Algebra 1 Scope and Sequence Standards Trajectory

Transcription

1 Algebra 1 Scope and Sequence Standards Trajectory Course Name Algebra 1 Grade Level High School Conceptual Category Domain Clusters Number and Quantity Algebra Functions Statistics and Probability Modeling Standards The Real Number System (N-RN) Quantities (N-Q) Seeing Structure in Expressions (A-SSE) Arithmetic with Polynomials and Rational Expressions (A-APR) Creating Equations (A-CED) Reasoning with Equations and Inequalities (A-REI) Interpreting Functions (F-IF) Building Functions (F-BF) Linear, Quadratic, and Exponential Models(F-LE) Interpreting Categorical and Quantitative Data (S-ID) Algebra 1 Common Core State Standards Use properties of rational and irrational numbers. (Additional) Reason quantitatively and use units to solve problems. (Supporting) Interpret structures of expressions. (Major) Write expressions in equivalent forms to solve problems. (Supporting) Perform arithmetic operations on polynomials. (Major) Understand relationship between zeros and factors of polynomials. (Supporting) Create equations that describe numbers or relationships. (Major) Understand solving equations as a process of reasoning and explain reasoning. (Major) Solve equations and inequalities in one variable. (Major) Solve systems of equations. (Additional) Represent and solve equations and inequalities graphically. (Major) Understand concept of a function and use function notation. (Major) Interpret functions that arise in applications in terms of context. (Major) Analyze functions using different representations. (Supporting) Build functions that model relationships between two quantities. (Supporting) Build new functions from existing functions. (Supporting) Construct and compare linear, quadratic, and exponential models and solve problems. (Supporting) Interpret expressions for functions in terms of situations they model. (Supporting) Summarize, represent, and interpret data on single count or measurement variables. (Additional) Summarize, represent, and interpret data on two categorical and quantitative variables. (Supporting) Interpret linear models. (Major) Denver Public Schools

2 Algebra 1 Scope and Sequence Standards Trajectory Colorado 21st Century Skills Mathematical Practices Invention Critical Thinking and Reasoning: Thinking Deeply, Thinking Differently Information Literacy: Untangling the Web Collaboration: Working Together, Learning Together Self-Direction: Own Your Learning Invention: Creating Solutions 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique others reasoning. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Unit of Study Length of Unit Time Frame 1: Statistics, Take 1 18 days August 26 September 19, : Proportional Reasoning 14 days September 20 October 9, : Solving and Writing Linear Equations 17 days October 10 November 5, : Statistics, Take 2 26 days November 6 December 20, : Systems of Equations and Inequalities 13 days January 7 24, : Exponential Growth 15 days January 27 February 14, : Functions 18 days February 19 March 14, : Transformations 14 days March 17 April 11, : Quadratic Models 33 days April 14 June 5, 2014 End-of-Year Fluency Recommendations Solve characteristic problems involving the analytic geometry of lines. (i.e., write equations of lines given point and slope) (A/G) Add, subtract, and multiply polynomials. (A-APR.1) Transform expressions and chunking (see parts of expressions as single objects). (A-SSE.1b) Denver Public Schools

3 Algebra 1, Unit 1: Statistics, Take 1 Unit of Study 1: Statistics, Take 1 Length of Unit 18 days (August 26 September 19, 2013) Focusing Lens Standards Fluency Recommendations Inquiry Questions ELGs Concepts Modeling Content Standards Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting) N-Q.1: Use units as a way to understand problems and guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret scale and origin in graphs and data displays. Interpreting Categorical and Quantitative Data (S-ID) Summarize, represent, and interpret data on single count or measurement variables. (Additional) S-ID.1: Represent data with plots on the real number line (dot plots, histograms, box plots). S-ID.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S-ID.3: Interpret differences in shape, center, and spread in context of data sets, accounting for possible effects of extreme data points (outliers). Summarize, represent, and interpret data on two categorical and quantitative variables. (Supporting) S-ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in context of data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in data. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. Is there really a difference? How different is really different? Standards for Mathematical Practice Calculate and use measures of center and spread (range and interquartile range) to interpret differences in data sets. (ELG.MA.HS.S.1) Represent and interpret data displayed in dot plots, histograms, and box plots. (ELG.MA.HS.S.1) Shape, center, spread, data representations, outliers, statistical measures Denver Public Schools

4 Generalizations (Conceptual Understanding) Algebra 1, Unit 1: Statistics, Take 1 Guiding Questions to Build Conceptual Understanding My students Understand that Factual Conceptual Knowledge of shape, center, and spread facilitates comparison of two data sets. (S-ID.2-3) Analyzing a variety of data representations helps determine appropriate measures of center and spread to describe data sets. (S-ID.1) Outlier influence is an important consideration when selecting and interpreting statistical measures. (S-ID.3) Two-way frequency tables provide the necessary structure to make conclusions about the association of categorical variables. (S-ID.5) What is the difference between mean and median? What is the relationship between mean and median in skewed data? How can we use technology to find center and spread for data sets? What can be inferred about two data sets with large differences in measures of spread? What is the best way to display data? How does our data display choice affect which information will be conveyed? What is an outlier? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do) What is categorical data? What do joint, marginal, and conditional relative frequencies mean? Why is mean by itself not a complete summary of data sets? How can summary statistics or data displays be accurate but misleading? When would median be a more appropriate measure of center than mean? How can summary statistics or data displays be accurate but misleading? Why is it important to analyze data spread? Why do outliers affect some measures of center more than others? Why do outliers affect some measures of spread more than others? Why is it appropriate to use a two-way frequency table with categorical data? Represent data with plots on the real number line (dot plots, histograms, box plots). (S-ID.1) Use statistics appropriate to data distributions shapes to compare centers (median, mean) and spreads (range, interquartile range, standard deviation) of two or more different data sets. (S-ID.2) Interpret differences in shape, center, and spread in the context of data sets, accounting for possible effects of extreme data points (outliers). (S-ID.3) Summarize categorical data for two categories in two-way frequency tables, interpret relative frequencies in the context of data (including joint, marginal, and conditional relative frequencies), and recognize possible associations and trends in data. (S-ID.5) Denver Public Schools

5 Algebra 1, Unit 1: Statistics, Take 1 Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples. Given two data sets, compare and contrast the data by describing shape, center, and spread. Academic Vocabulary Technical Vocabulary Represent, quantities, fit, assess, accuracy, recognize, trends, interpret, shape, center, spread, comparison, data, representations, communicate, difference, findings, predictions Variables; relative frequency; joint, marginal, and conditional frequencies; mean, median, interquartile range, dot plot, histogram, box plot, two-way frequency tables, categorical, association, outliers, statistical measures, skewed distribution, skewed, quartiles, range, standard deviation, measures of center, measures of spread Resources Textbook Core Lessons Additional Core Lessons Core Instructional Task Discovering Algebra 1.1: Dot plots 1.2: Measures of center 1.3: Box plots and interquartile range 1.4: Histograms 1.0a: Two-Way Tables 1.4a: Describing Distributions 1.4b: Standard Deviation Representing Data Using Box Plots Technology Advanced Data Grapher: Performance/ Learning Task Misconceptions Notes Haircut Costs: (S-ID.2) Students might not recognize the difference between categorical and quantitative data and confuse when to use bar graphs, histograms, dot plots, and box plots. Students might not account for differences in scales when comparing distributions of two data sets. Students might not sort data values before finding the median. Students might think box plot interval lengths are related to the number of subjects in each interval. As you plan for the unit, focus on comparing data sets. Outliers are addressed in Lesson 1.4a: Describing Distributions. Note that outliers are defined in Discovering Algebra, on page 58, question 10. If you need additional background about the statistics in this unit, refer to the Draft High School Progression on Statistics and Probability, an excellent resource for understanding the statistics in the CCSS: Denver Public Schools

6 Algebra 1, Unit 2: Proportional Reasoning Unit of Study 2: Proportional Reasoning Length of Unit 14 days (September 20 October 9, 2013) Focusing Lens Relationships Standards Fluency Recommendations Content Standards The Real Number System (N-RN) Use properties of rational and irrational numbers. N-RN.3: Explain why the sum or product of two rational numbers is rational, that the sum of a rational number and an irrational number is irrational, and that the product of a nonzero rational number and an irrational number is irrational. Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting) N-Q.1: Use units as a way to understand problems and guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and origin in graphs and data displays. N-Q.2: Define appropriate quantities for the purpose of descriptive modeling. N-Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Reasoning with Equations and Inequalities (A-REI) Represent and solve equations and inequalities graphically. (Major) A-REI.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. Inquiry Question How do ratios connect with unit conversion? Standards for Mathematical Practice ELGs Concepts Use units as a way to understand problems. (ELG.MA.HS.N.3) Create equations to model and solve proportional relationships, including data presented in tables. (ELG.MA.HS.A.7) Proportion, ratio, variation, unit conversion, direct variation, inverse variation Denver Public Schools

7 Generalizations (Conceptual Understanding) Algebra 1, Unit 2: Proportional Reasoning Guiding Questions to Build Conceptual Understanding My students Understand that Factual Conceptual Proportional relationships can be modeled with linear equations of the form y = kx or y = k/x (A-CED.2) Precision with units is key to dimensional analysis. (N-Q.1) Sums and products of rational numbers remain in the set of rational numbers. (N-RN.3) Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do) What is a proportional relationship? How do we know when a proportional relationship involves direct variation or inverse variation? What is dimensional analysis? How is dimensional analysis used in science? What is the sum of two irrational numbers? What is the sum of a rational and an irrational number? What is the product of a rational and irrational number? What is the product of two rational numbers? What is the product of two irrational numbers? Why are proportional relationships important? How is dimensional analysis related to precision? Why is the sum or product of two rational numbers always rational? Why are the sum and products of irrational numbers with rational numbers always irrational? Change measurement units through conversion factors and dimensional analysis. (N-Q.1) Analyze proportional relationships to determine whether they represent direct variations or inverse variations. Write equations for proportional relationships representing direct variations or inverse variations. (A-CED.2) Graph proportional relationships. (A-CED.2) Explain why the sum or product of two rational numbers is rational. (N-RN.3) Explain why the sum of a rational number and an irrational number is irrational. (N-RN.3) Explain why the product of a nonzero rational number and an irrational number is irrational. (N-RN.3) Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples. Academic Vocabulary Technical Vocabulary Compare and contrast the constant of variation, the graph, and the equation of a direct variation and an inverse variation. Compute, identify, represent, explain, simulate, random Proportion, constant of variation, ratio, proportional relationship, coordinate graphs, direct variation, directly proportional, inverse variation, inversely proportional, rational, irrational Denver Public Schools

8 Algebra 1, Unit 2: Proportional Reasoning Resources Textbook Core Lessons Additional Core Lessons Instructional Task Technology Performance/ Learning Tasks Misconceptions Notes Discovering Algebra 2.1: Proportions 2.2: Capture-Recapture 2.3: Proportions and Measurement Systems 2.4: Direct Variation 2.5: Inverse Variation Runner s World: (N-Q) Operations with Rational and Irrational Numbers: (N-RN) Students might think all in/out tables operate the same as direct variation tables. Students might extend properties of rational and irrational numbers and conclude (through limited examples) that the sum of any two irrational numbers is also irrational (e.g., (2 + 3) + (2-3) = 4, a rational number). Students might not realize the importance of unit conversions along with the computation when solving problems involving measurement. Students might express answers to a greater degree of precision than required when using a calculating device s display of eight to 10 decimal places. Emphasize the graph of an equation in two variables is the set of all the equation s solutions. The performance/learning task for N-RN, Operations with Rational and Irrational Numbers, is essential for student engagement because it addresses the necessary understandings for N-RN.3. Consider beginning the unit with this task. Students studied both rational and irrational numbers in middle school and will work more with rational numbers in this unit. Denver Public Schools

9 Algebra 1, Unit 3: Solving and Writing Linear Equations Unit of Study 3: Solving and Writing Linear Equations Length of Unit 17 days (October 10 November 5, 2013) Focusing Lens Standards Fluency Recommendations Inquiry Questions ELGs Concepts Structure Content Standards Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.1: Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as single entities. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.1: Create equations and inequalities in one variable and use them to solve problems. A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Reasoning with Equations and Inequalities (A-REI) Understand solving equations as a process of reasoning and explain reasoning. (Major) A-REI.1: Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct viable arguments to justify solution methods. Solve equations and inequalities in one variable. (Major) A-REI.3: Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Building Functions (F-BF) Build functions that model relationships between two quantities. (Supporting) F-BF.1: Write functions that describe relationships between two quantities. a. Determine explicit expressions, recursive processes, or steps for calculations from context. 2. Reason abstractly and quantitatively. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Standards for Mathematical Practice Solve characteristic problems involving analytic geometry of lines (i.e., write equations of lines given point and slope). (A/G) Transform expressions and chunking (see parts of expressions as single objects). (A-SSE.1b) Why do some equations have a unique solution while others have no solution? How many solutions can a linear equation have? Interpret complicated expressions by analyzing structures of expressions to solve equations; evaluate expressions. (ELG.MA.HS.A.1) Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (ELG.MA.HS.A.7) Solve linear equations in one variable; justify processes and solutions. (ELG.MA.HS.A.8) Write arithmetic sequences both recursively and explicitly, use to model situations, and translate between the two forms. (ELG.MA.HS.F.4) Solving equations, linear equations, recursive patterns, rate of change Denver Public Schools

10 Generalizations (Conceptual Understanding) Algebra 1, Unit 3: Solving and Writing Linear Equations Guiding Questions to Build Conceptual Understanding My students Understand that Factual Conceptual Creating equivalent algebraic equations provides the necessary foundation for solving linear equations in one variable. (A-REI.1) Linear relationships can be described using multiple representations. (A-CED.2) Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do) What is an example of a one-variable linear equation with no solution? What is an example of a one-variable linear equation with infinite solutions? How does creating equivalent expressions lead to solving one-variable linear equations? Which methods can be used to represent linear relationships? How does the context of the linear relationship help to interpret the rate of change and initial value of the linear function? How can graphs, tables, and equations determine the rate of change and initial value? How can a one-variable linear equation have no solutions or infinite solutions? How does the context of the problem affect the reasonableness of a solution? Why do we represent linear relationships using different representations? Solve linear equations, including equations with coefficients represented by letters. (A-REI.3) Write recursive routines or formulas to describe linear relationships. (F-BF.1a) Write linear equations in intercept form. (A-CED.2) Graph linear equations. (A-CED.2) Rearrange formulas to highlight quantities of interest. (A-CED.4) Explain each step in solving equations. (A-REI.1) Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples. Solve linear equations in one variable and explain the logic in each step. Academic Vocabulary Technical Vocabulary Evaluate, convert, undoing process, recursive, rule, intercept, rate of change Order of operations, recursive sequence, sequence, term, recursive routine, starting value, linear relationship, additive inverse, like terms, y-intercept Denver Public Schools

11 Algebra 1, Unit 3: Solving and Writing Linear Equations Resources Textbook Core Lessons Additional Core Lessons Instructional Task Discovering Algebra 2.7: Evaluating Expressions 2.8: Undoing Operations 3.1: Recursive Sequences 3.2: Linear Plots 3.3: Time-Distance Relationships 3.4: Linear Equations and the Intercept Form 3.5: Linear Equations and Rate of Change 3.6 Solving Equations Using the Balancing Method Technology Algebra Balance Scales: Performance/ Learning Task Same Solutions?: (A-REI.1 2) Misconception Students may struggle with the arithmetic of negative numbers due to over-memorization of rules for integer operations. Notes Emphasize that graphs of equations in two variables are the set of all the equations solutions. Include an informal discussion of functions throughout Lessons 3.4 and 3.5 using the vocabulary: relation, input, output, function. A formal definition of function and function notation is introduced in Unit 7. Include problems focused on A-CED.4; see Equations and Formulas: and question 11, page 202, in Discovering Algebra. Denver Public Schools

12 Algebra 1, Unit 4: Statistics, Take 2 Unit of Study 4: Statistics, Take 2 Length of Unit 26 days (November 6 December 20, 2013) Focusing Lenses Standards Communication and Modeling Content Standards Quantities (N-Q) Reason quantitatively and use units to solve problems. (Supporting) N-Q.2: Define appropriate quantities for the purpose of descriptive modeling. Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.2: Use the structure of an expression to identify ways to rewrite it. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.4: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given verbal descriptions of the relationships. Key features include: intercepts; intervals where function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.6: Calculate and interpret average rates of change of functions (presented symbolically or as tables) during specified intervals. Estimate rates of change from graphs. Linear, Quadratic, and Exponential Models(F-LE) Interpret expressions for functions in terms of situations they model. (Supporting) F-LE.5: Interpret parameters in linear functions or exponential functions in terms of the context. Interpreting Categorical and Quantitative Data (S-ID) Summarize, represent, and interpret data on two categorical and quantitative variables. (Supporting) S-ID.6: Represent data on two quantitative variables on scatter plots and describe how variables are related. a. Fit functions to data; use functions fitted to data to solve problems in context of data. Use given functions or choose functions suggested by context. Emphasize linear, quadratic, and exponential models. b. Informally assess fits of functions by plotting and analyzing residuals. c. Fit linear functions for scatter plots that suggest linear associations. Interpret linear models. (Major) S-ID.7: Interpret slopes (rates of change) and intercepts (constant term) of linear models in the context of data. S-ID.8: Compute (using technology) and interpret correlation coefficients of linear fits. S-ID.9: Distinguish between correlation and causation. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically. Standards for Mathematical Practice Denver Public Schools

13 Fluency Recommendation Algebra 1, Unit 4: Statistics, Take 2 Solve characteristic problems involving analytic geometry of lines (i.e., write equations of lines given point and slope). (A/G) Inquiry Question How can mathematics help us make predictions and decisions? ELGs Concepts Use structures of equations and expressions to identify ways to write equivalent ones (e.g., point-slope, slope-intercept). (ELG.MS.HS.A.1) Create equations in two variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (ELG.MA.HS.A.7) Interpret key features (e.g., slopes, intercepts) of functions from equations, graphs, situations, or tables. (ELG.MA.HS.F.2) Represent data on two quantitative variables on scatter plots and write linear functions to model data. (ELG.MA.HS.S.2) Representations, correlation, causation, association, slope, y-intercept, residual Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that Factual Conceptual Correlation does not imply causation. (S-ID.9) Correlation coefficients can determine linear model usefulness for describing data and making predictions. (S-ID.8) Mathematicians focus on slope and y-intercept when interpreting linear models in the context of data. (S-ID.7) Linear models describe situations with constant rates of change (slope). (S-ID.6c) What is the difference between correlation and causation? How do we find correlation coefficients on a graphing calculator? What are residuals and how do we calculate them? How do we determine whether we have strong or weak linear correlations? How do we quantify the strength of correlations? What do slope and intercept of linear models mean? What is slope? How can we tell if a situation has a constant rate of change? How can results of statistical investigations be used to support arguments? Why is it important to know correlation strength for data sets? Why does correlation not imply a causal relationship? Linear models are not always the best choice for all data sets. Why? How do slope and y-intercept help interpret linear models? Why can we only model situations having constant rates of change with linear functions? Denver Public Schools

14 Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do) Algebra 1, Unit 4: Statistics, Take 2 Graph linear functions and show intercepts. (A-CED.2) Represent data on two quantitative variables on scatter plots and describe how variables are related. (S-ID.6) Informally assess fits of functions by plotting and analyzing residuals. (S-ID.6b) Fit linear functions for scatter plots that suggest linear associations. (S-ID.6c) Interpret slope (rate of change) and intercept (constant term) of linear models in the context of data. (S-ID.7) Compute (using technology) and interpret correlation coefficients of linear fits. (S-ID.8) Distinguish between correlation and causation. (S-ID.9) Define appropriate quantities for the purpose of descriptive modeling. (N-Q.2) Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples. Academic Vocabulary Technical Vocabulary Correlation does not imply causation. Represent, quantities, fit, recognize, trends, interpret, data, representations, communicate, findings, predictions Variables, scatter plot, correlation, causation, association, correlation coefficient, linear, slope, y-intercept, parameter Resources Textbook Core Lessons Additional Core Lessons Instructional Task Technology Performance/ Learning Task Discovering Algebra 4.1: A Formula for Slope 4.2: Writing a Linear Equation to Fit Data 4.3: Point-Slope Form of a Linear Equation 4.4: Equivalent Algebraic Equations 4.5: Writing Point-Slope Equations to Fit Data 4.6: More on Modeling (Using Q-points is an optional method to determine lines of fit.) 4.7: Applications of Modeling 4.5a: Correlation 4.5b: Residuals 4.7a: Correlation and Causation Fast Food Sandwiches Advanced Data Grapher: (use to create scatter plot) Line of Best Fit: Coffee and Crime: (S-ID.6 9) Denver Public Schools

15 Misconceptions Notes Algebra 1, Unit 4: Statistics, Take 2 Students might apply the distributive property inappropriately, so emphasize the distributive property of multiplication over addition. Students might think residual plots should show patterns. Students might think 45-degree lines in scatter plots of two numerical variables always indicate a slope of 1, but this is only when the two variables have the same scaling. Average rates of change could be discussed when creating lines of fit by calculating rates of change, if different points were used to create the line of fit. Use property vocabulary as students solve equations. F-LE.5 refers to parameters of linear functions. Parameters are constants that can take on various values in function families. For example, given y = mx + b, constants m and b are called parameters. The term parameter should be used in classroom discourse. Limit calculator use of linear regression until Lesson 4.5a. When students are asked to create mathematical models, they should create equations to represent the mathematical situations and define appropriate variables, even when students are not prompted to do so. Denver Public Schools

16 Algebra 1, Unit 5: Systems of Equations and Inequalities Unit of Study 5: Systems of Equations and Inequalities Length of Unit 13 days (January 7 24, 2014) Focusing Lens Modeling Standards Fluency Recommendations Content Standards Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.3: Represent constraints by equations and/or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or nonviable options in a modeling context. Reasoning with Equations and Inequalities (A-REI) Solve systems of equations. (Additional) A-REI.5: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A-REI.6: Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphically. (Major) A-REI.10: Understand that graphs of equations in two variables are the set of all the solutions plotted in the coordinate plane, often forming a curve (which could be a line). A-REI.11: Explain why x-coordinates of the points where graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find solutions approximately (e.g., using technology to graph functions, make tables of values, or find successive approximations). Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. A-REI.12: Graph solutions to linear inequalities in two variables as a half-plane (excluding the boundary in the case of a strict inequality) and graph the solution sets to systems of linear inequalities in two variables as the intersection of corresponding half-planes. 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. 5. Use appropriate tools strategically. Standards for Mathematical Practice Inquiry Question How might we determine when a hybrid car would be a better buy, compared to a less expensive, nonhybrid car? ELGs Concepts Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or nonviable options in terms of the context. (ELG.MA.HS.A.7) Solve systems of two linear equations exactly (algebraically) and approximately (graphically) and justify processes and solutions. (ELG.MA.HS.A.10) Graph solutions to linear inequalities in two variables as half-planes and identify solutions to systems of two linear inequalities graphed in the plane. (ELG.MA.HS.A.11) Constraint, equations, inequalities, solutions, intersection, systems of equations and inequalities Denver Public Schools

17 Generalizations (Conceptual Understanding) Algebra 1, Unit 5: Systems of Equations and Inequalities Guiding Questions to Build Conceptual Understanding My students Understand that Factual Conceptual The points on graphs of equations represent the set of all solutions for a context. (A-REI.10) When solving systems of linear equations, the type of solution set (one solution, no solutions, or infinite solutions) can be determined both graphically and algebraically. (A-REI.6) Characteristics of equations in systems determine the most efficient strategy for finding solutions. (A-REI.6) The intersection of two half-planes provides a means to visualize and represent the solution to a system of linear inequalities. (A-REI.12) Mathematicians evaluate mathematical solutions for their relevance to a model; not all solutions to a system are viable in context. (A-CED.3) How can we determine from a graph if an ordered pair is part of the solution set of an equation? How do graphs represent all solutions to an equation? What do different types of solutions for a system of linear equations look like on a graph? How are solutions to systems of equations visualized or approximated on a graph? Is it possible for a system of equations to have no solution? What would it look like on a graph, and what would it look like when algebraically solving the system? What are the different types of solution processes for solving systems of linear equations? How can we use a calculator to determine the solution to systems of equations? What would a graph showing a system of linear inequalities with no solution look like? What are characteristics of nonviable solutions? How do we know when a solution will be viable? Why is it important to understand units of the problem variables when determining solutions to the problem? How does the graph of a pair of lines describe the possible solution sets for a system of a pair of linear equations? How do we decide which method to use when given a system of equations? What are the advantages to each method and what are the limitations? Why is substitution sometimes more efficient than elimination for solving a system of linear equations algebraically and vice versa? Why are solutions to linear inequalities better represented graphically than algebraically? Why would we model a context with an inequality rather than an equation? Why is it important to evaluate all solutions within the original context? Denver Public Schools

18 Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do) Algebra 1, Unit 5: Systems of Equations and Inequalities Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. (A-REI.5) Solve systems of linear equations in two variables exactly and approximately. (A-REI.6) Explain why x-coordinates of the points where graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find approximate solutions using technology to graph functions, make tables of values, or find successive approximations of linear functions. (A-REI.11) Create linear equations in two variables to represent relationships between quantities and graph equations on coordinates with labels and scale. (A-CED.2) Graph solutions to linear inequalities in two variables as a half-plane. (A-REI.12) Graph solution sets to systems of two linear inequalities in two variables as the intersection of the corresponding half-planes. (A-REI.12) Represent constraints by equations and inequalities, or by systems of equations and inequalities, and interpret solutions as viable or nonviable within a modeling context. (A-CED.3) Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples. Academic Vocabulary Technical Vocabulary Resources Textbook Core Lessons Additional Core Lessons Instructional Task The intersection of two linear equations is their solution set. If the lines do not intersect, there are no solutions. If the lines are the same, an infinite number of solutions exist. Intersection, efficiency, characteristics, solutions, one solution, no solutions, infinite solutions, viable, nonviable, approximation, constraints, relevance, context Systems of equations, linear equations, solution set, graphically, algebraically, inequalities, system of inequalities, half-plane, model, elimination, substitution, function, linear Discovering Algebra 5.1: Solving Systems of Equations 5.2: Solving Systems of Equations Using Substitution 5.3: Solving Systems of Equations Using Elimination 5.5: Inequalities in One Variable 5.6: Graphing Inequalities in Two Variables 5.7: Systems of Inequalities Notebooks and Pens Technology Linear Programming: (scroll down page to application) Performance/ Learning Task Fishing Adventures 3: (A-REI) Denver Public Schools

19 Misconceptions Notes Algebra 1, Unit 5: Systems of Equations and Inequalities Students might confuse the rule of changing a sign of an inequality when multiplying or dividing by a negative number with changing the sign of an inequality when one or two sides of the inequality is negative. Students might believe the graph of a function is simply a line or curve connecting the dots, without recognizing that the graph represents all solutions to the equation. Students should recognize when linear systems have exactly one solution, no solutions, or infinitely many solutions from graphs and from the systems equations (see Discovering Algebra, page 279, question 11). Denver Public Schools

20 Algebra 1, Unit 6: Exponential Growth Unit of Study 6: Exponential Growth Length of Unit 15 days (January 27 February 14, 2014) Focusing Lens Standards Modeling Content Standards Seeing Structure in Expressions (A-SSE) Interpret structures of expressions. (Major) A-SSE.1: Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(a + r) n as the product of P and a factor not depending on P. Write expressions in equivalent forms to solve problems. (Supporting) A-SSE.3: Choose and produce equivalent forms of expressions to reveal and explain properties of the quantity represented by the expressions. c. Use properties of exponents to transform expressions for exponential functions. Creating Equations (A-CED) Create equations that describe numbers or relationships. (Major) A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Interpreting Functions (F-IF) Interpret functions that arise in applications in terms of the context. (Major) F-IF.4: For functions that model relationships between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given verbal descriptions of the relationships. Key features include: intercepts; intervals where function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Building Functions (F-BF) Build functions that model relationships between two quantities. (Supporting) F-BF.1: Write functions that describe relationships between two quantities. a. Determine explicit expressions, recursive processes, or steps for calculations from context. Linear, Quadratic, and Exponential Models(F-LE) Construct and compare linear, quadratic, and exponential models and solve problems. (Supporting) F-LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F-LE.2: Construct linear and exponential functions, including arithmetic and geometric sequences, with graphs, descriptions of relationships, or two input-output pairs (include reading these from tables). F-LE.3: Observe, using graphs and tables, that quantities increasing exponentially eventually exceed quantities increasing linearly, quadratically, or (more generally) as polynomial functions. Interpret expressions for functions in terms of situations they model. (Supporting) F-LE.5: Interpret parameters in linear or exponential functions in terms of context. (F-LE.5) Denver Public Schools

21 Fluency Recommendations 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 4. Model with mathematics. 5. Use appropriate tools strategically. Algebra 1, Unit 6: Exponential Growth Inquiry Question What are the consequences of a population that grows exponentially? Standards for Mathematical Practice ELGs Concepts Use properties of exponents to transform expressions for exponential functions. (ELG.MA.HS.A.2) Graph exponential functions showing key features, including intercepts and end behavior. (ELG.MA.HS.F.3) Determine explicit expressions, recursive processes, or steps for calculation from context. (ELG.MA.HS.F.4) Recognize situations in which one quantity grows or decays by a constant percentage rate over equal intervals. (ELG.MA.HS.F.6) Construct exponential functions including geometric sequences with graphs, descriptions, or tables. (ELG.MA.HS.F.6) Observe, with graphs and tables, that quantities increasing exponentially eventually exceed quantities increasing linearly. (ELG.MA.HS.F6) Growth, decay, constant rate of change, constant rate of growth, linear functions, exponential functions, arithmetic and geometric sequences Generalizations (Conceptual Understanding) Guiding Questions to Build Conceptual Understanding My students Understand that Factual Conceptual Linear and exponential functions provide the means to model constant rates of change and growth, respectively. (F-LE.1) Quantities increasing exponentially eventually exceed quantities increasing linearly or quadratically. (F-LE.3) Linear and exponential functions model arithmetic and geometric sequences, respectively. (F-LE.2) How do we determine whether situations can be modeled by linear functions, exponential functions, or neither? How do we determine from equations whether exponential functions model growth or decay? What are typical situations modeled by linear and exponential functions? How many points of data do we need to determine whether functions are linear or exponential? How does the rate of growth in linear and exponential functions differ? How can we determine when an exponential function will exceed a linear function? How do we know whether sequences are arithmetic or geometric? How can we determine slope and y-intercept of arithmetic sequences? How can we determine ratios for geometric sequences? How are differences between linear and exponential functions visible in equations, tables, and graphs? Why does a common difference indicate a linear function and a common ratio indicate an exponential function? Why can so many situations be modeled by exponential growth? Why is it important to consider limitations of exponential models? Why do linear and exponential functions model so many situations? Why is the domain of a sequence a subset of integers? Denver Public Schools

22 Parameters of equations interpretation must consider real-world contexts. (F-LE.5) The generation of equivalent exponential functions by applying properties of exponents sheds light on a problem context. (A-SSE.3) Algebra 1, Unit 6: Exponential Growth What is a coefficient? How do we choose coefficients given a set of data? How do properties of exponents simplify exponential expressions? Why does a number raised to the power of zero equal one? Why are coefficients sometimes represented with letters? Why does changing coefficients affect a model? How do exponential patterns explain negative exponents? Key Knowledge and Skills (Procedural Skill and Application) My students will be able to (Do) Use properties of exponents to transform expressions for exponential functions with integer exponents. (A-SSE.3c) Create equations in one variable and use them to solve problems; include equations arising from linear and exponential functions with integer exponents. (A-CED.2) Recognize situations in which quantities grow or decay by constant percentage rate per unit interval relative to another. (F-LE.1c) Recognize situations in which one quantity changes at a constant rate per unit internal relative to another. (F-LE.1b) Construct linear and exponential functions, including arithmetic and geometric sequences, with graphs, relationship descriptions, or two input-output pairs. (F-LE.2) Interpret parameters in linear or exponential (domain of integers) functions in terms of real-world context. (F-LE.5) Prove that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. (F-LE.1a) Critical Language: Academic and technical vocabulary to be used in oral and written classroom discourse Algebra 1 students demonstrate ability to apply and comprehend critical language through the following examples. Academic Vocabulary Technical Vocabulary A linear function has a constant rate of change, while an exponential function has a constant rate of growth. Transform, model, create, interpret, situations, real-world contexts, growth, decay, relationships, tables, graphs Quantity, constant rate of change, constant rate of growth, linear functions, exponential functions, exponentially, linearly, quadratically, arithmetic sequence, geometric sequence, explicit, recursive, initial value, parameter, common difference, common ratio, parameter, coefficient, key features Resources Textbook Core Lessons Additional Core Discovering Algebra 6.1: Recursive Routines 6.2: Exponential Equations 6.3: Multiplication and Exponents 6.4: Scientific Notation for Large Numbers 6.5: Looking back with Exponents 6.6: Zero and Negative Exponents 6.7: Fitting Exponential Models to Data Denver Public Schools

23 Lessons Instructional Task Comparing Investments Algebra 1, Unit 6: Exponential Growth Technology Modeling: (scroll down page to application) Performance/ Learning Tasks Misconceptions Comparing Exponentials: (F-LE) Students might believe arithmetic and geometric sequences are the same, and they might not be able to recognize the difference. Students might believe the laws of exponents work for all operations. Students might believe any number to the zero power is zero. Notes In working with sequences, use the language of arithmetic sequence (linear function) and geometric sequence (exponential function). Denver Public Schools