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Solving Linear Equations Unit Guide Math 8 Big Idea (Cluster): Analyze and solve linear equations and pairs of simultaneous linear equations (8.EE.7). Edited 5/18/14 Renton School District Domain: Expressions and Equations (8.EE.7) Big Idea (Cluster): Analyze and solve linear equations and pairs of simultaneous linear equations Standard 8.EE.7 Analyze and solve linear equations and pairs of simultaneous linear equations. Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 2 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7) Relevant Math Practices and Student Actions MP1 Make sense of problems and persevere in solving them. Perseveres when solving problems Understands what the problem asks and the relationship among the problem’s parts Looks for an entry point into solving the problem Uses multiple strategies and representations Checks process, answers and ask “Does this make sense?” MP 2 Reason abstractly and quantitatively. Interprets problems in context Uses representations to make meaning of problems Translates a problem from situation to equation Creates a situation for a given equation Explains connections between equation and situation Flexibly uses properties of operations and place value MP 3 Construct viable arguments and critique the reasoning of others. Makes conjectures and evaluates their accuracy Justifies conclusions with mathematical evidence and responds to arguments of others Communicates and defends mathematical reasoning using objects, drawings, diagrams, actions, examples and counterexamples Asks clarifying and probing questions to improve argument Attends to, questions, and evaluates the reasoning of others MP 4 Model with mathematics. Describe, analyze, and solve problems arising in everyday life, society and workplace Makes assumptions and approximations to make problems easier Identifies quantities necessary to solve a problem and uses representations to map their relationships MP 5 Use appropriate tools strategically. Selects tools strategically for visualizing, exploring, comparing, predicting, and solving problems Uses technological tools and resources to pose problems, solve problems and deepen understanding MP 6 Attend to precision. Communicates mathematical thinking accurately both orally and in writing Understands the meaning of mathematical symbols and vocabulary and uses them appropriately Labels consistently and accurately graphs and diagrams Specifies the units of measure when labeling Calculates accurately and efficiently Pursues a level of precision appropriate to the context of the problem Formulates questions and explanations carefully MP 7 Look for and make use of structure. Looks for, identifies, develops and generalizes patterns and relationships Makes connections to prior mathematical knowledge to solve new problems Breaks down complex problems into manageable parts Views complicated quantities both as single objects or compilations of several objects and uses operations to make sense of problems MP 8 Look for and express regularity in repeated reasoning. Notices repeated calculations and looks for general methods and shortcuts to solve a problem Identifies patterns to develop algorithm, formula, or calculation Evaluates reasonableness of intermediate and final results 3 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7) SBAC Required Evidence (Claim 1) Student identifies and writes examples of linear equations in one variable with one solution, infinitely many solutions, or no solution (SBAC Grade 8 Claim 1 Target D, required evidence #1). Student solves linear equations in one variable with rational number coefficients, including equations with solutions that require expanding expressions using distributive property and collecting like terms (SBAC Grade 8 Claim 1 Target D, required evidence #2). A calculator is an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of standards, read the Grade 8 Claim 1 item specifications for Target D. This cluster of standards will also be assessed through Claim 2 Problem Solving (8.EE.C) and Claim 3 Communicating Reasoning (8.EE.7 and 8.EE.8a) and Claim 4 Modeling and Data Analysis (8.EE.C). The “8.EE.C” means Grade 8 Expressions and Equations (EE) domain the third cluster (C). Vocabulary Mathematically proficient students communicate precisely by engaging in discussions about their reasoning using appropriate mathematical language. Students should learn the following terms with increasing precision within the cluster. The bolded terms will be used on Smarter Balanced assessment items. coefficient combining like terms constant distributive property linear equation ordered pair property of equality rational number slope solution variable x-coordinate y-coordinate y-intercept 4 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7) Domain: Expressions and Equations (8.EE.7) Big Idea (Cluster): Analyze and solve linear equations and pairs of simultaneous linear equations Standard 8.EE.7 Analyze and solve linear equations and pairs of simultaneous linear equations. Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. See Grade 8 Common Core Flip Book pages 17-20 for explanations and examples of this standard. Learning Objectives Solve linear equations in one variable, including equations whose solutions require expanding expressions using the distributive property and collecting like terms (SBAC Grade 8 Claim 1 Target D Achievement Level Descriptor, Level 3). Solve mathematical and real-world linear equations in over variable. Show a linear equation has one solution, no solution, or infinitely many solutions by transforming the equation into simpler form (SBAC Grade 8 Claim 1 Target D Achievement Level Descriptor, Level 3). Know the conditions for which a linear equation will have a one solution, no solution of infinitely many solutions. Classify a linear equation as either having no solution, one solution, or infinitely many solutions. Write linear equation with either no solution, one solution, or infinitely many solutions compared to a given equation. Compare equations that have no solution, infinitely many solutions, and one solution. 5 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7) SBAC Required Evidence (Claim 1) Connections to Prior Learning Student identifies and writes examples of linear equations in one variable with one solution, infinitely many solutions, or no solution (SBAC Grade 8 Claim 1 Target D, required evidence #1). Student solves linear equations in one variable with rational number coefficients, including equations with solutions that require expanding expressions using distributive property and collecting like terms (SBAC Grade 8 Claim 1 Target D, required evidence #2). 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. A calculator is an allowable tool for this cluster of standards on an assessment. For more information on the assessment of this set of standards, read the Grade 8 Claim 1 item specifications for Target D. This cluster of standards will also be assessed through Claim 2 Problem Solving (8.EE.C) and Claim 3 Communicating Reasoning (8.EE.7 and 8.EE.8a) and Claim 4 Modeling and Data Analysis (8.EE.C). Connections to Curricular Materials Moving Straight Ahead investigation 3 introduces solving linear equations in one variable but not to the depth required by 8.EE.7. The investigation is more aligned to Grade 6 and 7 equation standards but still lacking the use of rational numbers beyond integers for those grade levels. Engage NY Grade 8 Module 4 lessons 1-9 can be used to supplement this standard if building resources are not already developed. If you have prior developed resources, check for alignment with 8.EE.7a. Students will need to classify, compare, write and solve linear equations with no solution, a unique solution and infinitely many solutions. 7.EE.4 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px +q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Additional Resources/Technology Resources Engage NY Grade 8 Module 4 Lessons 3, 6 and 7 Linear Equations in one variable module (NPS) Expressions and Equations progression of learning document Solving Linear Equations in One Variable task (MAP) Khan Academy Common Core Student Practice site for 8.EE.7 6 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7) The Engage NY Grade 8 Module 4 is being used to supplement this standard where CMP2 curricular materials do not provide resources aligned to Common Core Standard 8.EE.7. Please be selective in the resources within the Engage NY modules you use for your students in order to meet proficiency or higher on this standard. The Engage NY is fairly absent of the Standards for Mathematical Practice which should be equally weighted with content on a daily basis in classroom instruction. For the time-being, teachers will use the Engage NY resources as a foundation to access these standards. But, teachers are highly encouraged to improve the recommended tasks to be less direct instruction based, have students engage in rich mathematical talk and provide opportunities for students to work collaboratively. The Engage NY modules are a temporary solution as we transition to the Common Core State Standards for Mathematics. Solving Linear Equations Unit Pacing Guide Supplemental CCSS-M Aligned Lessons Engage NY Grade 8 Module 4 Lesson 1 Engage NY Grade 8 Module 4 Lesson 2 Engage NY Grade 8 Module 4 Lesson 3 Engage NY Grade 8 Module 4 Lesson 4 Engage NY Grade 8 Module 4 Lesson 5 (optional) Engage NY Grade 8 Module 4 Lesson 6 Engage NY Grade 8 Module 4 Lesson 7 Engage NY Grade 8 Module 4 Lesson 8 Engage NY Grade 8 Module 4 Lesson 9 (extension) PLC-developed common assessment until 2015-2016 Alignment Notes Students will write mathematical statements using symbols to represent numbers Students will know the properties of linear and non-linear expressions in x and identify expressions as linear and non-linear. Students will know that a linear equation is a statement of equality between two expressions. CCSS-M 7.EE.4 8.EE.7 8.EE.7 8.EE.7 Students will extend the use of the properties of equality to solve linear equations having rational coefficients. Students will apply knowledge of geometry to writing and solving linear equations 8.EE.7 Students will transform equations into simpler forms using the distributive property and combining like terms. Students will also experience linear equations with no solution. Students will learn the conditions for which a linear equation will have a unique solution, no solution or infinitely many solutions. Students will rewrite and solve equations that do not appear to be linear equations using properties of equality. Students will rewrite an exponential expression that represents a series as a linear equation. 8.EE.7 8.EE.7 8.EE.7 8.EE.7 8.EE.7 For understanding the content required to be assessed, see the rubrics at the end of the unit guide 8.EE.7 which is based on the SBAC Grade 8 Claim 1 item specifications for Target D. Based on the SBAC item specifications for Target D, a calculator is an allowable tool during the assessment. *SBAC ALD Level 3 or Level 4 means Smarter Balanced Achievement Level Descriptors for entering into a Level 3 or Level 4. Hard copies of these lessons can be found at http://staff.rentonschools.us/renton/secondary-math/math-8-ccss-m-resources . The following resources were used to create this curriculum guide: SBAC Item Specifications Claim 1 Target D, Grade 8 Common Core State Standards Flip Book compiled by Melissa Hancock and Expressions and Equations progression of learning document. 7 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7) Standard 8.EE.7 Analyze and solve linear equations and pairs of simultaneous linear equations. Solve linear equations in one variable. a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 4 In addition to being proficient on the standard, student can demonstrate one or more of the following: Applies understanding of standard to unfamiliar situations and/or to solve complex problems Use precise and relevant communication to justify mathematical thinking Connects knowledge to other learning targets and/or advance problem sets. For example, analyze a student’s conjecture on how to classify equations in one variable and justify or refute the conjecture with a chain of reasoning. 8.EE.7 3 Transform a given equation into simpler forms in order to demonstrate a system of linear equations has one solution, no solution or infinitely many solutions (x=a, a=a, or a=b) Write and solve linear equations in one variable with rational coefficients including equations using distributive property and collecting like terms that have one solution, no solution or infinitely many solutions. Rewrite expanded equations using distributive property or by collecting like terms 2 Write and solve linear equations in one variable with integer coefficients that have one solution, no solution or infinitely many solutions. 1 With help, minimal success solving linear equations in one variable with integer coefficients that have one solution, no solution or infinitely many solutions. For more detail on the assessment of the standard, read the SBAC Grade 8 Item Specifications-Target D. This rubric was constructed from the SBAC Claim 1 item specifications and SBAC Achievement Level Descriptors. These rubrics should be further developed at the PLC level. Students will also be assessed on this standard through Claim 2 Problem Solving (8.EE.C) and Claim 3 Communicating Reasoning (8.EE.7 and 8.EE.8a) and Claim 4 Modeling and Data Analysis (8.EE.C) which is not reflected in the above rubric. 8 This document is a draft and will continue to develop as we learn more about the Common Core State Standards and the SBAC assessment. (8.EE.7)