Analyzing and Solving Pairs of Simultaneous Linear Equations

Transcription

1 Analyzing and Solving Pairs of Simultaneous Linear Equations Mathematics Grade 8 In this unit, students manipulate and interpret equations in one variable, then progress to simultaneous equations. They look at the structure of equations (SMP.7) to predict whether there is one solution, no solution, or infinite solutions. Students see the relationships between equations and between algebraic and graphical representations. They extend prior knowledge of the properties of equality, representing quantities as variables, and writing simple equations to construct and solve systems of two linear equations that model real-world and mathematical situations (SMP.4). Students will be able to analyze the context of equations and problems in order to determine which method for solving simultaneous equations is most efficient (SMP.3). Before starting this unit, students need to complete the unit on Proportions, Lines, and Equations (8.EE.5, 8.EE.6) These Model Curriculum Units are designed to exemplify the expectations outlined in the MA Curriculum Frameworks for English Language Arts/Literacy and Mathematics incorporating the Common Core State Standards, as well as all other MA Curriculum Frameworks. These units include lesson plans, Curriculum Embedded Performance Assessments, and resources. In using these units, it is important to consider the variability of learners in your class and make adaptations as necessary. Draft 8/ 2013 Page 1 of 137

2 Table of Contents Unit Plan Lesson Lesson Lesson Lesson Lesson 5.86 Lesson 6.98 Appendix: Optional Pre-Lesson. 107 Unit Resources CEPA Draft 8/ 2013 Page 2 of 137

3 Stage 1 Desired Results ESTABLISHED GOALS 8.EE.7 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 the given 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. 8.EE.8 Analyze and solve linear equations and pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution Transfer Students will be able to independently use their learning to Apply mathematical knowledge to analyze and model mathematical relationships in the context of a situation in order to make decisions, draw conclusions, and solve problems. Meaning UNDERSTANDINGS Students will understand that U1 Linear equations in one variable are used to model real-world situations. They may result in one solution, infinitely many solutions, or no solution. U2 Simultaneous equations are used to model real-world situations. Solutions may result in one common solution, an infinite number of solutions or no common solution. Students will know 1. The properties of operations and the properties of equality can be used to write and solve equivalent equations with rational number coefficients. 2. Simultaneous linear equations can be solved using multiple methods (tables, graphing, algebraic methods). 3. The solution of simultaneous linear equations solved algebraically corresponds to the point(s) of intersection of their graphs. 4. Academic Vocabulary: model, coefficient, ESSENTIAL QUESTIONS 1. How can equations be used to represent real-world and mathematical situations? 2. Given a particular problem situation, how do we determine which method of solving simultaneous equations will be the most useful? Acquisition Students will be able to 1. Interpret a word problem, define the variable(s), and write the equation(s). 2. Use the properties of operations (associative, distributive, commutative, and inverse) to transform equations. 3. Use the properties of equality to transform equations (see 2011 MA Curriculum Framework for Mathematics, p. 185) 4. Solve problems involving linear equations in one variable. 5. Analyze solutions to draw conclusions in the context of the problem. 6. Analyze when a linear equation in one variable has one Draft 8/ 2013 Page 3 of 137

4 because 3x + 2y cannot simultaneously be 5 and 6. 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. SMP.3 Construct viable arguments and critique the reasoning of others. SMP.4 Model with mathematics. SMP.7 Look for and make use of structure. elimination; graphical; linear; point of intersection; simultaneous; slopeintercept form; substitution; system; one, infinite, no solution; identity; infinite solution (x=a); infinite solutions (a=a); or no solution (a = b). 7. Produce examples of linear equations that represent one solution, infinite solutions, and no solutions. 8. Solve problems involving simultaneous equations using multiple methods (tabular, graphical, and algebraic). 9. Write systems of equations to represent mathematical and real-world problems. 10. Estimate solutions to simple cases by inspection of equations and/or graphs. 11. Prove that the solution of simultaneous equations solved algebraically is the point(s) of intersection of the lines graphed in a plane. 12. Justify the accuracy of solutions. Stage 2 - Evidence Evaluative Criteria Assessment Evidence See CEPA Rubric. CURRICULUM EMBEDDED PERFOMANCE ASSESSMENT (PERFORMANCE TASKS) PT Powering Up Patriot School Students act as technology consultants, using simultaneous equations to develop a recommendation for the purchase of laptops and tablets for a school. They prepare a report (including a cost analysis and an itemized budget) for the principal, superintendent, and members of the school committee. OTHER EVIDENCE: OE Formative Assessments: Lesson 1 Writing Equations to Represent Situations Lesson 3 Ticket to Leave Number of Solutions One, None, or Infinitely Many Lesson 4 Creating Systems of Equations Lesson 5 Substitution Cards Stage 3 Learning Plan Draft 8/ 2013 Page 4 of 137

5 Summary of Key Learning Events and Instruction 1. Solving equations in one variable. 2. Categorizing linear equations as having one solution, no solutions, or infinite solutions. 3. Solving linear equations through graphing. 4. Solving simultaneous equations by substituting for one of the variables. 5. Solving simultaneous equations by eliminating one of the variables. 6. Connecting learning through a multi-step problem that requires knowledge of linear equations and systems of linear equations. 7. Curriculum Embedded Performance Assessment (CEPA). Appendix: Optional Pre-lesson: Modeling real-world and mathematical situations with equations in one variable Draft 8/ 2013 Page 5 of 137