Algebra 1, Quarter 2, Unit 2.1. Linear Functions. Overview

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1 Algebra 1, Quarter 2, Unit 2.1 Linear Functions Overview Number of instructional days: 10 (2 assessments) (1 day = 45 minutes) Content to be learned Demonstrate conceptual understanding of linear functions and relations. (1 day) Demonstrate the specific characteristics of functions including domain, range, increasing, decreasing, and intercepts. 2 days) Use the concepts of constant, variable, and average rate of change. (2 days) Identify x and y intercepts. (1 day) Demonstrate the connection among graphs, tables, equations, and function notation. (2 days) Essential questions How can you describe and represent functions? What real-world applications are represented by functions? What does slope indicate about the relationship between the independent and dependent variables? Mathematical practices to be integrated Model with mathematics. Use functions to model problem situations. Analyze the relationships of specific characteristics of functions. Look for and make use of structure. Demonstrate understanding of functions by using the structure of functions. Use the structure of functions to determine connections among graphs, tables, and equations. In what kind of real-world situations would the domain and range of a linear function be restricted? What information is displayed by the equation of a line? Cumberland, Lincoln, and Woonsocket Public Schools C-17

2 Algebra 1, Quarter 2, Unit 2.1 Final, July 2011 Linear Functions (10 days) Written Curriculum Grade-Span Expectations M(F&A) 10 2 Demonstrates conceptual understanding of linear and nonlinear functions and relations (including characteristics of classes of functions) through an analysis of constant, variable, or average rates of change, intercepts, domain, range, maximum and minimum values, increasing and decreasing intervals and rates of change (e.g., the height is increasing at a decreasing rate); describes how change in the value of one variable relates to change in the value of a second variable; or works between and among different representations of functions and relations (e.g., graphs, tables, equations, function notation). (State) Clarifying the Standards Prior Learning Students have been working with linear relationships since fourth grade. They have identified, described, and compared situations that represent constant rates of change. In middle school, students were introduced to problem-solving situations involving slope and constant versus varying rates of change. In addition, students have had to distinguish between linear and nonlinear relationships. Current Learning This unit is a reinforcement of understanding linear functions and relations. It is an introduction to specific characteristics such as domain, range, intercepts, and increasing and decreasing intervals. Students work between and among different representations of functions and relations. Future Learning In grade 10, the average rate of change (slope) will be introduced, as well as applications of slope. This GSE will also be revisited in grades 11, 12, and advanced math, where students will represent and analyze functions in several ways, analyze characteristics of functions (exponential, logarithmic, and trigonometric), and apply knowledge of functions to interpret situations. Additional Research Findings A Research Companion to Principles and Standards for School Mathematics indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities (pp ). Cumberland, Lincoln, and Woonsocket Public Schools C-18

3 Algebra 1, Quarter 2, Unit 2.2 Equations of Lines Overview Number of instructional days: 9 (1 day assessment) (1 day = 45 minutes) Content to be learned Find the equation of a line, given two points on the line and use in problem situations. (2 days) Find the equation of a line, given one point on the line and the slope of the line from a contextual situation. (2 days) Find the equation of a line, given the y-intercept and the slope of the line and use in problem situations. (2 days) Find the equation of a line, given the graph of the line and use in problem situations. (2 days) Essential questions How do you determine the independent and dependent variables when x and y are not used in the function? What information do you need to find the equation of a line? What do the parts of an equation tell you about the graph? Mathematical practices to be integrated Reason abstractly and quantitatively. Make sense of relationships of points on a line and the equation of the line. Contextualize a problem situation given information regarding the equation of a line. Attend to precision. Communicate precisely to others the process for determining the equation of a line. Make explicit use of definitions How does the equation of a line help with making predictions in the real world? How can you determine which method for writing the equation of a line is best to use? Cumberland, Lincoln, and Woonsocket Public Schools C-19

4 Algebra, Quarter 2, Unit 2.2 Final, July 2011 Equations of Lines (9 days) Written Curriculum Grade-Span Expectations M(F&A) 10 4 Demonstrates conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or provides the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (State) Clarifying the Standards Prior Learning Students began demonstrating conceptual understanding of equality in the lower elementary grades by finding what makes an open sentence true through adding, subtracting, or multiplying. In upper elementary grades students solved one-step linear equations involving whole numbers through adding, subtracting, multiplying, or dividing. In middle school, students showed equivalence between two expressions by using models or different representations of the expressions; solving formulas for a variable requiring one transformation (e.g., d = rt; d/r = t); solving multi-step linear equations with integer coefficients; translating a problem situation into an equation; showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution; and by informally solving problems involving systems of linear equations in a context. Current Learning This content should be taught at the reinforcement level. Students find the equation of a line when given specific information about the line and solve problems involving equations of lines. Students apply the knowledge from this unit in future units this year. Future Learning This GSE will be revisited in grades 10, 11, 12, and advanced math, where the students will learn about solving a variety of more complex functions. Additional Research Findings A Research Companion to Principles and Standards for School Mathematics indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities (pp ). Cumberland, Lincoln, and Woonsocket Public Schools C-20

5 Algebra 1, Quarter 2, Unit 2.3 Solving Systems of Linear Equations Overview Number of instructional days: 8 (1 day assessment) 1 day = 45 minutes Content to be learned Solve systems of linear equations using a variety of methods (graphing, substitution, elimination). (2 days) Connect real-world problem scenarios to a mathematical representation, showing that a system of equations could have one solution, no solution, or infinite solutions. (2 days) Use an efficient method (graphing, substitution, elimination) to solve a system of linear equations formed from a problem scenario. (3 days) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Use systems of equations to plan a solution pathway for a problem situation. Monitor and evaluate progress in solving a problem and change course if necessary. Use appropriate tools strategically. Determine appropriate tools to use for solving systems of linear equations. Detect possible errors by strategically using estimation and other mathematical knowledge. Essential questions What is the meaning of a solution to a system of linear equations? How does a graphical solution to a system of linear equations relate to an algebraic solution? Why does the substitution method for solving a system of linear equations lead to the same solution as the elimination method? How can you determine which method for solving a system of linear equations is best to use given a problem scenario? What are examples of real-world situations that can be modeled by a system of linear equations? Cumberland, Lincoln, and Woonsocket Public Schools C-21

6 Algebra 1, Quarter 2, Unit 2.3 Final, July 2011 Solving Systems of Linear Equations (8 days) Written Curriculum Grade-Span Expectations M(F&A) 10 4 Demonstrates conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or provides the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (State) Clarifying the Standards Prior Learning Students began demonstrating conceptual understanding of equality in grades K 3 by finding what makes an open sentence true through adding, subtracting, or multiplying. In grades 4 6, students began solving one-step linear equations involving whole numbers through adding, subtracting, multiplying, or dividing. In middle school (grades 6-8), students showed equivalence between two expressions by using models or different representations of the expressions; solving formulas for a variable requiring one transformation (e.g., d = rt; d/r = t); solving multi-step linear equations with integer coefficients; translating a problem situation into an equation; showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution; and by informally solving problems involving systems of linear equations in a context. Current Learning Students demonstrate conceptual understanding of equality by solving problems involving algebraic reasoning about equality. Translating problem situations into equations is reinforced. Students are also introduced to solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically. They provide graphical interpretations of solution(s) in problem-solving situations and solve problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. Future Learning Students will demonstrate conceptual understanding of equality by solving equations, systems of equations, or inequalities and interpreting the solutions algebraically and graphically; by factoring, completing the square, using the quadratic formula, and graphing quadratic functions to solve quadratic equations; by solving and interpreting solutions of equations involving polynomial, rational, and radical expressions; and by analyzing the effect of simplifying radical or rational expressions on the solution set of equations involving such expressions. Additional Research Findings According to A Research Companion to Principles and Standards for School Mathematics, Stasis and change presents a conceptually rich theme across the grades K 2 curriculum. It has the potential to tie together patterns, functions, and algebra (pp ). Cumberland, Lincoln, and Woonsocket Public Schools C-22

7 Algebra 1, Quarter 2, Unit 2.4 Solving Word Problems Using Systems of Linear Equations Overview Number of instructional days: 9 (3 days review/ task/assessment) (1 day = 45 minutes) Content to be learned Interpret the solution to a system of linear equations in the context of a problem situation. (2 days) Connect real-world problem scenarios to a linear mathematical representation. (2 days) Use an efficient method (graphing, substitution, elimination) to solve a system of linear equations formed from a problem scenario. (2 days) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Use systems of equations to plan a solution pathway for a problem situation. Monitor and evaluate progress in solving a problem and change course if necessary. Model with mathematics. Apply mathematics previously learned to solve problems involving systems of linear equations. Identify important quantities in a problem situation and use tools such as graphs to solve the problem. Essential questions How can you determine when a problem scenario can be modeled by a system of linear equations? What is the meaning of the solution of a system of linear equations in the context of a problem scenario? How can you determine which method for solving a system of linear equations is best to use given a problem scenario? What are examples of real-world situations that can be modeled by a system of linear equations? Cumberland, Lincoln, and Woonsocket Public Schools C-23

8 Algebra 1, Quarter 2, Unit 2.4 Final, July 2011 Solving Word Problems Using Systems of Linear Equations (9 days) Written Curriculum Grade-Span Expectations M(F&A) 10 4 Demonstrates conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or provides the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (State) Clarifying the Standards Prior Learning Students began demonstrating conceptual understanding of equality in the lower elementary grades by finding what makes an open sentence true through adding, subtracting or multiplying. In upper elementary grades, students began solving one-step linear equations involving whole numbers through adding, subtracting, multiplying or dividing. In middle school, students showed equivalence between two expressions, using models or different representations of the expressions; solving formulas for a variable requiring one transformation (e.g., d = rt; d/r = t); solving multi-step linear equations with integer coefficients; translating a problem situation into an equation; showing that two expressions are or are not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution; and by informally solving problems involving systems of linear equations in a context. Current Learning This unit is a continuation of the content from the previous unit, focusing on solving word problems. Students demonstrate conceptual understanding of equality by solving problems involving algebraic reasoning about equality. Translating problem situations into equations is reinforced. Students solve special-case word problems including, but not limited to, coin, age, mixture, break-even, and rates. Future Learning In grades 11, 12, and advanced math, students will demonstrate conceptual understanding of equality by solving equations and systems of equations or inequalities and interpreting the solutions algebraically and graphically; by factoring, completing the square, using the quadratic formula, and graphing quadratic functions to solve quadratic equations; by solving and interpreting solutions of equations involving polynomial, rational, and radical expressions; and by analyzing the effect of simplifying radical or rational expressions on the solution set of equations involving such expressions. Additional Research Findings According to A Research Companion to Principles and Standards for School Mathematics, Stasis and change presents a conceptually rich theme across the grades K 12 curriculum. It has the potential to tie together patterns, functions, and algebra (pp ). Cumberland, Lincoln, and Woonsocket Public Schools C-24

Relations and Functions

Algebra 1, Quarter 2, Unit 2.1 Relations and Functions Overview Number of instructional days: 10 (2 assessments) (1 day = 45 60 minutes) Content to be learned Demonstrate conceptual understanding of linear