Algebra 2, Quarter 1, Unit 1.1. Probability. Overview

Transcription

1 Algebra 2, Quarter 1, Unit 1.1 Probability Overview Number of instructional days: 4 (1 day = minutes) Content to be learned Compute the theoretical and experimental probabilities for a sample spaces containing equally and non-equally likely outcomes. Calculate the probability of an event. State the differences between independent and dependent events using lists and tables. Determine the union, intersection, and the complement of events. Determine the complement and the negation of a single event. Solve problems involving conditional probability. Essential questions In what real-world situations would you encounter conditional probability? How does one determine whether events are independent or dependent? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to the solution of the problem What is the relationship between theoretical and experimental probability? Why is the probability of dependent events different from the probability of independent events? 1

2 Algebra 2, Quarter 1, Unit 1.1 Probability (4 days) Written Curriculum Common Core State Standards for Mathematical Content Conditional Probability and the Rules of Probability S-CP Understand independence and conditional probability and use them to interpret data S-CP.1 S-CP.2 S-CP.3 S-CP.4 S-CP.5 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Use the rules of probability to compute probabilities of compound events in a uniform probability model S-CP.6 S-CP.7 Find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. 2

3 Algebra 2, Quarter 1, Unit 1.1 Probability (4 days) Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Clarifying the Standards Prior Learning In elementary school, students found the mean, median, mode, and range from a data set. Students were introduced to probability concepts by investigating simple situations. In middle grades, students investigated problems involving change and compound events. Current Learning Students compute and compare results of theoretical and experimental probability and independent and dependent events. Students find unions, intersections, and complements of events. Future Learning In a probability or statistic course, students will extend their knowledge to include studying discrete and continuous probability distributions and their impact on hypothesis testing and confidence intervals. Additional Findings None at this time. 3

4 Algebra 2, Quarter 1, Unit 1.1 Probability (4 days) 4

5 Algebra 2, Quarter 1, Unit 1.2 Statistical Studies Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Make inferences using the different ways of sampling. Recognize that increasing sample size or varying the ways the sample is chosen affects the results. Use mean and standard deviation to fit a set of data to a normal distribution and use it to establish population percentage (not finding standard deviation). Recognize, by using a graph or a table of data, that some data do not fit a normal curve. Use calculators, spreadsheets, or tables to establish areas under the normal curve. Use a sample survey to estimate a mean and margin of error from the survey data. Compare control data to treatment data. Evaluate different methods of sampling results to determine the best representation. Essential questions What statistical measures and procedures could you use to compare the data in the control and treatment groups? What real-life examples can be modeled using a normal distribution? What real-life situations are not modeled by the normal distribution and why? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Explain correspondences between equations, tables, and graphs. Ask Does this make sense? Use appropriate tools strategically. Use a calculator, spreadsheet, computer algebra system, or statistical package. Use technological tools to explore and deepen understanding of concepts. Attend to precision. Label axes to clarify the correspondence with quantities in a problem. Make explicit use of definitions. How do measures of dispersion differ from measures of central tendency? Provide examples of where one or both are used? Why is it necessary to examine different sampling techniques for a particular case study? What characteristics of a survey do you suspect might make for a greater margin of error? 5

6 Algebra 2, Quarter 1, Unit 1.2 Statistical Studies (10 days) Written Curriculum Common Core State Standards for Mathematical Content Making Inferences and Justifying Conclusions S-IC Understand and evaluate random processes underlying statistical experiments S-IC.1 S-IC.2 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model? Make inferences and justify conclusions from sample surveys, experiments, and observational studies S-IC.3 S-IC.4 S-IC.5 S-IC.6 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. Evaluate reports based on data. Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on a single count or measurement variable S-ID.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. 6

7 Algebra 2, Quarter 1, Unit 1.2 Statistical Studies (10 days) Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7

8 Algebra 2, Quarter 1, Unit 1.2 Statistical Studies (10 days) Clarifying the Standards Prior Learning In early elementary grades, students answered questions related to data given in words, diagrams, or verbally, and scribed responses using models and tally charts. Students also analyzed trends using more, less, or equal. Representations using pictographs with one-to-one correspondence and tables were added and students explored representations of line plots interpreted bar graphs, made predictions, and analyzed trends using most frequent (mode), least frequent, largest, or smallest. In later elementary grades, pictographs and circle graphs were interpreted, and students analyzed data and justified conclusions to make predictions and solve problems. Students also used measures of central tendency (median or mode) or range. Additionally, line graphs were also interpreted and students used measures of central tendency (mean, median, or mode) or range to analyze situations and solve problems. In middle school grades, stem-and-leaf plots were added, and dispersion (range) was added to analyze situations, students interpreted scatterplots representing discrete linear relationships or histograms. Students identified outliers in data and analyzed situations to determine their effect on mean, median, or mode, and evaluated the sample from which the statistics were developed (bias). Also, scatterplots and box-and-whisker plots (quartile values) were analyzed, and an estimated line of best fit was determined. Students evaluated samples to determine if a survey was random, nonrandom, or biased. In high school courses, students analyzed data to formulate and justify conclusions, make predictions, and solve problems. Students estimated a regression line and found correlation (strong positive, strong negative, or no correlation. They also solved problems involving conceptual understanding of the sample. Current Learning Students recognize sampling techniques to analyze data, make inferences, and formulate, justify, and critique conclusions. They analyze measures of dispersion including standard deviation and central tendency for the normal distribution. Future Learning Students will use and encounter statistics in the field of economics, business, research, science, medicine, engineering, sports, and information technology. Additional Findings None at this time. 8

9 Algebra 2, Quarter 1, Unit 1.3 Series and Sequences Overview Number of instructional days: 6 (1 day = minutes) Content to be learned Identify arithmetic and geometric sequences. Write an arithmetic and geometric sequence recursively. Find the nth term of arithmetic and geometric sequences by writing an explicit formula and recognizing that the domain is a subset of the integers. Write a series in summation notation. Derive the formula for the sum of an arithmetic series. Derive the formula for the sum of a finite geometric series. ++ Note: FOR ALL SITUATIONS UNDER CONTENT TO BE LEARNED, INCLUDE REAL- LIFE APPLICATIONS. Possible contextual examples include: arena seating, rabbit population, bouncing ball, Fibonacci sequence, and mortgage payments. Essential questions What are the differences between arithmetic and geometric sequences? Why is the initial condition in a recursive definition important? What are the similarities and differences between arithmetic and geometric sequences defined explicitly or recursively? Mathematical practices to be integrated Model with mathematics. Make assumptions and approximations to simplify a complicated situation. Analyze relationships mathematically to draw conclusions. Reflect on whether results make sense. Attend to precision. State the meaning of the symbols chosen, including using the equal sign consistently and appropriately. Make explicit use of definitions. Look for and express regularity in repeated reasoning. Maintain oversight of the process, while attending to the details. Evaluate the reasonableness of intermediate results. What real-world applications are modeled by arithmetic and geometric series? What is the primary distinction between a sequence and a series? What are the advantages of using summation notation? 9

10 Algebra 2, Quarter 1, Unit 1.3 Series and Sequences (6 days) Written Curriculum Common Core State Standards for Mathematical Content Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems A-SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. Interpreting Functions F-IF Understand the concept of a function and use function notation F-IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1. Building Functions F-BF Build a function that models a relationship between two quantities F-BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Common Core Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 10

11 Algebra 2, Quarter 1, Unit 1.3 Series and Sequences (6 days) 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In early elementary school, students were introduced to patterns by finding the next one, two, or three elements; patterns were represented in models, tables, or sequences. Student also found missing elements in a numerical pattern. In the middle elementary grades, students studied nonlinear patterns and wrote rules in words or symbols to find the next case. Patterns were studied in problem situations and expressed in equation form. Patterns were expressed using words or symbols to express the generalization of a linear relationship. In the middle grades, students extended this writing of patterns to nonlinear relationships. In the high school courses, students solved problems involving patterns as well as the method of differences. Current Learning Students identify arithmetic and geometric sequences and find the nth term. Students compute sums and partial sums of arithmetic and geometric series. Future Learning In precalculus, students will review finite series and sequences. They will expand their knowledge to infinite sequences and series. The study of sequences and series ultimately leads to an analysis of limits, integral calculus, engineering calculus, and applications of higher calculus. Students will compute limits for various functions, the integral as a net accumulator, and Taylor series for various functions. Furthermore, students will calculate the area under a curve and the volume of a curve using partitioning, slicing, and cross-sections. 11

12 Algebra 2, Quarter 1, Unit 1.3 Series and Sequences (6 days) Additional Findings None at this time. 12

13 Algebra 2, Quarter 1, Unit 1.4 Using and Interpreting Function Models, Including Regression Models Overview Number of instructional days: 6 (1 day = minutes) Content to be learned Find intercepts; intervals of increasing, decreasing, positive, and negative; relative maximums and minimums; symmetries; end behavior; and domain values that make sense in application problems using a graph. Find the above features using a table. Sketch a graph of a function given a description. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Calculate and interpret the average rate of change of a function given over a specified interval from an equation or table. (For example, find the average rate of change of f(x) = x 2 + 5x from x 1 = 3 and x 2 = 6. Average = f (3) f (6) 3 6 Estimate the rate of change from a graph. Choose the best model using residuals and/or r 2 given a table of data (use the graphing calculator and focus on linear, quadratic, and exponential regression). Mathematical practices to be integrated Use appropriate tools strategically. Use technological tools to explore and deepen understanding of concepts. Model with mathematics. Interpret mathematical results in the context of a given situation. 13

14 Algebra 2, Quarter 1, Unit 1.4 Using and Interpreting Function Models, Including Regression Models (6 days) Essential questions What are the differences when finding the essential characteristics of functions from a graph to a table? What are the advantages of being able to sketch a graph from its description? In what cases would finding the average rate of change over a specific interval be useful? What are the differences between using r 2 and residuals to find best-fit models? Written Curriculum Common Core State Standards for Mathematical Content Building Functions F-BF Build a function that models a relationship between two quantities [Include all types of functions studied] F-BF.1 Write a function that describes a relationship between two quantities. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models] F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Interpret functions that arise in applications in terms of the context [Emphasize selection of appropriate models] F-IF.6 F-IF.9 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 14

15 Algebra 2, Quarter 1, Unit 1.4 Using and Interpreting Function Models, Including Regression Models (6 days) Linear, Quadratic, and Exponential Models F-LE Interpret expressions for functions in terms of the situation they model F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. Interpreting Categorical and Quantitative Data S-ID Summarize, represent, and interpret data on two categorical and quantitative variables S-ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Interpret linear models S-ID.7 S-ID.8 S-ID.9 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Compute (using technology) and interpret the correlation coefficient of a linear fit. Distinguish between correlation and causation. Common Core Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 15

16 Algebra 2, Quarter 1, Unit 1.4 Using and Interpreting Function Models, Including Regression Models (6 days) 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning In grade 5, students graphed points on a coordinate plane and solved real-world and mathematical problems. In grade 6, students were introduced to variables. They constructed and analyzed tables in equivalent ratios and used equations to describe relationships between quantities. In grade 8, students were introduced to the concept of a function. They translated among tabular and graphical representations. Slope as a rate of change was introduced. In grade 9, students mastered line of best fit for linear functions and were introduced to correlation. Students also graphed quadratic and exponential functions. Current Learning Students expand their knowledge of function characteristics to include intervals of increasing and decreasing and end behavior. They use functions to model problem situations. Students master working between tables and graphs. They analyze parts of graphs to calculate rate of change and average rate of change. They also use a graphing calculator to run linear, quadratic, and exponential regression equations and then use residuals to determine the best model. Future Learning In future statistics courses, students will use regression equations and residuals to make predictions about data. Additional Findings A Research Companion to Principles and Standards for School Mathematics discusses functions and solutions on a graph (pp ). Principles and Standards for School Mathematics discusses why students learn algebra to expand their repertoire of functions (p. 297). 16

17 Algebra 2, Quarter 1, Unit 1.5 Systems of Equations Overview Number of instructional days: 6 (1 day = minutes) Content to be learned Review solving linear 2x2 systems algebraically and graphically. (1 day) Solve 3x3 systems algebraically; show the graphs. (2 days) Solve simple systems algebraically and graphically involving a line and circle as well as line and parabola. (2 days) Mathematical practices to be integrated Make sense of problems and persevere in solving them. Make conjectures about the form and meaning of a solution. Plan a solution pathway rather than simply jumping into a solution attempt. Consider analogous problems and try special cases, as well as simpler forms of the original problem, in order to gain insight into a solution. Use appropriate tools strategically. Make sound decisions about when and what tools might be helpful to solve a problem. Analyze graphs of functions and solutions generated using a graphing calculator. Essential questions How can you tell if a graph has one solution, no solution, or infinitely many solutions? How do you determine which algebraic method is best suited for a given system of equations? How are solutions to systems of equations solved algebraically related to solutions to systems of equations solved graphically? How does solving 2x2 systems differ from solving 3 x3 systems? 17

18 Algebra 2, Quarter 1, Unit 1.5 Systems of Equations (6 days) Written Curriculum Common Core State Standards for Mathematical Content Reasoning with Equations and Inequalities A-REI Understand solving equations as a process of reasoning and explain the reasoning 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 a viable argument to justify a solution method. Solve systems of equations A-REI.5 A-REI.6 A-REI.7 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. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 18

19 Algebra 2, Quarter 1, Unit 1.5 Systems of Equations (6 days) 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. Clarifying the Standards Prior Learning In elementary and middle school, students solved linear equations and applied substitution properties to simplify expressions and solve equations. Students in Algebra 1 learned to solve and graph systems of linear equations. Current Learning Students extend their knowledge of 2x2 systems to algebraically solve 3x3 linear systems. In addition, students solve simple systems, graphically and algebraically, of a line and a circle and of a line and a parabola. Future Learning The use of 2x2 and 3x3 systems will be further investigated in the next course. Also, students will study matrices and vector analyses in future courses. Students will also study conic sections as a loci of points formed from the intersection of plane and double-napped cone in precalculus. In calculus, students will solve problems with parametric equations. Additional Findings None at this time. 19

20 Algebra 2, Quarter 1, Unit 1.5 Systems of Equations (6 days) 20

21 Algebra 2, Quarter 1, Unit 1.6 Quadratic Functions and the Complex Number System Overview Number of instructional days: 8 (1 day = minutes) Content to be learned Introduce the Complex Number System 2 ( i = 1 and i = 1) and know that every number has the form a + bi with a and b real. Add, subtract, and multiply complex numbers using the commutative, associative, and distributive properties. (Note: Use all types of complex numbers, e.g., (2 + 3i)(5 + 3i); (3 + 5 )( ); ( -4)(3 + -2) ) Solve quadratic equations with real coefficients that have complex solutions. Factor a quadratic function to find the zeros, extreme value, and symmetry of the graph of the function. Use completing the square to find the zeros, extreme value (vertex form), and symmetry of the graph of a function. Essential questions Why do imaginary numbers exist? After performing operations with complex numbers, what determines when the expression is in its simplest form? What are the advantages of learning how to solve quadratic equations by using the process of completing the square and factoring? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Analyze givens, constraints, and relationships. Consider simpler forms of quadratic problems to gain insight into its solution. Understand the approaches to solving complex problems. Attend to precision. Solve quadratic equations with a degree of precision appropriate for the problem situation. Use clear definitions when explaining complex numbers. What are the advantages of manipulating a quadratic function into different forms? What are the implications of complex numbers in the solutions of problems? 21

22 Algebra 2, Quarter 1, Unit 1.6 Quadratic Functions and the Complex Number System (8 days) Written Curriculum Common Core State Standards for Mathematical Content The Complex Number System N-CN Perform arithmetic operations with complex numbers. N-CN.1 N-CN.2 Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients] N-CN.7 N-CN.8 Solve quadratic equations with real coefficients that have complex solutions. (+) Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x 2i). Interpreting Functions F-IF Analyze functions using different representations [Focus on using key features to guide selection of appropriate type of model function] F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Seeing Structure in Expressions A-SSE Interpret the structure of expressions [Polynomial and rational] 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(1+r) n as the product of P and a factor not depending on P. 22

23 Algebra 2, Quarter 1, Unit 1.6 Quadratic Functions and the Complex Number System (8 days) Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning Students have solved problems by computation since grade 2. In algebra 1, students learned how to factor, and they identified and understood the relationship between zeros and factors of polynomials. Current Learning Students reinforce the concept of nth roots to expand into imaginary numbers. They master addition, subtraction, and multiplication of complex numbers as an extension of the real number system. Students learn to solve quadratics with complex solutions and use factoring and completing the square to find zeros, extreme values, and symmetry of a graph. Advanced topics may include complex number identities. Future Learning In precalculus, students will expand their knowledge of complex numbers by performing division and will learn to write and graph the rectangular and polar form of complex numbers. 23

24 Algebra 2, Quarter 1, Unit 1.6 Quadratic Functions and the Complex Number System (8 days) Additional Findings Principles and Standards for School Mathematics discusses quadratic functions, their properties, and transformations that occur. There is information about changing the quadratic function into vertex form (pp ). Beyond Numeracy discusses the quadratic formula being the first theorem proved in high school algebra. The book also discusses approximating versus exact roots and links this content to physics problems (pp ). It also discusses the origins of i and the expansion of the number system to complex numbers. It also shows that the fundamental theorem of algebra works, in that solutions may be complex numbers (pp ). 24