving, making connections, and generalizing. 2012, TESCCC 05/07/12 page 5 of 8
inequalities in problem situations. The student is expected to: A.4A Find specific function values, simplify polynomial expressions, transform and solve equations, and factor as necessary in problem situations. Readiness Standard Manipulate, Solve, Transform EQUATIONS Distinction between expressions and equations and the difference between simplifying and solving (TxCCRS) Types of equations Linear Transformation of equations using properties of equality Note: In middle school, students solve equations in one variable using concrete models, pictorial models, and algebraic symbols. A.8 Linear functions. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to: A.8A Analyze situations and formulate systems of linear equations in two unknowns to solve problems. Supporting Standard I. Numeric Reasoning B1 Perform computations with real and complex numbers. (Algebra 1 only requires the students to compute with real numbers.) II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. II. Algebraic Reasoning B1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions). II. Algebraic Reasoning C1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations. Analyze, Formulate, Use, Solve SYSTEMS OF EQUATIONS FOR PROBLEM SITUATIONS Problem situations represented by 2 x 2 systems of linear equations Two unknowns Two equations VII. Functions C2 Develop a function to model a situation. IX. Communication and Representation B2 Summarize and interpret mathematical information provided orally, visually, or in written form within the given context. 2012, TESCCC 05/07/12 page 6 of 8
A.8B Solve systems of linear equations using concrete models, graphs, tables, and algebraic methods. Readiness Standard A.8C Interpret and determine the reasonableness of solutions to systems of linear equations. Supporting Standard Use, Solve SYSTEMS OF EQUATIONS Distinction between expressions and equations and the difference between simplifying and solving (TxCCRS) 2 x 2 systems of equations to represent problem situations Methods for solving linear 2 x 2 system of equations with and without technology Concrete and pictorial models Tables Common points on tables Graphs Identification of possible solutions in terms of points of intersection Algebraic methods Substitution Linear combinations (elimination) Relationships and connections between the methods of solution Representation of the solution as a point of intersection Interpretation of the point of intersection in terms of the problem situation II. Algebraic Reasoning A1 Explain and differentiate between expressions and equations using words such as solve, evaluate, and simplify. II. Algebraic Reasoning C1 Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations. IX. Communication and Representation B1 Model and interpret mathematical ideas and concepts using multiple representations. Analyze, Interpret, Determine REASONABLENESS OF SOLUTIONS TO SYSTEMS OF EQUATIONS Justification of solutions to systems of equations with and without technology Tables Graphs Verbal description Substitution of solutions into original function Justification of reasonableness of solution in terms of the problem situation or data collection I. Numeric Reasoning C1 Use estimation to check for errors and reasonableness of 2012, TESCCC 05/07/12 page 7 of 8
solutions. The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, 74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum. School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum. School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency. http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 ELPS# C(1) 1C C(3) 3H Subsection C: Cross-curricular second language acquisition essential knowledge and skills. Cross-curricular second language acquisition/learning strategies. The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: Use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary. Cross-curricular second language acquisition/speaking. The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to: Narrate, describe, and explain with increasing specificity and detail as more English is acquired. 2012, TESCCC 05/07/12 page 8 of 8